In a bimagic square of primes of order 11, if we want to use 121 consecutive primes, then the smallest "good" set starts from 823: 823, 827, 829, 839,... 1669 For the modulo reasons on sums well seen below by Christoph. Christian. -----Message d'origine----- De : math-fun-bounces+cboyer=club-internet.fr@mailman.xmission.com [mailto:math-fun-bounces+cboyer=club-internet.fr@mailman.xmission.com] De la part de Pacher Christoph Envoyé : vendredi 17 novembre 2006 22:04 À : math-fun Objet : RE: [math-fun] First known bimagic square of primes answering my own question
and one obvious question: are you sure that no bimagic square of primes can exist with 121 consecutive primes starting from 3? ;-)
Ok, it canNOT exist: Of course two necessary conditions to hold are order divides Sum[Prime[n],{n,2,order^2+1}] order divides Sum[Prime[n]^2,{n,2,order^2+1}] (here the sum is over the first order^2 odd primes). this does not hold for 11, but even more: besides order = 1 and 2 it does NOT hold for any order <= 5000, interesting. The first condition is however satisfied for order=1,2,12,35,215,225,398,2097,... (tested up to 5000). BTW: this sequence is not in the OEIS... The 2nd is true for order=1,2,4,8,14,16,32,44,172,173,344,430,712,944,2744,... (tested up to 8144), strange 44, isn't it: 44+300,900,2700, but not 8144! again the sequence is not in the OEIS... Christoph