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%I #15 Apr 03 2023 10:36:13
%S -1,-1,1,0,-1,1,-1,0,-1,-1,-1,0,-1,3,-1,-1,-1,1,-1,0,-1,-1,-1,0,-1,-1,
%T -1,-1,-1,1,-1,0,-1,-1,-1,-1,-1,3,-1,-1,-1,1,-1,0,-1,-1,-1,0,-1,-1,-1,
%U -1,-1,1,-1,-1,-1,-1,-1,0,-1,5,-1,-1,-1,-1,-1,0,-1,-1,-1,0,-1,3,-1,-1,-1,-1,-1,0,-1,-1,-1,0,-1,-1,-1,-1,-1,1,-1,-1,-1,-1,-1,-1
%N Least nonnegative integer y such that Kronecker(y^2 - 4, n) == -1 and (x+2)^(n+1) == 5 -+ 2*y (mod n, mod x^2 +- y*x + 1), or -1 if there is no such y.
%C Related to the 4.X Selfridge Conjecture by P. Underwood, which states that a(n)=-1 iff n > 5 is prime. (It seems that n=5 is the only prime that has a(n) >= 0.)
%C Records are [n, a(n]): [0, -1], [2, 1], [13, 3], [61, 5], [109, 6], [1009, 9], [2689, 11], [8089, 15], [33049, 17], [53881, 21], [87481, 27],[483289, 29], [515761, 35], [1083289, 39], [3818929, 45], ... See A205534.
%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) A205535(n)={/*isprime(n) &&*/for(y=0,n, 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));-1} /* the upper search bound is motivated from experimental data, which suggests that y << n, cf. A205534. If we admit the conjecture, we can prefix the for() loop with "isprime(n) &&". */
%K sign
%O 0,14
%A _M. F. Hasler_, Jan 28 2012
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