

A057128


Numbers n such that 3 is a square mod n.


8



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
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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^2n+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^23P+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


LINKS



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



PROG

(Sage)
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
(PARI) isok(n) = issquare(Mod(3, n)); \\ Michel Marcus, Feb 19 2016


CROSSREFS



KEYWORD

nonn


AUTHOR



STATUS

approved



