

A007520


Primes == 3 (mod 8).
(Formerly M2882)


46



3, 11, 19, 43, 59, 67, 83, 107, 131, 139, 163, 179, 211, 227, 251, 283, 307, 331, 347, 379, 419, 443, 467, 491, 499, 523, 547, 563, 571, 587, 619, 643, 659, 683, 691, 739, 787, 811, 827, 859, 883, 907, 947, 971, 1019, 1051, 1091, 1123, 1163, 1171, 1187
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OFFSET

1,1


COMMENTS

Primes of the form 3x^2 + 2xy + 3y^2 with x and y in Z.  T. D. Noe, May 07 2005
Also, primes of the form X^2 + 2Y^2, X=xy, Y=x+y.  Zak Seidov, Dec 06 2011
Each term is the sum of no fewer than three positive squares.  T. D. Noe, Nov 15 2010
Smallest terms expressible as sum of three distinct positive squares: 59 = 1^2 + 3^2 + 7^2, 83 = 3^2 + 5^2 + 7^2, 107, 131, 139, 179, 211, 227, 251, 283, 307.  Zak Seidov, Dec 06 2011
Except for the first term it appears that the terms of the sequence are also primes of the form 2k+1 such that 3*(2k+1) divides 2^k+1.  Hilko Koning, Dec 06 2019
Theorem (Legendre symbol): With p an odd prime and a an integer coprime to p the Legendre symbol L(a/p) = 1 if a is a quadratic nonresidue (mod p) and L(2/p) = 1 if p == +3 (mod 8).
Theorem (Euler's criterion): L(a/p) == a^((p1)/2) (mod p) so with a = 2 and prime p = 2k + 1 then 1 == 2^k (mod (2k+1)). So prime numbers 2k+1 = +3 (mod 8) are the prime numbers 2k+1  2^k+1.
If 2k+1 == 3 (mod 8) then k is even and 2^k+1 is not divisible by 3 and if 2k+1 == +3 (mod 8) then k is odd and 2^k+1 is divisible by 3.
Hence prime numbers 2k+1 == 3 (mod 8) are prime numbers such that 3*(2k+1)  2^k+1. Or, including the first term of the sequence, prime numbers 2k+1 with k odd such that 2k+1  2^k+1.
(End)


REFERENCES

N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).


LINKS

M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards, Applied Math. Series 55, Tenth Printing, 1972 [alternative scanned copy].
Alexander Kalmynin, On Novák numbers, arXiv:1611.00417 [math.NT], 2016. See P0 in Theorem 7 p. 11.


MAPLE

option remember;
local a;
if n = 1 then
return 3;
end if;
a := nextprime(procname(n1)) ;
while modp(a, 8) <> 3 do
a := nextprime(a) ;
end do:
a ;
end proc:


MATHEMATICA

p=3; k=0; nn=1000; Reap[While[k<nn, If[PrimeQ[p], k++; Sow[p]]; p=p+8]][[2, 1]] (* Zak Seidov, Dec 06 2011 *)
Select[Prime[Range[200]], Mod[#, 8]==3&] (* Harvey P. Dale, Apr 05 2023 *)


PROG



CROSSREFS



KEYWORD

nonn,easy


AUTHOR



STATUS

approved



