A057128 Numbers n such that -3 is a square mod n.

1, 2, 3, 4, 6, 7, 12, 13, 14, 19, 21, 26, 28, 31, 37, 38, 39, 42, 43, 49, 52, 57, 61, 62, 67, 73, 74, 76, 78, 79, 84, 86, 91, 93, 97, 98, 103, 109, 111, 114, 122, 124, 127, 129, 133, 134, 139, 146, 147, 148, 151, 156, 157, 158, 163, 169, 172, 181, 182, 183, 186, 193

Offset

1,2

Comments

The fact that there are no numbers in this sequence of the form 6k+5 leads to the result that all prime factors of central polygonal numbers (A002061 of the form n^2-n+1) are either 3 or of the form 6k+1. This in turn leads to there being an infinite number of primes of the form 6k+1, since if P=product[all known primes of form 6k+1] then all the prime factors of 9P^2-3P+1 must be unknown primes of form 6k+1.

Numbers that are not multiples of 8 or 9 and for which all prime factors greater than 3 are congruent to 1 mod 6. - Eric M. Schmidt, Apr 21 2013

Numbers that divide at least some member of A117950. - Robert Israel, Feb 19 2016

Links

Eric M. Schmidt, Table of n, a(n) for n = 1..1000

Example

a(7)=13 since -3 mod 13=10 mod 13=6^2 mod 13.

Maple

select(t -> numtheory:-quadres(-3, t) = 1, {$1..1000}); # Robert Israel, Feb 19 2016

Mathematica

Select[Range[200], IntegerQ[PowerMod[-3, 1/2, #]]&] // Quiet (* Jean-François Alcover, Mar 05 2019 *)

Prog

(Sage)
def A057128(n) :
    if n%8==0 or n%9==0: return False
    for (p, m) in factor(n) :
        if p % 6 not in [1, 2, 3] : return False
        return True
# Eric M. Schmidt, Apr 21 2013
(PARI) isok(n) = issquare(Mod(-3, n)); \\ Michel Marcus, Feb 19 2016

Crossrefs

Includes the primes in A045331 and these (primes congruent to {1, 2, 3} mod 6) are the prime factors of the terms in this sequence. Cf. A008784, A057125, A057126, A057127, A057129.

Cf. A117950.

Sequence in context: A239014 A030705 A305929 * A018534 A018276 A057732

Adjacent sequences: A057125 A057126 A057127 * A057129 A057130 A057131

Keyword

nonn

Author

Henry Bottomley, Aug 10 2000

Status

approved