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 _______________________________________________________________ | MathPages /*\ http://www.seanet.com/~ksbrown/ | | / \ | |___________/_____\_____________________________________________|