A205526 - OEIS

%I #13 Apr 03 2023 10:36:12

%S 1,3,1,3,1,3,1,4,1,1,4,3,1,3,1,1,1,5,3,1,3,4,1,1,3,1,3,1,6,1,3,1,1,4,

%T 1,4,3,3,1,1,1,5,1,3,1,4,4,3,1,5,1,1,5,1,1,1,1,4,3,1,3,1,3,1,3,1,4,3,

%U 1,5,1,1,3,3,4,1,1,3,1,5,1,6,1,3,4,1,1,3,1,3,1,1,3,1,4,1,1,1,3,6,3,1,1,1,4,3,1,1,1,5,3,3,1,4,4,1,3,1,1,1,5,3,1,1,4,1,6,1,3,3,4,1

%N Least positive integer y such that Kronecker(y^2 - 4, p(n)) == -1 and (x+2)^(p(n)+1) == 5 -+ 2*y (mod p(n), mod x^2 +- y*x + 1).

%C This is an alternate version of A205531, which should be considered as the main entry. One has A205526(n)=A205531(n) whenever the latter is nonzero.

%C Related to the 4.X Selfridge Conjecture by P. Underwood, which states that p is prime iff such an y exists.

%C Records are [p, y] = [A205532(n), A205534(n)] = [2, 1], [3, 3], [19, 4], [61, 5], [109, 6], [1009, 9], [2689, 11], [8089, 15], [33049, 17], [53881, 21], [87481, 27], [483289, 29], [515761, 35], [1083289, 39], [3818929, 45], ...

%H P. Underwood, <a href="https://t5k.org/bios/page.php?id=181">4.X Selfridge Conjecture</a> (on "Prime Pages" profile), Jan 2012.

%o (PARI) a(n)={n=prime(n);for(y=1,1e7, kronecker(y^2-4,n)==-1 | next;

%o Mod(x+Mod(2,n),x^2-y*x+1)^(n+1)==5+2*y | next; Mod(x+Mod(2,n),x^2+y*x+1)^(n+1)==5-2*y & return(y))}

%K nonn

%O 1,2

%A _M. F. Hasler_, Jan 28 2012