A208293 Numbers n such that (n^2+1)/26 is prime.

21, 31, 109, 125, 135, 151, 161, 229, 281, 291, 359, 369, 385, 525, 541, 551, 619, 629, 645, 671, 681, 749, 759, 801, 879, 941, 1009, 1019, 1035, 1149, 1165, 1175, 1399, 1425, 1435, 1529, 1539, 1555, 1565, 1581, 1669, 1685, 1695, 1799, 1851, 1919, 1945, 1971

Offset: 1

Author

Wolfdieter Lang, Feb 27 2012

Keywords

• nonn

Comments

The corresponding primes (n^2+1)/26 are given in A208292(n).

a(n) is the smallest positive representative of the class of

nontrivial solutions of the congruence x^2==1 (Modd A208292(n)), if n>=2. The trivial solution is the class with representative x=1, which also includes -1. For Modd n see a comment on A203571. For n=1: a(1) = 21 == 13 (Modd 17), and 13 is the smallest positive solution >1.

The unique class of nontrivial solutions of the congruence x^2==1 (Modd p), with p an odd prime, exists for any p of the form 4*k+1, given in A002144. Here a subset of these primes is covered, the ones for k=k(n)=(a(n)^2-25)/(4*26). These values are 4, 9, 114, 150, 175, 219, ...

Examples

			a(3)=109 because (109^2+1)/26 = 457 is prime.
  109 = sqrt(26*457-1) = sqrt(8*1485+1).
		

Links

• Vincenzo Librandi, Table of n, a(n) for n = 1..10000

Crossrefs

Cf. A208292, A207337, A207339, A129307, A027862, A002731.

Programs

Mathematica

Select[Range[10000], PrimeQ[(#^2 + 1)/26] &] (* T. D. Noe, Feb 28 2012 *)

PARI

is(n)=isprime((n^2+1)/26) \\ Charles R Greathouse IV, May 22 2017

Formula

a(n) = sqrt(26*A208292(n)-1) = sqrt(8*A208294(n)+1), n>=1.