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A121411 Positive integers k for which there are primes of the form a^2+k^n=b^2+k^m with positive integers (a,b,m,n) and a > b. 0

%I

%S 2,5,6,8,10,12,13,17,18,20,21,22,26,28,30,32,33,37,38,40,42,45,46,48,

%T 50,52,53,56,58,60,61,62,65,66,68,70,72,76,77,78,80,82,85,86,88,90,92,

%U 93,96,97,98

%N Positive integers k for which there are primes of the form a^2+k^n=b^2+k^m with positive integers (a,b,m,n) and a > b.

%C The sequence is "hard" in the sense that it not known how to prove that the necessary conditions are sufficient for the existence of primes.

%H David Broadhurst and Mike Oakes, <a href="https://listserv.nodak.edu/cgi-bin/wa.exe?A2=NMBRTHRY;a5fc6134.0607">Primes of the form a^2 + k^n = b^2 + k^m</a>.

%H David Broadhurst and Mike Oakes, <a href="http://physics.open.ac.uk/~dbroadhu/cert/dbmothrm.txt">proof</a> of the necessity the conditions given for the conjectured generating method.

%F Conjecturally, a(n) is the n-th positive nonsquare integer that is not congruent to -1 mod 4, nor to -1 mod 5, nor to -7 mod 16.

%e a(5455)=9998 because it was possible to find primes of the form a^2 + k^n = b^2 + k^m with positive integers (a,b,k,m,n), a > b, k < 10^4 and k satisfying the proved necessary conditions of the conjectured generating method.

%o (PARI) {ls=[];for(k=1,10^4,if(!issquare(k)&&(k+1)%4&&(k+1)%5&&(k+7)%16, ls=concat(ls,k)));print(ls)}

%K hard,nonn

%O 1,1

%A _David Broadhurst_, Jul 29 2006

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Last modified September 23 20:42 EDT 2021. Contains 347617 sequences. (Running on oeis4.)