OFFSET
1,1
COMMENTS
Prime powers q that are congruent to 1 or 7 modulo 8.
Odd prime powers q such that 2^((q-1)/2) = 1 in F_q.
Prime powers q such that x^2 - 2 splits into different linear factors in F_q[x].
Contains the powers of primes congruent to 1 or 7 modulo 8 and the even powers of primes congruent to 3 or 5 modulo 8.
Proposition 1: Suppose that q is not a power of 2, gcd(a,q) = 1, then a is a square in F_q if and only if the Jacobi symbol Jacobi(a,q) = 1.
Proof: a is a square if and only if a^((q-1)/2) == 1 (mod p). We have a^((q-1)/2) = (a^((p-1)/2))^((q-1)/(p-1)) == Jacobi(a,p)^((q-1)/(p-1)) (mod p). Write q = p^e, then by definition, we have Jacobi(a,q) = Jacobi(a,p)^e, so it remains to prove that (q-1)/(p-1) - e = Sum^{e-1}_{i=0} (p^i - 1) is always even, which is obvious.
A trivial corollary would be that if q is a square, then every integer a coprime to q is always a square in F_q (since Jacobi(a,q) = 1 in this case). Indeed, since F_q is the unique quadratic extension of F_{sqrt(q)}, every quadratic polynomial with coefficients in F_{sqrt(q)} splits in F_q.
Proposition 2: Suppose that a == 1 (mod 4), gcd(a,q) = 1, then x^2 - x - (a-1)/4 splits into different linear factors in F_q[x] if and only if Jacobi(q,a) = 1 (or Kronecker(a,q) = 1).
Proof: Proposition 1 deals with the case where q is odd. For even q, we have x^2 - x - (a-1)/4 = x^2 + x + 1, which is reducible over F_q[x] if and only if q is an even power of 2.
LINKS
Jianing Song, Table of n, a(n) for n = 1..10000
EXAMPLE
9 is a term since 2 = -1 = (+-i)^2 in F_9 = F_3(i).
PROG
(PARI) isA366526(n) = isprimepower(n) && (n%8==1 || n%8==7)
CROSSREFS
KEYWORD
nonn,easy
AUTHOR
Jianing Song, Oct 12 2023
STATUS
approved