Top 20 Arithmetic Progression of Primes
by Chris Caldwell
The Prime Pages keeps a list of the 5000 largest known primes, plus a few each of certain selected archivable forms and classes. These forms are defined in this collection’s home page. This page is about one of those forms.
Definitions and Notes
Are there infinitely many primes in most arithmetic progressions? Certainly not if the common difference has a prime factor in common with one of the terms (for example: 6, 9, 12, 15, …). In 1837, Dirichlet proved that in all other cases the answer was yes:
- Dirichlet’s Theorem on Primes in Arithmetic Progressions
- If a and b are relatively prime positive integers, then the arithmetic progression a, a+b, a+2b, a+3b, … contains infinitely many primes.
Prime Arithmetic Progression
An arithmetic progression of primes is a set of primes of the form for fixed and and consecutive , i.e., . For example, 199, 409, 619, 829, 1039, 1249, 1459, 1669, 1879, 2089 is a 10-term arithmetic progression of primes with difference 210.
It had long been conjectured that there exist arbitrarily long sequences of primes in arithmetic progression (Guy 1994). As early as 1770, Lagrange and Waring investigated how large the common difference of an arithmetic progression of primes must be. In 1923, Hardy and Littlewood (1923) made a very general conjecture known as the k-tuple conjecture about the distribution of prime constellations, which includes the hypothesis that there exist infinitely long prime arithmetic progressions as a special case. Important additional theoretical progress was subsequently made by van der Corput (1939), who proved than there are infinitely many triples of primes in arithmetic progression, and Heath-Brown (1981), who proved that there are infinitely many four-term progressions consisting of three primes and a number that is either a prime or semiprime. Continue reading
Waves – Lesson 1 – The Nature of a Wave
From Physics Classroom
Categories of Waves
Waves come in many shapes and forms. While all waves share some basic characteristic properties and behaviors, some waves can be distinguished from others based on some observable (and some non-observable) characteristics. It is common to categorize waves based on these distinguishing characteristics.
Longitudinal versus Transverse Waves versus Surface Waves
One way to categorize waves is on the basis of the direction of movement of the individual particles of the medium relative to the direction that the waves travel. Categorizing waves on this basis leads to three notable categories: transverse waves, longitudinal waves, and surface waves. Continue reading
Does Military Sonar Kill Marine Wildlife?
From Scientific American, June 10, 2009
Dear EarthTalk: Is it true that military sonar exercises actually kill marine wildlife? — John Slocum, Newport, RI
Unfortunately for many whales, dolphins and other marine life, the use of underwater sonar (short for sound navigation and ranging) can lead to injury and even death. Sonar systems—first developed by the U.S. Navy to detect enemy submarines—generate slow-rolling sound waves topping out at around 235 decibels; the world’s loudest rock bands top out at only 130. These sound waves can travel for hundreds of miles under water, and can retain an intensity of 140 decibels as far as 300 miles from their source.
These rolling walls of noise are no doubt too much for some marine wildlife. While little is known about any direct physiological effects of sonar waves on marine species, evidence shows that whales will swim hundreds of miles, rapidly change their depth (sometime leading to bleeding from the eyes and ears), and even beach themselves to get away from the sounds of sonar. Continue reading
How does sonar work?
From Science Wire
Sonar is simply making use of an echo. When an animal or machine makes a noise, it sends sound waves into the environment around it. Those waves bounce off nearby objects, and some of them reflect back to the object that made the noise. It’s those reflected sound waves that you hear when your voice echoes back to you from a canyon. Whales and specialized machines can use reflected waves to locate distant objects and sense their shape and movement.
The range of low-frequency sonar is remarkable. Dolphins and whales can tell the difference between objects as small as a BB pellet from 50 feet (15 meters) away, and they use sonar much more than sight to find their food, families, and direction. Continue reading
All in the FUor Family
FU Orionis resides near the imaginary shoulder of the great hunter constellation of Orion, at a location of about 3 degrees NW of Betelgeuse, and less than a degree east of the small planetary nebula NGC 2022. Although FU Ori can now be seen fluttering around 9th magnitude, its appearance hasn’t always been so brilliant. FU Ori’s rise to fame began in 1937 when the star appeared from relative obscurity at 16th magnitude and brightened by a factor of over 100 in 100-200 days. Accompanying the phantom star was a bright nebulosity, which glowed as a reflection of light from the luminous star. After increasing by 6 magnitudes in a period of a year, FU Ori continued to linger around maximum magnitude, where it has resided in a high state along with its ghostly nebula association ever since.
Since only one eruptive episode had been detected and because nothing was known of its pre-outburst state, the 1937 uprising of FU Ori was first thought to be a nova event. Although spectroscopic evidence suggested otherwise, interest in FU Ori and its nature seemed to wane with time. Continue reading
Orgy scenes, such as the one depicted in Roses of Heliogabalus by Alma Tadema, did not exist, according to Dr Alastair Blanshard.
Classical orgies are a myth of our own making
By Kate Rossmanith, 2006
A University researcher has dispelled a myth which has validated the saucy exploits of libertines for centuries: the widespread existence of the Roman orgy.
According to Alastair Blanshard, a Greek history researcher from the School of Philosophical and Historical Inquiry, there was no such thing. [Eyes wide shut. 7M]
“I’m sorry if there are any suburban swingers out there, but whatever you’re doing, it’s certainly not classical,” Dr Blanshard told the audience at a public lecture in the Nicholson Museum.
Dr Blanshard is studying how modern culture imagines antiquity, and how it perpetuates stories about the sex lives of ancient civilisations. He began unearthing whatever evidence he could find of illicit affairs, juicy encounters and frenzied group sex in classical Greece and Imperial Rome. Continue reading