login
Odd primes p such that 21 is a square mod p.
4

%I #29 Feb 03 2023 16:18:55

%S 3,5,7,17,37,41,43,47,59,67,79,83,89,101,109,127,131,151,163,167,173,

%T 193,211,227,251,257,269,277,293,311,331,337,353,373,379,383,419,421,

%U 457,461,463,467,479,487,499,503

%N Odd primes p such that 21 is a square mod p.

%C These primes correspond to the representation of the two classes of discriminant 21 of binary quadratic forms with principal reduced forms [1, 3, -3] and [3, 3, -1]. The first class represents the primes given in A141159 (or A139492). The second class gives the prime 3 (which divides 21), and primes congruent to 2 (mod 3) and also to 3, 5, 6 (mod 7). The solution of x^2 - 21 == 0 (mod p) leads to the representative primitive parallel forms for discriminant 21 and representation of primes p. - _Wolfdieter Lang_, Jun 19 2019

%C Prime factors of A082111 and excluding the 3, prime factors of A004538. - _Klaus Purath_, Jan 04 2023

%H Vincenzo Librandi, <a href="/A038893/b038893.txt">Table of n, a(n) for n = 1..1000</a>

%t Select[Prime[Range[100]], JacobiSymbol[21, #] != -1 &] (* _Vincenzo Librandi_, Sep 07 2012 *)

%o (PARI) isok(p) = (p>2) && isprime(p) && issquare(Mod(21, p)); \\ _Michel Marcus_, Jun 19 2019

%Y Cf. A139492, A141159.

%K nonn

%O 1,1

%A _N. J. A. Sloane_.

%E Name clarified by _Michel Marcus_, Jun 22 2019