From - Sat Sep 6 21:04:19 1997
From: ksbrown@seanet.com (Kevin Brown)
Newsgroups: sci.math
Subject: Concordant Primes
Date: Sat, 06 Sep 1997 06:06:27 GMT
Some time ago I posted an elementary proof that the pair of equations
a^2 + b^2 = c^2 a^2 + pb^2 = d^2
have no solution in integers a,b,c,d for any prime p such that p
is congruent to 3, 5, 9, 11, or 13 (mod 16) and every odd prime
divisor of p-1 is congruent to 3 (mod 4). (A prime p is said to be
"concordant" if such solutions exist.) This was a fairly strong
proposition, because all but 18 of the primes less than 1000 were
either ruled out by this proposition or are known to have known
solutions. The 18 exceptions were
103 131 191 223 271 311 431 439 443
593 607 641 743 821 863 929 971 983
Of these, the Birch/Swinnerton-Dyer conjecture suggests that 16 have
no solutions, but two of them, 863 and 983, ought to have solutions
according to the BSD conjecture. Subsequently, David Einstein showed
that 863 is in fact concordant by finding a solution. Just recently
Alan MacLeod found another solution 863 AND a solution for 983, so
this seems to complete the list of concordant primes less than 1000
(assuming the BSD conjecture). Alan's solutions (found using programs
from John Cremona) are
p = 863:
a = 21697973611729663760123617224693905231
b = 140467357958644482600871394399613917520
p = 983:
a = 25612319152259738402372448240896341241531
b = 2927481175425024504484732240429126750140
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