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Primes with Kronecker symbol (p|14) = 1.
20

%I #37 Oct 11 2022 21:42:19

%S 3,5,13,19,23,59,61,71,79,83,101,113,127,131,137,139,151,157,173,181,

%T 191,193,227,229,233,239,251,263,269,281,283,293,307,337,349,359,397,

%U 401,419,431,449,457,461,463,467,487,509,523,563,569,587,599,617,619

%N Primes with Kronecker symbol (p|14) = 1.

%C Originally incorrectly named "Primes that are squares mod 14", which is sequence A045373. - _M. F. Hasler_, Jan 15 2016

%C Conjecture: primes congruent to {1, 3, 5, 9, 13, 15, 19, 23, 25, 27, 39, 45} mod 56. - _Vincenzo Librandi_, Jun 22 2016

%C From _Jianing Song_, Nov 21 2019:

%C Rational primes that decompose in the field Q(sqrt(-14)).

%C These are primes p such that either one of (a) p == 1, 2, 4 (mod 7), p == 1, 7 (mod 8) or (b) p == 3, 5, 6 (mod 7), p == 3, 5 (mod 8) holds. So the conjecture above is correct. (End)

%H Vincenzo Librandi, <a href="/A191017/b191017.txt">Table of n, a(n) for n = 1..1000</a>

%t Select[Prime[Range[200]], JacobiSymbol[#,14]==1&]

%o (Magma) [p: p in PrimesUpTo(619) | KroneckerSymbol(p, 14) eq 1]; // _Vincenzo Librandi_, Sep 11 2012

%o (PARI) is(p)=kronecker(p, 14)==1&&isprime(p) \\ _Michel Marcus_, Jun 23 2016 after A191032

%Y Cf. A191018, A191020, A191061, A274504.

%K nonn,easy

%O 1,1

%A _T. D. Noe_, May 24 2011

%E Definition corrected (following an observation by _David Broadhurst_) by _M. F. Hasler_, Jan 15 2016