text:
Journal topic
Atmos. Chem. Phys., 20, 5309–5326, 2020
https://doi.org/10.5194/acp-20-5309-2020
Atmos. Chem. Phys., 20, 5309–5326, 2020
https://doi.org/10.5194/acp-20-5309-2020

Research article 06 May 2020

Research article | 06 May 2020

# Transformation and ageing of biomass burning carbonaceous aerosol over tropical South America from aircraft in situ measurements during SAMBBA

Transformation and ageing of biomass burning carbonaceous aerosol over tropical South America from aircraft in situ measurements during SAMBBA
William T. Morgan1, James D. Allan1,2, Stéphane Bauguitte3, Eoghan Darbyshire1, Michael J. Flynn1, James Lee4, Dantong Liu1, Ben Johnson5, Jim Haywood5,6, Karla M. Longo7,a, Paulo E. Artaxo8, and Hugh Coe1 William T. Morgan et al.
• 1Department of Earth & Environmental Sciences, University of Manchester, Manchester, UK
• 2National Centre for Atmospheric Science, University of Manchester, Manchester, UK
• 3Facility for Airborne Atmospheric Measurements, Cranfield University, Cranfield, UK
• 4Department of Chemistry, University of York, York, UK
• 5Met Office, Exeter, UK
• 6College of Engineering, Mathematics and Physical Sciences, University of Exeter, Exeter, UK
• 7National Institute for Space Research (INPE), São José dos Campos, Brazil
• 8Physics Institute, University of São Paulo, São Paulo, Brazil
• anow at: NASA Goddard Space Flight Center and USRA/GESTAR, Greenbelt, MD, USA

Correspondence: Hugh Coe (hugh.coe@manchester.ac.uk)

Abstract

We present a range of airborne in situ observations of biomass burning carbonaceous aerosol over tropical South America, including a case study of a large tropical forest wildfire and a series of regional survey flights across the Brazilian Amazon and Cerrado. The study forms part of the South American Biomass Burning Analysis (SAMBBA) project, which was conducted during September and October 2012. We find limited evidence for net increases in aerosol mass through atmospheric ageing combined with substantial changes in the chemical properties of organic aerosol (OA). Oxidation of the OA increases significantly and rapidly on the scale of 2.5–3 h based on our case study analysis and is consistent with secondary organic aerosol production. The observations of limited net enhancement in OA coupled with such changes in chemical composition imply that evaporation of OA is also occurring to balance these changes. We observe significant coatings on black carbon particles at source, but with limited changes with ageing in both particle core size and coating thickness.

We quantify variability in the ratio of OA to carbon monoxide across our study as a key parameter representing both initial fire conditions and an indicator of net aerosol production with atmospheric ageing. We observe ratios of 0.075–0.13 $\mathrm{µ}\mathrm{g}\phantom{\rule{0.125em}{0ex}}{\mathrm{sm}}^{-\mathrm{3}}\phantom{\rule{0.125em}{0ex}}{\mathrm{ppbv}}^{-\mathrm{1}}$ in the west of our study region over the Amazon tropical forest in air masses less influenced by precipitation and a value of 0.095 $\mathrm{µ}\mathrm{g}\phantom{\rule{0.125em}{0ex}}{\mathrm{sm}}^{-\mathrm{3}}\phantom{\rule{0.125em}{0ex}}{\mathrm{ppbv}}^{-\mathrm{1}}$ over the Cerrado environment in the east (where sm−3 refers to standard metre cubed). Such values are consistent with emission factors used by numerical models to represent biomass burning OA emissions. Black carbon particle core sizes typically range from mean mass diameters of 250 to 290 nm, while coating thicknesses range from 40 to 110 nm in air masses less influenced by precipitation. The primary driver of the variability we observe appears to be related to changes at the initial fire source. A key lesson from our study is that simply aggregating our observations as a function of atmospheric ageing would have been misleading due to the complex nature of the regional aerosol and its drivers, due to the many conflating and competing factors that are present.

Our study explores and quantifies key uncertainties in the evolution of biomass burning aerosol at both near-field and regional scales. Our results suggest that the initial conditions of the fire are the primary driver of carbonaceous aerosol physical and chemical properties over tropical South America, aside from significant oxidation of OA during atmospheric ageing. Such findings imply that uncertainties in the magnitude of the aerosol burden and its impact on weather, climate, health and natural ecosystems most likely lie in quantifying emission sources, alongside atmospheric dispersion, transport and removal rather than chemical enhancements in mass.

1 Introduction

Biomass burning represents a significant source of aerosol particles on the global scale and thus has a substantial impact on the Earth system. At the regional level, where large-scale and seasonal burning practices are conducted annually, significant anthropogenic perturbations may occur. The Brazilian Amazon rainforest and Cerrado are such regions, where an annual burning season typically running from August to October results in the build-up of a large atmospheric aerosol burden that can affect climate (e.g. Andreae2004), weather (e.g. Kolusu et al.2015), human health (e.g. Reddington et al.2015) and the regional ecosystem . The co-emission of large amounts of both gas- and particle-phase pollutants from biomass burning complicates assessments of its impact as they undergo physical and chemical processes downwind.

Biomass burning is the largest source of black carbon (BC) particles when considering the global scale . One of the key complicating factors in relation to BC is the co-emission and subsequent mixing with other chemical components (e.g. Bond et al.2013); aerosol components that predominantly scatter solar radiation can increase the absorption by BC (e.g. Bond et al.2006), with the mixing state of the particles being vital . Furthermore, the relative abundance of BC compared to other (scattering) components in the aerosol will strongly govern the radiative impact from a specific emission source . illustrated that absorption by biomass burning smoke over the Amazon can inhibit cloud formation or even induce so-called cloud “burn-off”. presented average vertical profiles of BC coating thickness across Brazil, with significantly thinner coatings in the east than the west of the Amazon basin.

As well as BC, biomass burning produces abundant emissions of organic carbon species in the gas and particle phase (e.g. van der Werf et al.2017). As well as direct emissions of primary organic aerosol (POA), biomass burning produces significant emissions of semi-volatile or intermediate-volatility non-methane organic carbon species that may potentially play a role in secondary organic aerosol (SOA) formation . A number of field studies have attempted to quantify changes in organic aerosol (OA) production downwind of fire sources, which is a key parameter for global models . reviewed a large body of work, including the chemistry of fresh smoke and particle evolution, with large variability in terms of composition and the timescale of growth rates and the complexity of the “photochemical soup” that embodies aged smoke plumes. Furthermore, they note the difficulty in comparing regional smoke with individual fires, as the former is subject to a vast array of complex drivers, while the latter is often relatively better constrained in terms of their specific burn characteristics and evolution. Many such studies have used carbon monoxide as a largely inert chemical tracer on the scale of anticipated changes in biomass burning emissions through ageing and then compared this with OA mass concentrations downwind to investigate any net changes. Previous airborne studies have reported both an increase in OA mass and a decrease , with the majority of studies reporting no detectable net addition of OA mass . Ground-based studies have also reported conflicting results, with reporting enhancements in plumes from two prescribed fires in the US, reporting no net enhancement for wildfire emissions in the US and while southern African biomass burning plumes have shown net enhancements with ageing . One consistent trend within such studies is the increasing oxidation of the organic aerosol with atmospheric ageing (Shrivastava et al.2017, and references therein). Such observations imply that SOA is being produced, and a number of studies have hypothesised that this is frequently balanced by the dilution and evaporation of POA mass to explain the limited net changes observed in the field. Furthermore, in studies where net enhancements are observed, the magnitude of the enhancements are smaller than those observed in urban or biogenic emissions .

The goal of this paper is to characterise the transformation and ageing of biomass burning aerosol over the Amazon basin, with a focus on the carbonaceous component. The experimental study was conducted during the airborne component of the South American Biomass Burning Analysis (SAMBBA) during September/October 2012. SAMBBA aimed to investigate the impact of biomass burning in the region on the Earth system and on human health. SAMBBA represented the first airborne deployment of an aerosol mass spectrometer and single-particle soot photometer in Brazil, providing previously unobtainable characterisation of non-refractory aerosol species and BC-containing particles and their microphysical properties with high time resolution and sensitivity. Such measurements allow us to investigate the composition of OA and mixing state of BC as a function of atmospheric ageing for a case study analysis of a large fire plume as well as regional-scale measurements across the Amazon basin. The analysis presented here serves as the bridge between near-source fire emissions characterised by and a regional-scale synthesis by . Compared to previous and subsequent biomass burning seasons, 2012 was a relatively normal year when compared to the past decade , which has seen reduced deforestation compared to the historical record, albeit with less significant reductions in fire count (e.g. Aragão et al.2018). As such, the observations presented here provide a characterisation of biomass burning aerosol under relatively typical conditions.

2 Method

## 2.1 Instrumentation

All measurements presented here were conducted on the UK Facility for Airborne Atmospheric Measurement (FAAM) British Aerospace 146 (BAe-146) atmospheric research aircraft. As a whole, SAMBBA (SAMBBA2014) was composed of 18 science flights conducted between 14 September and 3 October 2012. For the purposes of our regional analysis, we investigate nine of these flights, which focussed on boundary layer sampling of the regional biomass burning haze (see , for a broader discussion and context for SAMBBA). The flights and their operating regions are summarised in Table 1. The primary base of operations was Porto Velho in Rondônia state.

Table 1Flight summary of the operations included in this study. All flights were conducted during 2012. Local time (LT) is UTC4. Take-off and land times include airport used: PVH – Porto Velho; PMW – Palmas. The phases correspond to the synoptic meteorological conditions during the study, where relatively dry conditions were prevalent in Phase 1 (P1), before the period when the monsoonal transition was being established in Phase 2 (P2).

The aerosol instrumentation used in this study sampled via Rosemount inlets . These inlets have been shown to be satisfactory for sub-micrometre aerosol measurements , which is typical of the SAMBBA dataset as a whole based on size distribution measurements. Nafion driers were used to dry the aerosol sample, which in combination with ram heating as the sampled air enters the aircraft and decelerates, reduced the measured sample relative humidity to a range from 20 % to 60 %. Losses associated with the Nafion driers represent an additional uncertainty in our measurements, which we do not account for. All concentrations are reported at standard temperature and pressure (STP, 273.15 K and 1013.25 hPa respectively) and are denoted with an “s” in their unit where appropriate; for example sm−3 refers to standard metre cubed.

A Droplet Measurement Technologies (DMT, Boulder, CO, USA) single-particle soot photometer (SP2, Stephens et al.2003) measured BC physical properties during this study. The SP2 measures what is generally referred to as refractory BC or rBC, as defined by . Details of the instrument setup on the FAAM research aircraft and data processing relevant to this study have been detailed elsewhere . The incandescence signal of the instrument was calibrated using Aquadag BC particle standards (Aqueous Deflocculated Acheson Graphite, manufactured by Acheson Inc., USA) to calculate BC mass using a scaling factor of 0.75 to account for differences between the reference BC standard and ambient BC. As in previous studies, we assume a 30 % uncertainty in the SP2 BC mass (e.g. McMeeking et al.2010, 2012). Following the method presented by and a rBC spherical equivalent core diameter, Dc, is derived and related to the particle diameter, Dp, which represents both the rBC core and its associated coating. By assuming a full concentric encapsulation of the spherical core, the coating thickness of single rBC particles is estimated using a core refractive index of 2.26+1.26i and coating refractive index of 1.50+0i.

An Aerodyne Research Inc. (ARI, Billerica, MA, USA) compact time-of-flight aerosol mass spectrometer measured non-refractory OA and inorganic component mass (sulfate, nitrate, chloride and ammonium). In terms of OA, the AMS measures all non-refractory organic matter (OM) rather than just organic carbon (OC), and we thus refer to OM when discussing the AMS data specifically while using OA when referring to the organic component more broadly. Details on the instrument setup on the FAAM research aircraft and calibration protocols have been detailed elsewhere . Measured mass concentrations are subject to an uncertainty of approximately 30 % . In addition to the standard operating procedure of the AMS, we collected data at 1 s time resolution during discrete fire plume sampling by employing the fast mass spectrum mode of the instrument .

Additional information regarding quality assurance procedures for the SP2 and AMS during SAMBBA can be found in .

Carbon monoxide mixing ratios were measured using a vacuum ultraviolet (VUV) fast fluorescence CO analyser, with measurement uncertainties of approximately 2 % .

## 2.2 Background concentration calculations

Excess mixing ratios and concentrations of individual species x, denoted as Δx, are necessary in order to investigate chemically driven changes as well as other processes such as dilution and wet removal. For the following regional-scale analysis, we use the fifth percentile for each species during a straight-and-level run (SLR) as the ambient background values of species x to determine Δx. For the case study analysis, we identified the smoke plumes manually based on the time series of CO, OM and rBC and then determined the ambient background values while sampling outside the smoke plumes for each cross-plume SLR using the same method as the regional-scale analysis. Tropospheric mixing can lead to changes in the background air composition, which can lead to uncertainties in the determination of Δx ; our sampling and method aims to mitigate for such changes as our SLRs are relatively short (10–30 min) and within the atmospheric boundary layer, typically sampling a fairly homogenous haze burden over a single SLR or flight. This limits large changes in mixing plus we manually inspect our time series and background values to identify clear shifts due to changing air masses, e.g. large-scale spatial gradients (B734) or wet scavenging (B739), and recalculate background values over shorter flight segments if necessary. For the case study analysis, our measurements are very close to source, and we observed constant background concentrations throughout our plume intercepts. As a result, we expect uncertainties in the determination of Δx to be small.

3 Tropical forest fire case study

The following section presents a case study of a tropical forest fire sampled on flight B737 on 20 September 2012 in Rondônia state. Take-off was at 14:45 UTC (10:45 local time), lasting 3 h 45 min. The state is characterised by tropical moist broadleaf forest, as well as extensive deforestation. A large smouldering tropical forest fire was sampled and is shown in Fig. 1 alongside the flight track of the aircraft during the low-level sampling of the fire and smoke plume. reported near-source measurements of the fire, concluding that the fire was likely natural in origin as it was located well away from deforestation areas and was in a national park many kilometres from the nearest road. They reported a modified combustion efficiency (MCE, Ward and Radke1993) of 0.79±0.02, which is relatively low compared to typical deforestation fires; for example reported a value of 0.87 for such fires in Brazil. The fire was characterised by substantial emissions of carbon monoxide and very low emissions of rBC compared to prior literature on Brazilian deforestation fires and global tropical forest fires, as well as being largely composed of OM (97.1 % of the sub-micrometre mass, Hodgson et al.2018).

Figure 1Spatial overview of case study analysis for flight B737. (a) Photograph taken from the aircraft of the fire, courtesy of William T. Morgan. (b) Flight track of B737 in relation to the wider study region. (c) Low-level flight track of the case study, indicating both cross-plume intercepts and the along-plume straight-and-level run (SLR). Also shown are Moderate Resolution Imaging Spectroradiometer (MODIS) hotspot data from the Terra overpass coincident with our flight sampling. The 10 km2 box represents the scale for the flight track.

The in situ plume sampling was conducted as a sequence of crosswind intercepts downwind of the fire at approximately 2500 m above sea level, an along-plume SLR at the same altitude and a series of overpass intercepts directly above the fire at an altitude of approximately 1800 m above sea level. As the fire was located on a 900 m high plateau, the near-source sampling was conducted approximately 900 m above ground-level. These flight sections are illustrated in Figs. 1 and 2.

Figure 2Time series of altitude, carbon monoxide (CO), organic matter (OM), refractory black carbon (rBC) and inorganic aerosol components during the case study analysis for flight B737. Downwind cross-plume intercepts, the along-plume straight-and-level run (SLR), and fire overpass intercepts across the plume are indicated by the dashed boxes. The fire itself was located on a 900 m high plateau.

We do not have a quantitative estimate of fire size for the B737 case study, although, based on MODIS hotspot data corresponding to our sample location, the maximum fire size is approximately 5 km2 and was likely smaller during our sampling. Furthermore, based on the velocity of the aircraft and the width of cross-plume intercepts, we estimate that the plume was approximately 3–4 km wide in the near field before expanding to approximately 21 km when 56 km downwind of the fire. The atmospheric stability profile tends towards instability in the flight conditions during our case study, with absolutely unstable air below approximately 2 km where the lapse rate is 10.9 K km−1 and conditionally unstable air from 2 to 3 km where the lapse rate is 7.44 K km−1.

The time series of gas- and particle-phase species shown in Fig. 2 illustrates the significant enhancements in their concentrations during plume intercepts, as well as the gradual increase in their concentrations as the aircraft approached the fire. CO mixing ratios were in excess of 5000 ppb on the approach to the fire, climbing to over 15 000 ppb when directly above it during the overpass intercepts. OM mass concentrations reached almost 800 µg sm−3 on the along-plume SLR and over 3500 µg sm−3 during the above-fire intercepts.

Measurements from the along-plume SLR are shown in Fig. 3 relative to the distance from the fire, as well as the approximate age of the plume at the point of sampling. The distance from the fire is calculated as the great circle distance of the aircraft from the latitude and longitude of the fire (approximately 11.0 S, 63.6 W). The age of the plume is calculated using the average wind speed, which was 6.2±1.7m s−1. We note there was a small gradient in wind speed along the length of the plume of 0.02 $\mathrm{m}\phantom{\rule{0.125em}{0ex}}{\mathrm{s}}^{-\mathrm{1}}{\mathrm{km}}^{-\mathrm{1}}$, corresponding to an average increase of 1.2 m s−1 along the length of the plume. We omit this from our calculations due to the variability in wind speed also observed, while noting that the latter plume ages reported are potentially biased towards higher values. The plume extended approximately 65 km downwind based on the concentrations reaching regional background values, which equates to approximately 3 h in terms of plume age.

Figure 3Various aerosol chemical and physical parameters as a function of plume length and age for the case study analysis from flight B737. Organic matter (OM), nitrate and sulfate are normalised by carbon monoxide (CO) to account for dilution of the smoke plume downwind. Further details on the calculations are provided in Sects. 2.2 and 3. Red star markers on the left-hand side of the figure are averaged across the fire overpass intercepts across the plume with the bars denoting the 2σ standard deviation range around the mean value (OM, nitrate and sulfate ratios are taken from ). Individual data points are shown as red circles from the along-plume straight-and-level run (SLR), with a linear regression slope included to illustrate any apparent trends. Slopes of the linear regression are given on the right-hand side of the figure along with their 95 % confidence interval. The correlation coefficient, r, is also provided.

The evolution of the ΔOM : ΔCO ratio along the length of the plume indicates negligible net change in OM mass downwind of the fire, with the ratio exhibiting a small net decline over the course of the measurements and a low correlation coefficient of 0.16. Compared to the near-source measurements directly above the fire, the ΔOM : ΔCO ratio is slightly enhanced in the near field, although the variability in the above-fire ratio is large relative to the difference. Small net enhancements in sulfate and nitrate relative to CO are observed along the length of the plume, with correlation coefficients of 0.36 and 0.48 respectively. In addition, the along-plume measurements are enhanced relative to the above-fire intercepts. The relative intensity of the organic signal at mz 44 to the total organic mass, which corresponds to the ${\mathrm{CO}}_{\mathrm{2}}^{+}$ ion and is denoted as f44, is used as an indicator for the level of oxidation of the organic aerosol. We observed a substantial increase in f44 along the length of the plume from approximately 0.05 to 0.15, with a correlation coefficient of 0.90. This equates to an increase in O:C of approximately 0.29 to 0.73 over the course of the sampling, which is calculated using the equation from . We observe a small reduction in coating thickness of the rBC-containing particles of $-\mathrm{4.9}±\mathrm{1.4}$nm h−1 with a correlation coefficient of 0.28. The coating thickness observed during the along-plume sampling is lower than that from the above-fire intercepts, although the variability is large in the latter. We do not observe any change in the ΔrBC : ΔCO ratio along the length of the plume sampling; based on the cross-plume intercepts, the mean ratio was 0.098±0.013$\mathrm{µ}\mathrm{g}\phantom{\rule{0.125em}{0ex}}{\mathrm{sm}}^{-\mathrm{3}}\phantom{\rule{0.125em}{0ex}}{\mathrm{ppmv}}^{-\mathrm{1}}$, which was lower than the above-fire intercept value of 0.37±0.22$\mathrm{µ}\mathrm{g}\phantom{\rule{0.125em}{0ex}}{\mathrm{sm}}^{-\mathrm{3}}\phantom{\rule{0.125em}{0ex}}{\mathrm{ppmv}}^{-\mathrm{1}}$ reported in . We observe a minor absolute decrease in rBC core diameter along the plume ($-\mathrm{4.4}±\mathrm{2.3}$nm h−1), with a correlation coefficient of 0.16; the geometric mean mass diameter along the plume was 249±49nm, compared to 270±30nm from the above-fire intercepts. The lower values in the plume run compared to the above-fire intercepts for these rBC parameters may point towards a small shift in the fire conditions, although the variability in the plume is large.

Figure 4Aerosol mass spectrometer organic mass spectra from different segments of the case study analysis from flight B737. The above-fire mass spectrum is from sampling directly above the fire during a cross-plume intercept, with the two downwind mass spectra measured during the along-plume straight-and-level run (SLR), while the background mass spectrum is in the regional aerosol haze away from the main fire plume study region.

Figure 4 shows examples of average OM mass spectra at different stages of the plume's evolution, with approximately 1-order-of-magnitude decreases in concentration. Consistent with the f44 evolution above, the organic aerosol becomes increasingly oxidised downwind of the fire and closely resembles the background regional aerosol after approximately 2.5 h of ageing. Above the fire, mz 43 dominates (7.6 %), corresponding to CH and CHO ions, with further enhancements from other hydrocarbon peaks, especially mz 29, 41, 55 and 57. mz 60, which is associated with levoglucosan and other anhydrous sugars, is also elevated (3.9 % of the organic signal); levoglucosan is often reported as a tracer for biomass burning aerosol. To illustrate the evolution of the mass spectra, we calculate the mean absolute difference between the plume spectra and the background as the square root of the sum of the square differences between their organic peak intensities. We also calculate the linear correlation coefficient between them. The mean absolute difference and correlation between the mass spectrum and the background is 0.0023 and 0.66 respectively, illustrating their similarity. Approximately 1 h downwind, a similar pattern is observed but now with increased signal at mz 44 (6.2 %), while maintaining the signal at mz 43 (7.5 %) and reduced signal at the hydrocarbon peaks noted above. The contribution of mz 60 has reduced to 2.7 % at this point. The mean absolute difference and correlation between the mass spectrum and the background is 0.0016 and 0.87 respectively, illustrating their similarity. After 2.5 h, the contribution from the hydrocarbon peaks has reduced substantially and mz 44 dominates the organic mass spectrum (13.3 %). The contribution of mz 44 increases further in the background organic aerosol (15.7 %). The mean absolute difference and correlation between the mass spectrum and the background is 0.00064 and 0.99 respectively, illustrating their similarity. The contribution of mz 60 has diminished further after 2.5 h (1.0 %), while being close to zero in the background (0.4 %).

The evolution in the organic mass spectra is further illustrated in Fig. 5, where f44 is compared with f43 and f60 for the plume run and regional background aerosol. Relative to the increase in f44 along the plume and its eventually comparable magnitude to the regional background, f43 initially decreases within approximately the first 45 min of the plume's evolution, before a partial increase and stable magnitude up to the 2 h mark. f43 then increases over the rest of the plume run until reaching the regional background value. The points fall within the f44 and f43 “triangle” space reported by prior studies focussed on organic aerosol less influenced by biomass burning . Comparing f44 and f60 illustrates a gradual reduction in f60 as the plume ages and becomes increasingly oxidised downwind, falling within the space reported by and , who reported a similar linear progression.

Figure 5Comparison of f44 vs. f43 and f60 during the case study analysis for B737, where fx refers to the fraction of the organic aerosol mass signal at a given mass-to-charge ratio measured by the aerosol mass spectrometer. Also shown is regional haze data during the same flight. Points from the along-plume straight-and-level run (SLR) are coloured according to the approximate plume age. Dashed lines in the f44 vs. f43 show the triangle space reported by prior studies focussed on organic aerosol less influenced by biomass burning . Dashed lines in the f44 vs. f60 are from previous studies on biomass burning organic aerosol ageing, with the upper line from and the lower line from .

4 Regional biomass burning haze analysis

The following section examines regional biomass burning flights during SAMBBA to investigate the ageing and evolution of the carbonaceous aerosol on the regional scale. Following the case study in Sect. 3, we relate the evolution of the regional OM based on changes in f44 as an indicator of the age of the biomass burning smoke sampled. We couple this with the ratio of rBC to CO as an indicator of the air mass history based on the assumption that both are relatively inert tracers that are strongly controlled by the initial conditions at source; the ratio also provides an indication for the influence of precipitation, which would reduce the rBC mass concentration to a larger extent than CO. Including this ratio in our analysis framework provides a means of isolating net changes in OM mass concentration during ageing from changes driven by air mass history.

We focus on boundary layer regional haze, which is determined based on the procedure outlined in , where near-source plumes were identified based on a series of threshold concentrations for multiple pollutants and then flagged separately. This allows us to exclude such plumes from the wider regional haze that we are interested in here.

Figure 6Flight tracks of the aircraft while sampling within regional haze that are included in our regional analysis. Scale in lower right-hand corner pertains to the enlarged map focussed on our study region. See Sect. 4 and Table 1 for further details.

Figure 6 illustrates the geographical scope of the flight campaign, with the majority of flights sampling within Rondônia state, as well as western Mato Grosso and Tocantins. Deforestation and degradation fires in tropical forest environments are the primary source within Rondônia and Mato Grosso, while Cerrado fires dominate in Tocantins. As well as the geographical regions identified, the analysis can be split into distinct meteorological phases following and . From 14 to 22 September (flights B731–B737), relatively dry conditions were prevalent and were characteristic of the dry season. Following this period, the monsoonal transition was being established and was characterised by increased precipitation across the western and southern Amazon basin. The number of fires was consequently reduced during this transition phase. Further details and analysis of the meteorological fields are available in .

Figure 7Sub-micrometre aerosol chemical composition overview for the regional analysis with the data split into individual flight operations in the regional boundary layer aerosol haze. Data are from straight-and-level runs (SLRs) only with the bars denoting mean concentrations and the text above each bar providing the mass fraction as a percentage. B742 took place over Tocantins, while B746 was primarily over Mato Grosso with some measurements over Rondônia. The rest of the flights were conducted over Rondônia.

Figure 7 summarises the aerosol chemical composition for each flight using data from SLRs in boundary layer regional haze. Flight B731 in Rondônia state was the most polluted, with total sub-micrometre mass concentrations of 46 µg sm−3. Lower average concentrations are observed across the remainder of the flights, with B734 in Rondônia and B742 in Tocantins being the next greatest in terms of total sub-micrometre mass concentrations of 18 and 17 µg sm−3 respectively. OM dominates the sub-micrometre chemical composition, ranging from 75 % of the total on flight B746 to 86 % of the total on flight B739. Sulfate mass fractions ranged from 3.2 % to 9.7 % and are typically larger than nitrate mass fractions, which generally fell between 1.7 % and 3.9 % with flight B740 as an outlier with 7.7 %. Chloride mass fractions were low, ranging from 0.2 % to 0.6 %. Based on ion balance calculations of the inorganic aerosol species, the aerosol was typically neutralised. rBC mass concentrations varied from 2.0 % to 6.1 % on flights within Rondônia state, with the largest concentrations on flights B731 and B734. Note the data coverage for the SP2 was more limited on flight B731 than the other flights. Average rBC mass concentrations (1.5 µg sm−3) were greatest on flight B742 in Tocantins, contributing 8.7 % of the total sub-micrometre mass concentration. The largest contribution by rBC was 12.4 % on flight B746, which sampled within both western Mato Grosso and Rondônia.

Figure 8Comparison of organic matter (OM) with carbon monoxide (CO) across individual flights from the regional analysis. Points are coloured according to the ratio of refractory black carbon (rBC) to carbon monoxide (CO) except for B731 where limited rBC data were available. The black dashed line shows the 0.1 $\mathrm{µ}\mathrm{g}\phantom{\rule{0.125em}{0ex}}{\mathrm{sm}}^{-\mathrm{3}}\phantom{\rule{0.125em}{0ex}}{\mathrm{ppbv}}^{-\mathrm{1}}$ as a consistent baseline for context across all flights, with the solid black lines showing the linear regression for either the whole flight or smaller segments where two lines are shown for a single flight. Red text next to the linear regression lines is the slope of the line of best fit in $\mathrm{µ}\mathrm{g}\phantom{\rule{0.125em}{0ex}}{\mathrm{sm}}^{-\mathrm{3}}\phantom{\rule{0.125em}{0ex}}{\mathrm{ppbv}}^{-\mathrm{1}}$.

Figure 8 illustrates the relationship between excess concentrations of OM and CO for each flight. For additional context, the points are coloured by the ratio of rBC to CO. Broadly speaking, there is a strong linear relationship between OM and CO, with correlation coefficients ranging from 0.57 to 0.98. However, the ratio varies both between and within flights. Variability within individual flights, e.g. B734, B737, B739 and B746, is coincident with differences in the ratio of rBC to CO, which likely reflects differences in air mass history across the region(s) sampled on the flight.

Figure 9Comparison of the ratio organic matter (OM) to carbon monoxide (CO) vs. f44 across individual flights from the regional analysis. f44 refers to the fraction of the organic aerosol mass signal at a mass-to-charge ratio of 44 measured by the aerosol mass spectrometer. Points are coloured according to the ratio of refractory black carbon (rBC) to carbon monoxide (CO) except for B731 where limited rBC data were available. The grey dashed line shows the 0.1 $\mathrm{µ}\mathrm{g}\phantom{\rule{0.125em}{0ex}}{\mathrm{sm}}^{-\mathrm{3}}\phantom{\rule{0.125em}{0ex}}{\mathrm{ppbv}}^{-\mathrm{1}}$ as a consistent baseline for context across all flights.

Figure 9 shows the relationship between the ratio of OM and CO compared with f44 to examine whether the ratio changes with variability in OM oxidation and ageing. Within an individual flight, we observe a limited relationship between the ratio and the level of oxidation of the OM, with predominantly low correlation coefficients ranging from 0.09 to 0.09, except for B734 (0.51) and B740 (0.26). On some flights (B731, B734, B745) there are enhancements for f44 greater than approximately 0.16, although in the case of B734 and B745 they appear at least partially related to a change in the ratio of rBC and CO; the limited SP2 data coverage for B731 precludes analysis, although we note that there is also a reduction in the ratio at greater f44 values. In addition, the uncertainty in the ratio of ΔOM to ΔCO will tend to be larger for low concentrations of either or both, leading to more extreme values; this predominantly occurs for the most aged air masses in terms of f44. Comparing across all flights, changes in the ratio of OM and CO compared to the level of oxidation appear related to changes in the ratio of rBC and CO, which suggests a link with air mass history and any perceived change in net condensation or evaporation of OM.

Figure 10Comparison of f44 vs. f43 from the regional analysis across individual flights, where fx refers to the fraction of the organic aerosol mass signal at a given mass-to-charge ratio measured by the aerosol mass spectrometer. Points are coloured according to the ratio of refractory black carbon (rBC) to carbon monoxide (CO) except for B731 where limited rBC data were available. Dashed lines show the triangle space reported by prior studies focussed on organic aerosol less influenced by biomass burning .

Figure 11Comparison of f44 vs. f60 from the regional analysis across individual flights, where fx refers to the fraction of the organic aerosol mass signal at a given mass-to-charge ratio measured by the aerosol mass spectrometer. Points are coloured according to the ratio of refractory black carbon (rBC) to carbon monoxide (CO) except for B731 where limited rBC data were available. Dashed lines are from previous studies on biomass burning organic aerosol ageing, with the upper line from and the lower line from .

Figures 10 and 11 examine f44 compared with f43 and f60 in a similar manner to Fig. 5 in Sect. 3. The majority of the regional haze data shown in Fig. 10 fall within the triangle space reported by prior studies focussed on OM less influenced by biomass burning . Some flights display a broader range of values suggesting that the flights sampled a more diverse range of air masses in terms of their chemistry and ageing. The behaviour described in Sect. 3 in relation to the evolution during the early stages of the plume's age is present in flights B731, B742 and B745, which suggests sampling of fresher biomass burning smoke on those flights; on flights B742 and B745, such features are distinct in terms of the ratio of rBC and CO as well. In terms of the f44 and f60 space shown in Fig. 11, the regional sampling is predominantly confined to lower f60 values, as well as displaying the linear tendency noted for the case study in Sect. 3 and previous work (e.g. Cubison et al.2011; Lack et al.2013).

Figure 12Histograms of refractory black carbon coating thickness from the regional analysis across individual flights.

Figure 12 presents histograms of the median rBC coating thickness across the individual flights, which appears to vary appreciably from flight to flight, as well as within some individual flights. The lowest coating thicknesses we observe are in the range from 10 to 20 nm on our case study flight, with regional measurements typically ranging from 40 to 120 nm; such observations indicate that the vast majority of rBC-containing particles within the boundary layer are at least partially coated. We found no clear and consistent relationship between coating thickness and f44 across the dataset. The broad bimodal-like structure in coating thickness in flight B737 could be linked with differences in the ratio of rBC and CO, with the thicker coatings of 80–100 nm associated with a larger ratio; conversely, the thinner coatings of less than 60 nm are coincident with the smaller ratios observed. However, there was no clear pattern in this linkage in flights B740 and B744, which also had a bimodal-like structure in coating thickness. Regional-scale variability in rBC coating thickness appears to be predominantly driven by fire-source and/or air mass differences rather than ageing of the aerosol population after emission. We also observed no clear link between the physical size of the rBC core and f44, with geometric mean mass diameters typically between 250 and 290 nm.

5 Discussion

Whether considering the case study analysis in Sect. 3 or regional analysis in Sect. 4, we observe either limited or no net enhancement in the ratio of OM to CO. However, we do observe substantial increases in f44, which is interpreted as an indicator for the O:C content of the OM. Such a trend with atmospheric ageing is consistent with SOA being produced downwind of source following dilution but that this is approximately balanced by the loss of POA emitted at source. A number of studies have observed such features in other biomass burning environments and hypothesised such a process . An additional feature of our observations of the single plume case study is the apparent plateau in f44 approximately 2.5 h downwind of source; we also observe a similar plateau in net ozone production and reduction in nitrogen dioxide downwind. After reaching this plateau, the level of oxidation is comparable to the regional background, with highly similar organic mass spectra as well. Such observations suggest that whatever chemical process drives the ageing of OM, it is relatively fast under the environmental conditions of our measurements. We note that our case study is likely of a natural fire and that it is highly smouldering compared to previous fires sampled in Brazil , which may have a bearing on our observations. While we cannot directly ascertain the regional evolution of f44 with atmospheric ageing, our results imply similar phenomena are present as f44 increases with decreasing OM concentrations as well as reaching a defined end point at approximately 0.20 in the f44 vs. f43 and f60 spaces.

examined the role of a number of factors that could control SOA production in ambient plumes, including fire area as a driver of dilution rate, mass emission flux and atmospheric stability. Based on , our estimated initial fire size and atmospheric stability conditions would lead to some evaporation of OM in the near field but with significant SOA production downwind that could balance the initial loss of particulate matter, which would be consistent with our observations of limited net enhancement in OM. Based on thermodynamic analysis of the SAMBBA experiment by , our regional sampling was typically conducted in unstable air, which is consistent with our observed limited net enhancement in OM.

Aqueous processing of biomass burning emissions has been identified as a potential source of SOA (e.g. Gilardoni et al.2016; Tomaz et al.2018), which is likely an important component in Amazonia. However, we are not in a position to assess the role of such processes through our observations. More oxidised OM is thought to be more hygroscopic (e.g. Jimenez et al.2009), so such processes have implications for the lifetime and radiative impact of biomass burning smoke.

Our regional analysis illustrates the importance of evaluating changes in the ageing of regional OM within a framework that also accounts for differences in air mass history. Differences in vegetation, fire dynamics and environmental conditions can result in significant diversity in the absolute and relative emissions of different pollutants from biomass burning that will manifest in the regional aerosol burden. We also observe significant variability from flight to flight, even within the same region, that is likely a consequence of differing meteorological conditions e.g. the influence of precipitation, as well as changes in fire dynamics. Were we to interpret our observed changes in the ratio of OM to CO as a function of f44, we would see enhancements of 2–3 in some instances that are most likely driven by differences in air mass history and fire dynamics rather than chemical processing – ΔrBC : ΔCO was key in illustrating such differences where enhancements in OM to CO were often easily distinguished by large changes in rBC to CO.

Our results indicate that uncertainties in the magnitude of the aerosol burden most likely lie in quantifying emission sources, alongside atmospheric dispersion, transport and removal rather than chemical enhancements in mass. Across our study, ΔOM : ΔCO ratios range from 0.029 to 0.13 $\mathrm{µ}\mathrm{g}\phantom{\rule{0.125em}{0ex}}{\mathrm{sm}}^{-\mathrm{3}}\phantom{\rule{0.125em}{0ex}}{\mathrm{ppbv}}^{-\mathrm{1}}$ in the west of the Amazon basin, with a value of 0.095 $\mathrm{µ}\mathrm{g}\phantom{\rule{0.125em}{0ex}}{\mathrm{sm}}^{-\mathrm{3}}\phantom{\rule{0.125em}{0ex}}{\mathrm{ppbv}}^{-\mathrm{1}}$ in the Cerrado environment sampled on flight B742. Numerical models that attempt to represent the magnitude of the atmospheric aerosol burden typically use fixed emission factors for distinct ecosystems; for example those used in the fourth version of the Global Fire Emissions Database (GFED4, van der Werf et al.2017) report ΔOM : ΔCO ratios of 0.10 and 0.09 $\mathrm{µ}\mathrm{g}\phantom{\rule{0.125em}{0ex}}{\mathrm{sm}}^{-\mathrm{3}}\phantom{\rule{0.125em}{0ex}}{\mathrm{ppbv}}^{-\mathrm{1}}$ for tropical forest and savannah fires respectively (assuming an OM:OC ratio of 1.6 for biomass burning OA following , and ). In the western Amazon basin, our central estimate for the ΔOM:ΔCO ratio when considering all flights together is 0.09–0.10 $\mathrm{µ}\mathrm{g}\phantom{\rule{0.125em}{0ex}}{\mathrm{sm}}^{-\mathrm{3}}\phantom{\rule{0.125em}{0ex}}{\mathrm{ppbv}}^{-\mathrm{1}}$, closely matching the value for tropical forests in GFED4. Based on the local- and synoptic-scale situations during flights B739 and B746, we suspect that the lower observed ratios on these flights are a consequence of wet removal; the lower ratio associated with limited rBC in B734 is less clear based on the large-scale synoptic situation, so we do not speculate on a cause in this case. If these three flights with lowered observed ΔOM : ΔCO ratios are excluded from the analysis, then our regional values range from 0.075 to 0.13 $\mathrm{µ}\mathrm{g}\phantom{\rule{0.125em}{0ex}}{\mathrm{sm}}^{-\mathrm{3}}\phantom{\rule{0.125em}{0ex}}{\mathrm{ppbv}}^{-\mathrm{1}}$ in the west of the Amazon basin, indicating significant variability compared to any assumed fixed emission factor. However, we note that variability of 25 %–30 % is much lower than the discrepancy reported between measurements and models quantifying the aerosol burden over tropical South America, where factors ranging from 1.5 to 6 are required to match satellite and ground-based observations of aerosol optical depth (Reddington et al.2016, and references therein).

In terms of inorganic species, we observe small net enhancements in sulfate and nitrate relative to CO in the case study analysis. We do not observe clear enhancements in nitrate at the regional scale, while we do observe minor absolute increases in sulfate on some flights.

Aside from the comparison between the above-fire intercepts and the along-plume sampling in the case study, we do not observe clear changes in rBC coating thickness with plume age. Our observations indicate that rBC is rapidly coated in the near field based on our case study, as well as other near-field sampling during SAMBBA, in contrast to many urban sources and environments (e.g. Liu et al.2017). This is consistent with previous measurements of North American wildfire emissions, which also showed significant coatings on near-field rBC particles . On the regional scale, we observe no clear link between coating thickness and f44, with variability in coating thickness appearing to be driven by fire-source and/or air mass differences. Prior measurements of biomass burning emissions over boreal Canada by reported that scavenging of rBC via wet deposition preferentially removed the largest and most coated particles. Such processes may explain some of our observed variability in coating thickness on the regional scale, although we observe limited variability in rBC core size across the dataset. Our measurements indicate that the vast majority of rBC-containing particles within the boundary layer are at least partially coated, with the lowest coating thicknesses of 10–20 nm observed in our case study flight and the majority of our regional measurements indicating thicknesses of 40–120 nm. Such coatings will lead to a lowering of rBC lifetime as rather than being hydrophobic the rBC-containing particles will be at least mildly hygroscopic, making them more susceptible to cloud activation and wet removal. Furthermore, such coatings have implications for the radiative impact of black-carbon-containing particles via enhanced absorption . The lack of change in coating thickness appears consistent with the limited enhancement in OM, although given that our results imply a process of both evaporation and condensation of OA-related species there are potentially complex particle dynamics occurring within the rBC-containing fraction that warrant further investigation.

6 Conclusions

We observe limited to no enhancement in OM mass production during atmospheric ageing of biomass burning over Brazil for both a case study of a likely natural tropical forest fire and regional sampling over the Amazon and Cerrado. Variability in the ratio of OM to CO is predominantly driven by regional differences likely related to changes at the initial fire source, as well as air mass differences likely as a consequence of wet scavenging of the aerosol. What enhancements we do observe are small in absolute terms compared to regional-scale variability across our study. Such variability at the regional scale can be significant, with flight-averaged ΔOM : ΔCO ratios ranging from 0.075 to 0.13 $\mathrm{µ}\mathrm{g}\phantom{\rule{0.125em}{0ex}}{\mathrm{sm}}^{-\mathrm{3}}\phantom{\rule{0.125em}{0ex}}{\mathrm{ppbv}}^{-\mathrm{1}}$ across our study region in cases where we suspect the influence of precipitation to be minor. We did not observe a systematic difference between the west of our study region and the Cerrado in terms of ΔOM : ΔCO, although we only have one flight in the latter region. While significant, we note that the scale of the variability is much smaller than typical factors required to match satellite and ground-based observations of aerosol optical depth to numerical model estimates of the aerosol burden. We do observe substantial changes in the chemical composition of OM, with significantly increased oxidation downwind, implying SOA formation that is being balanced by evaporation of OM. During our case study, we observed an increase in O:C of approximately 0.25±0.09h−1, reaching a plateau after approximately 2.5–3.0 h of atmospheric ageing and a comparable magnitude to the background regional aerosol. Such changes may enhance the hygroscopicity of the OA, and given its dominance of the aerosol burden (75 %–86 % of the sub-micrometre mass in our study), this will have implications for the life cycle and radiative impact of biomass burning aerosol in the region.

We observe limited changes in the microphysical properties of rBC subsequent to emission, with neither significant changes in particle core size nor significant changes in coating thickness. We observe substantial coatings on rBC-containing particles at source. Given the limited changes with ageing, our results suggest that any absorption enhancements will be dictated by the initial conditions in the near field and precipitation influences, rather than ageing, although we have not investigated particle morphology changes that may occur. Such coatings likely reduce the lifetime of rBC as they are likely to be at least mildly hygroscopic, especially compared to uncoated hydrophobic rBC particles.

The complex nature of the regional aerosol and its drivers implies that aggregating our observations from the entire study as a function of atmospheric ageing is unwise due to the many conflating and competing factors present. The continuing puzzle over the contrasting observations of the evolution of OM:OC ratios with atmospheric ageing remains, although our results appear consistent with the framework presented by . Further detailed quantification of the processes driving these should be further explored in the literature, as well as chamber and ambient studies specifically designed to probe such processes.

Overall, our results suggest that the initial conditions are the biggest driver of carbonaceous aerosol composition and physical properties in the region, aside from significant oxidation of OA during atmospheric ageing. Uncertainties in the magnitude of the aerosol burden and its impact most likely lie in quantifying emission sources, alongside atmospheric dispersion, transport and removal rather than chemical enhancements in mass.

Data availability
Data availability.

All raw time series data from the FAAM research aircraft are publicly available from the Centre for Environmental Data Analysis (http://www.ceda.ac.uk/, last access: July 2014), where the entire SAMBBA dataset may be accessed at https://catalogue.ceda.ac.uk/uuid/2ff89840a89840868acff801f8859451 (SAMBBA, 2014). AMS mass spectral features, SP2 size distribution and coating thickness data are available on request. Data masks for categorising flight patterns into plume-sampling and other sampling types (vertical profiles and SLRs) are currently available on request. Active fire data used in the paper are available publicly from NASA (see acknowledgements for further details).

Author contributions
Author contributions.

WTM analysed the data and wrote the paper. JDA, ED, JL and DL provided additional data analysis support, including data processing and quality assurance. SB and JL operated the gas-phase instruments, while JDA and MJF operated the aerosol instruments during the field campaign. BJ, JH, KML, PEA and HC led the planning of the field campaign and were co-principal investigators on the SAMBBA project.

Competing interests
Competing interests.

The authors declare that they have no conflict of interest.

Acknowledgements
Acknowledgements.

We would like to acknowledge the substantial efforts of the whole SAMBBA team before, during and after the project. Active fire data were produced by the University of Maryland and acquired from the online Fire Information for Resource Management System (FIRMS; https://earthdata.nasa.gov/data/near-real-time-data/firms/abouts; specific product: MCD14ML, last access: 4 June 2018).

Financial support
Financial support.

Eoghan Darbyshire was supported by a NERC studentship (grant nos. NE/J500057/1 and NE/K500859/1). This work was supported by the NERC SAMBBA project (grant no. NE/J010073/1). Paulo E. Artaxo was supported by FAPESP (Fundação de Amparo à Pesquisa do Estado de São Paulo; grant nos. 2017-17047-0 and INCT 2014/50848-9). This research was also supported by the LBA (the Large Scale Biosphere-Atmosphere Experiment in Amazonia) central office, operated by INPA (Instituto Nacional de Pesquisas Espaciais).

Review statement
Review statement.

This paper was edited by Sergey A. Nizkorodov and reviewed by three anonymous referees.

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text:
# Fermat's little theorem

For other theorems named after Pierre de Fermat, see Fermat's theorem (disambiguation).

Fermat's little theorem states that if p is a prime number, then for any integer a, the number a pa is an integer multiple of p. In the notation of modular arithmetic, this is expressed as

$a^p \equiv a \pmod p.$

For example, if a = 2 and p = 7, 27 = 128, and 128 − 2 = 7 × 18 is an integer multiple of 7.

If a is not divisible by p, Fermat's little theorem is equivalent to the statement that a p − 1 − 1 is an integer multiple of p, or in symbols

$a^{p-1} \equiv 1 \pmod p.$[1][2]

For example, if a = 2 and p = 7 then 26 = 64 and 64 − 1 = 63 is a multiple of 7.

Fermat's little theorem is the basis for the Fermat primality test and is one of the fundamental results of elementary number theory. The theorem is named after Pierre de Fermat, who stated it in 1640. It is called the "little theorem" to distinguish it from Fermat's last theorem.[3]

## History

Pierre de Fermat

Pierre de Fermat first stated the theorem in a letter dated October 18, 1640, to his friend and confidant Frénicle de Bessy as the following:[3]

If p is a prime and a is any integer not divisible by p, then a p - 1 - 1 is divisible by p.

Fermat did not prove his assertion, only stating:[4]

Et cette proposition est généralement vraie en toutes progressions et en tous nombres premiers; de quoi je vous envoierois la démonstration, si je n'appréhendois d'être trop long.

(And this proposition is generally true for all series and for all prime numbers; the proof of which I would send to you, if I did not fear it being too long.)[5]

Euler provided the first published proof in 1736 in a paper entitled "Theorematum Quorundam ad Numeros Primos Spectantium Demonstratio" in the Proceedings of the St. Petersburg Academy,[6] but Leibniz had given virtually the same proof in an unpublished manuscript from sometime before 1683.[3]

The term "Fermat's Little Theorem" was probably first used in print in 1913 in Zahlentheorie by Kurt Hensel:

Für jede endliche Gruppe besteht nun ein Fundamentalsatz, welcher der kleine Fermatsche Satz genannt zu werden pflegt, weil ein ganz spezieller Teil desselben zuerst von Fermat bewiesen worden ist."

(There is a fundamental theorem holding in every finite group, usually called Fermat's little Theorem because Fermat was the first to have proved a very special part of it.)

An early use in English occurs in A.A. Albert, Modern Higher Algebra (1937), which refers to "the so-called "little" Fermat theorem" on page 206.

### Further history

Main article: Chinese hypothesis

Some mathematicians independently made the related hypothesis (sometimes incorrectly called the Chinese Hypothesis) that 2p ≡ 2 (mod p) if and only if p is a prime. Indeed, the "if" part is true, and is a special case of Fermat's little theorem. However, the "only if" part of this hypothesis is false: for example, 2341 ≡ 2 (mod 341), but 341 = 11 × 31 is a pseudoprime. See below.

## Proofs

Several proofs of Fermat's little theorem are known. It is frequently proved as a corollary of Euler's theorem.

## Generalizations

Fermat's little theorem is a special case of Euler's theorem: for any modulus n and any integer a coprime to n, we have

$a^{\varphi (n)} \equiv 1 \pmod{n},$

where φ(n) denotes Euler's totient function (which counts the integers between 1 and n that are coprime to n). Euler's theorem is indeed a generalization, because if n = p is a prime number, then φ(p) = p − 1.

A slight generalization of Euler's theorem, which immediately follows from it, is: if a, n, x, y are integers with n positive and a and n coprime, then

If $x \equiv y \pmod{\varphi(n)}$, then $a^x \equiv a^y \pmod{n}$.

This follows as x is of the form y + φ(n)k, so

$a^x = a^{y + \varphi(n)k} = a^y (a^{\varphi(n)})^k \equiv a^y 1^k \equiv a^y \pmod{n}.$

In this form, the theorem finds many uses in cryptography and, in particular, underlies the computations used in the RSA public key encryption method.[7] The special case with n a prime may be considered a consequence of Fermat's little theorem.

Fermat's little theorem is also related to Carmichael's theorem, as well as to Lagrange's theorem in group theory.

The algebraic setting of Fermat's little theorem can be generalized to finite fields.

## Converse

The converse of Fermat's little theorem is not generally true, as it fails for Carmichael numbers. However, a slightly stronger form of the theorem is true, and is known as Lehmer's theorem. The theorem is as follows:

If there exists an a such that

$a^{p-1}\equiv 1\pmod p$

and for all primes q dividing p − 1

$a^{(p-1)/q}\not\equiv 1\pmod p$

then p is prime.

This theorem forms the basis for the Lucas–Lehmer test, an important primality test.

## Pseudoprimes

If a and p are coprime numbers such that a p − 1 − 1 is divisible by p, then p need not be prime. If it is not, then p is called a pseudoprime to base a (or a Fermat pseudoprime). F. Sarrus in 1820 found 341 = 11 × 31 as one of the first pseudoprimes, to base 2.

A number p that is a pseudoprime to base a for every number a coprime to p is called a Carmichael number (e.g. 561). Alternately, any number p satisfying the equality

$\gcd\left(\sum_{a=1}^{p-1} a^{p-1}, p\right)=1$

is either a prime or a Carmichael number.

## Notes

1. ^ Long 1972, pp. 87–88
2. ^ Pettofrezzo & Byrkit 1970, pp. 110–111
3. ^ a b c Burton 2011, p. 514
4. ^ Bergeron & Zhao (2004)
5. ^ Ore 1988, p. 272 for the English translation
6. ^ Ore 1988, p. 273
7. ^ Trappe, Wade; Washington, Lawrence C. (2002), Introduction to Cryptography with Coding Theory, Prentice-Hall, p. 78, ISBN 0-13-061814-4

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Chapter 235

## Abstract

Abstract

1. Retinitis pigmentosa is the name given to a set of heritable degenerations of the retina. Patients with retinitis pigmentosa typically experience night blindness and loss of the midperipheral visual field early in the disease. As the disease progresses, the visual field is reduced to a shrinking island of central vision (called tunnel vision) and scattered patches of far peripheral vision. In many cases, all useful vision is lost during middle age. In advanced cases and sometimes even in early cases, the fundus oculi exhibits the following features: thin retinal vessels, a pale optic nerve head, and clumps of intraretinal pigment. Related diseases that feature progressive degeneration of the retina include cone-rod degeneration, cone degeneration, and macular degeneration. In cone-rod degeneration, there is early loss of visual acuity due to a progressive panretinal degeneration that in early stages affects cone photoreceptors more severely than rod photoreceptors. In cone degeneration, cones but not rods degenerate. In macular degeneration (see Chap. 243), cones and rods degenerate in the macula but not elsewhere in the retina. Most forms of retinitis pigmentosa and related retinal degenerations affect only the eye, although in a minority of cases the retinal degeneration is one feature of a syndrome that includes other systemic abnormalities, such as retinitis pigmentosa and deafness in Usher syndrome. In young patients with the symptom of night blindness, it is important to differentiate retinitis pigmentosa from stationary night blindness. Both diagnostic categories feature defective vision in dim light as an early symptom, but in stationary night blindness, cone photoreceptors and daytime vision do not deteriorate substantially during the patient's lifetime.

2. Measuring the retina's electrical response to flashes of light is the best way to distinguish the early stages of progressive retinal degeneration such as retinitis pigmentosa from nonprogressive retinal diseases such as stationary night blindness. This noninvasive measurement, called the electroretinogram (ERG), can be used to diagnose young patients with retinitis pigmentosa and many other forms of retinal degeneration even before visual symptoms or funduscopic abnormalities are apparent.

3. The visual loss in retinitis pigmentosa and related retinal degenerations corresponds to the degeneration of photoreceptor cells of the retina. Molecular genetic and biochemical studies have revealed that the defects causing these cells to die can be placed into three categories depending on the mutant gene in each case: (a) a primary biochemical defect inherent in the photoreceptor cells (the rods, the cones, or both cell types), (b) a primary biochemical defect in neighboring retinal cells such as the retinal pigment epithelium, or (c) a peculiar sensitivity of the photoreceptors or the retinal pigment epithelium to a generalized metabolic defect. In stationary night blindness, rod photoreceptors malfunction because of defects inherent in them or perhaps in the cells they synapse with; however, the rod or cone photoreceptors do not degenerate at a rate symptomatically faster than that which accompanies normal aging.

4. Families with retinitis pigmentosa, cone-rod degeneration, cone degeneration, or stationary night blindness can demonstrate any of the known monogenic inheritance patterns: autosomal dominant, autosomal recessive, or X-linked. A syndromic form of retinitis pigmentosa, Kearns-Sayre syndrome, is transmitted with a maternal, or mitochondrial, inheritance pattern. In addition, in some families, retinitis pigmentosa is transmitted in a digenic fashion, where affected individuals are double heterozygotes for defects in both of two unlinked genes. There is nonallelic or locus heterogeneity within the categories of dominant (at least 10 loci implicated), recessive (over 22 loci), or X-linked retinitis pigmentosa (at least 3 loci). Stationary night blindness can be allelic to forms of retinitis pigmentosa.

5. Most retinitis pigmentosa genes remain unidentified. Of those which are known, some encode members of the rod phototransduction cascade (e.g., rhodopsin, the α or β subunits of rod cGMP-phosphodiesterase, and the α subunit of the rod cGMP-gated cation channel, and arrestin). Other identified genes encode proteins important to the structure of photoreceptor outer segments (RDS and ROM1) or encode proteins with unknown function (RPE65, TULP1, ABCR, RPGR, and RP2). Some genes causing syndromic forms of retinal degeneration also have been identified. Examples are the genes encoding myosin VIIa causing Usher syndrome type I, a presumed extracellular matrix protein causing Usher syndrome type II, α-tocopherol transfer protein causing retinitis pigmentosa and Friedrich-like ataxia, and phytanoyl-CoA α-hydroxylase causing Refsum disease. The identified retinitis pigmentosa genes account for an estimated 35 to 50 percent of all cases of the disease. Forms of cone-rod degeneration are caused by mutations in the gene encoding guanylate cyclase, the transcription factor crx, or the ABCR protein of unknown function. Cone degeneration can be caused by mutations in the gene encoding guanylate cyclase activator 1A. Rod monochromacy, a form of colorblindness due to absent or nonfunctional cone photoreceptors, can be caused by mutations in the cone cGMP-gated cation channel. Mutations in the following genes encoding proteins in the phototransduction cascade are responsible for some forms of autosomal dominant or recessive stationary night blindness: rhodopsin, the α subunit of rod transducin, the β subunit of rod cGMP-phosphodiesterase, rhodopsin kinase, and rod arrestin. A form of X-linked stationary night blindness is due to mutations in an L-type calcium channel.

6. Patients with abetalipoproteinemia (MIM 200100; see Chap. 115), Refsum disease (MIM 266510; see Chap. 132), or deficiency in α-tocopherol transfer protein can exhibit the signs and symptoms of retinitis pigmentosa as the first manifestations of disease. Because vision-saving treatments are available for these conditions, it is important to consider them in all newly diagnosed patients with retinitis pigmentosa. Also noteworthy in this regard is that at least some patients with a related hereditary retinal degeneration named gyrate atrophy (MIM 258870; see Chap. 83) may benefit from specifically modifying their diet. For the nonsyndromic forms of retinitis pigmentosa and for Usher syndrome, oral supplements of vitamin A have been reported to slow the rate of cone photoreceptor degeneration. No other therapy advocated to slow or stop the course of retinitis pigmentosa or related retinal degenerations has been tested in a large, double-blind clinical trial. However, a night vision scope can help to alleviate the symptom of night blindness in some patients. Also, acetazolamide has been reported to ameliorate temporarily the cystoid macular edema that reduces central vision in some cases, although this drug can have serious side effects.

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