A163185 Primes p such that the equation x^2 = -2 mod p has a solution, and ord_p(-2) is even.

17, 41, 73, 89, 97, 113, 137, 193, 233, 241, 257, 313, 337, 353, 401, 409, 433, 449, 457, 521, 569, 577, 593, 601, 641, 673, 761, 769, 809, 857, 881, 929, 937, 953, 977, 1009, 1129, 1153, 1201, 1217, 1249, 1289, 1297, 1321, 1361, 1409, 1433, 1489, 1601, 1609

Offset: 1

Author

Christopher J. Smyth, Jul 23 2009

Keywords

• nonn
• easy

Comments

Such primes are the exceptional p for which x^2 == -2 (mod p) has a solution, as x^2 == -2 (mod p) is soluble for *every* p with ord_p(-2) odd. But if ord_p(-2) is even and p - 1 = 2^r.j with j odd, then x^2 == -2 (mod p) is soluble if and only if ord_p(-2) is not divisible by 2^r.

More generally, the equation x^(2^k) == -2 (mod p) has a solution iff either ord_p(-2) is odd or (p == 1 (mod 2^(k+1)) and ord_p(-2) is even but not divisible by 2^(r-k+1)).

Proof: Choose primitive root g mod p with -2 == g^a (mod p), where a = (p-1)/ord_p(-2). Writing x = g^u, see that solving x^(2^k) == -2 (mod p) is equivalent to solving u*2^k + v*(p-1) = a for some integers u,v.

A necessary and sufficient condition for this is that gcd(2^k,p-1) | a. So for p-1 = j*2^r, j odd and ord_p(-2) = h*2^s, h odd, condition becomes min(k,r) <= r-s. If s = 0 (i.e., ord_p(-2) odd) this is always valid; for positive s we need k < r-s+1, or s < r-k+1.

Examples

			17 belongs to this sequence as 7^2 == -2 (mod 17) and ord_p(-2) = 8, even but <> 0 (mod 16).
		

Links

• Jinyuan Wang, Table of n, a(n) for n = 1..1000

Crossrefs

Cf. A033203 (all p for which x^2 == -2 (mod p) has a solution); .

Cf. A163183 (p with ord_p(-2) odd): a subsequence of A033203, whose complement in A163183 is the current sequence.

Programs

Maple

with(numtheory):k:=1: A:=NULL:p:=2: for c to 30000 do p:=nextprime(p);o:=order(-2,p);R:=gcd(2^100,p-1);if o mod 2=0 and p mod 2^(k+1) = 1 and o mod R/2^(k-1)<>0 then A:=A,p;;fi;od:A;

PARI

lista(nn) = forprime(p=3, nn, if(znorder(Mod(-2, p))%2==0 && []~!=polrootsmod(x^2+2, p), print1(p, ", "))); \\ Jinyuan Wang, Mar 24 2020

Extensions

More terms from Jinyuan Wang, Mar 24 2020