A097933 Primes p that divide 3^((p-1)/2) - 1.

11, 13, 23, 37, 47, 59, 61, 71, 73, 83, 97, 107, 109, 131, 157, 167, 179, 181, 191, 193, 227, 229, 239, 241, 251, 263, 277, 311, 313, 337, 347, 349, 359, 373, 383, 397, 409, 419, 421, 431, 433, 443, 457, 467, 479, 491, 503, 541, 563, 577, 587, 599, 601, 613

Offset: 1

Author

Cino Hilliard, Sep 04 2004

Keywords

• nonn

Comments

Rational primes that decompose in the field Q[sqrt(3)]. - N. J. A. Sloane, Dec 26 2017

For all primes p > 2 and integers gcd(x, y, p) = 1, x^((p-1)/2) +- y^((p-1)/2) is divisible by p. This is because (x^((p-1)/2) - y^((p-1)/2))(x^((p-1)/2) + y^((p-1)/2)) = x^(p-1) - y^(p-1) is divisible by p according to Fermat's Little Theorem (FLT). This sequence lists p that divides 3^((p-1)/2) - 1^((p-1)/2), and A003630 lists the '+' case.

Apart from initial terms, this and A038874 are the same. - N. J. A. Sloane, May 31 2009

Primes in A091998. - Reinhard Zumkeller, Jan 07 2012

Also, primes congruent to 1 or 11 (mod 12). - Vincenzo Librandi, Mar 23 2013

Conjecture: Let r(n) = (a(n) - 1)/(a(n) + 1) if a(n) mod 4 = 1, (a(n) + 1)/(a(n) - 1) otherwise; then Product_{n>=1} r(n) = (6/5) * (6/7) * (12/11) * (18/19) * ... = 2/sqrt(3). - Dimitris Valianatos, Mar 27 2017

Primes p such that Kronecker(12,p) = +1 (12 is the discriminant of Q[sqrt(3)]), that is, odd primes that have 3 as a quadratic residue. - Jianing Song, Nov 21 2018

Comment from Richard R. Forberg, Feb 07 2023: (Start)

Conjecture: These are the exclusive prime factors of the set of integers d > 1 such that there exist primitive Heronian triangles with sides {b, b+d, b+2d} for one or more integers b.

Also b is always > d. For d=11 the b values begin {15, 17, 65, 75, 267, 305, 1025, ...}. For d=1 (not prime, thus not listed) the b values are given by A016064. (End)

Examples

			For p = 5, 3^2 - 1 = 8 <> 3*k for any integer k, so 5 is not in this sequence.
For p = 11, 3^5 - 1 = 242 = 11*22, so 11 is in this sequence.
		

Links

• Reinhard Zumkeller, Table of n, a(n) for n = 1..10000
• Index to sequences related to decomposition of primes in quadratic fields

Crossrefs

Cf. A003630, A010051, A038874, A091998.

Programs

Haskell

a097933 n = a097933_list !! (n-1)
a097933_list = [x | x <- a091998_list, a010051 x == 1]
-- Reinhard Zumkeller, Jan 07 2012

Magma

[p: p in PrimesUpTo(1000) | p mod 24 in [1, 11, 13, 23]]; // Vincenzo Librandi, Mar 23 2013

Mathematica

Select[Prime[Range[300]], MemberQ[{1, 11, 13, 23}, Mod[#, 24]]&] (* Vincenzo Librandi, Mar 23 2013 *)
Select[Prime[Range[2,200]],PowerMod[3,(#-1)/2,#]==1&] (* Harvey P. Dale, Jun 02 2020 *)

PARI

/* s = +-1, d=diff */ ptopm1d2(n,x,d,s) = { forprime(p=3,n,p2=(p-1)/2; y=x^p2 + s*(x-d)^p2; if(y%p==0,print1(p","))) }

PARI

{a(n)= local(m, c); if(n<1, 0, c=0; m=0; while( cMichael Somos, Aug 28 2006 */