A205526 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).

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, 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, 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

Offset: 1

Author

M. F. Hasler, Jan 28 2012

Keywords

• nonn

Comments

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.

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

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], ...

Links

• P. Underwood, 4.X Selfridge Conjecture (on "Prime Pages" profile), Jan 2012.

Programs

PARI

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