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the problem of computing the stable homotopy groups of spheres is of fundamental importance in algebraic topology .
although this subject has been studied for a very long time , the adams spectral sequence is still the best way to do stemwise computations at the prime 2 .
bruner produced a computer - generated table of @xmath5 differentials in the adams spectral sequence @xcite .
he started with his theorem on the interaction between adams differentials and squaring operations @xcite to obtain several values of the @xmath5 differential .
then he methodically exploited all ( primary ) multiplicative relations , both forwards and backwards , to deduce more and more values of the @xmath5 differential . through at least the 80-stem ,
this procedure gives the vast majority of the values of the @xmath5 differential .
in bruner s approach , the first unknown value is @xmath6 , where @xmath7 is a certain element in the 52-stem .
this gives a precise sense to the claim that @xmath6 is  harder " than any previous @xmath5 differential .
mark mahowald communicated ( unpublished ) the following argument for the presence of the differential @xmath0 .
first , there is an element @xmath8 in the @xmath9-stem that detects @xmath10 ( * ? ? ?
* proposition 3.5.1 ) . since @xmath11
, we deduce that @xmath12 detects @xmath13 .
now @xmath14 , so @xmath12 detects @xmath15 .
the final step is the claim that @xmath16 is zero .
this final step turns out to be false ; @xmath16 is detected by @xmath17 ( * ? ? ?
* lemma 4.2.90 ) .
mahowald s mistake arose from an incorrect understanding of the toda bracket @xmath18 ( * ? ? ?
* lemma 4.2.91 ) .    in this note , we will repair the hole left by this mistake .
the proof relies on motivic calculations in a fundamental way . in other words
, our argument does not work if applied in the classical context .
we will first establish a massey product involving the element @xmath7 .
then we will apply the higher leibniz rule on the interaction of adams differentials with massey products @xcite .
the second author discovered this proof based on motivic calculations by the first author .
we adopt notation from @xcite without further explanation . for charts of the classical and motivic adams spectral sequences ,
see @xcite .    the higher leibniz rule of moss @xcite is an appealing result that has not been as useful in practice as one might expect . in applications
, indeterminacies often interfere .
the differential that we establish here is an unusually clear application of the higher leibniz rule .
our work suggests that an extension of bruner s program to methodically exploit all @xmath19-fold massey products is likely to lead to further new results about adams @xmath5 differentials .
the origin of this note lies in the first author s analysis of the adams spectral sequence through the 59-stem .
the first author was able to give careful arguments for every adams differential , with the sole exception of possible differentials on the element @xmath7 .
the second author was instrumental in finishing the last remaining differential , whose argument we present here .
the chief consequence of our adams differential calculation is the following results about the stable 51 and 52 stems .
the 2-primary order of the stable 51-stem is 128 . as a group ,
the 2-primary stable 51-stem is either @xmath20 or @xmath21 .
this follows immediately from the adams @xmath22-page .
the first @xmath23 lies in the image of @xmath24 and has a generator that is detected by @xmath25 .
the last @xmath26 is detected by @xmath27 .
the remaining elements are detected by @xmath28 , @xmath29 , and @xmath30 .
these elements assemble into @xmath23 if there is a hidden @xmath2 extension from @xmath29 to @xmath30 , and they assemble into @xmath31 if there is no hidden @xmath2 extension .    according to @xcite , there is no hidden @xmath2 extension in the stable 51-stem .
however , this claim is inconsistent with other claims in @xcite , as discussed in ( * ? ? ?
* remark 4.1.17 ) .
because of this uncertainty , we leave this 2 extension unresolved .
the 2-primary stable 52-stem is equal to @xmath32 .
this is immediate from the adams @xmath22-page .
according to @xcite , there are no possible hidden @xmath2 extensions .
the crucial calculation is the following lemma which is due to tangora @xcite .
this massey product is proven using the lambda algebra . since the elements @xmath7 and @xmath35 lie beyond the range of the published curtis tables @xcite , tangora used another massey product to deduce that the representative of @xmath7 in the lambda algebra has a leading term @xmath36 .
one can also use unpublished results of mahowald and tangora to see directly that @xmath35 is represented by @xmath37 .
then the massey product follows from @xmath38 .
one can not apply may s convergence theorem @xcite directly to get this massey product from the may differential @xmath39 .
the problem is that there is a  crossing " may differential @xmath40 that voids the hypotheses of may s convergence theorem .
corollary [ cor : g - bracket ] gives us the massey product @xmath43 in the motivic adams spectral sequence .
using the higher leibniz rule of moss @xcite , we have @xmath44 since there is no indeterminacy . since @xmath45 ( * ? ? ?
* lemma 3.3.12 ) , we have @xmath46 . in particular , this is nonzero .
then the only possibility is that @xmath0 .          in the classical adams spectral sequence , @xmath49 and @xmath50
are both zero .
this means that our proof is strictly motivic in nature .
we do not know a proof using only the classical adams spectral sequence .    using only classical information
, we suggest a possible argument that @xmath7 does not survive the classical adams spectral sequence .
start with the massey product @xmath51 and the classical differential on @xmath52 .
moss s convergence theorem @xcite then implies that @xmath53 is not a well - formed toda bracket .
the only possibility is that @xmath7 does not survive .
however , that still leaves two possible differentials : @xmath54 , or @xmath55 .
we do not know how to rule out the second alternative using only the classical adams spectral sequence .
mahowald s original argument for the differential @xmath6 used a faulty computation of the toda bracket @xmath56 .
partial information about this toda bracket is used in the construction of a kervaire invariant element @xmath57 in dimension @xmath58 @xcite .
our improved understanding of the toda bracket does not contradict any of the claims of @xcite .
e. b. curtis , p. goerss , m. mahowald and r. j. milgram .
calculations of unstable adams @xmath33 terms for spheres . algebraic topology ( seattle , wash . , 1985 ) , 208266 , lecture notes in math . , 1286 , springer , berlin , 1987 .
s. o. kochman .
stable homotopy groups of spheres , a computer - assisted approach .
lecture notes in mathematics 1423 , springer - verlag , 1990 .
j. p. may .
matric massey products .
j. algebra 12 ( 1969 ) , 533568 .
we establish a differential @xmath0 in the @xmath1-stem of the adams spectral sequence at the prime @xmath2 , which gives the first correct calculation of the stable 51 and 52 stems . 
 this differential is remarkable since we know of no way to prove it without recourse to the motivic adams spectral sequence . 
 it is the last undetermined differential in the range of the first author s detailed calculations of the @xmath3-stems for @xmath4 @xcite . 
 this note advertises the use of the motivic adams spectral sequence to obtain information about classical stable homotopy groups .