A270361 Odd primes p for which there exists an odd prime q < p such that p*q - 1 is a square.

13, 29, 53, 61, 73, 89, 97, 109, 137, 149, 157, 173, 233, 241, 277, 317, 349, 353, 373, 389, 397, 409, 433, 461, 521, 541, 569, 593, 617, 641, 653, 661, 673, 701, 709, 733, 757, 769, 773, 821, 829, 853, 877, 881, 929, 937

Offset: 1

Author

Richard R. Forberg, Mar 15 2016

Keywords

• nonn

Comments

Conjecture: For any odd prime p there is at most one odd prime q, with q < p, for which p*q-1 is square.

(Note: If p were not restricted to being prime, there could be multiple primes q which make p*q-1 a square, with increasing multiplicities for larger p. The upper limit on the multiplicity of solutions, for prime and nonprime p and q values, is given by A006278. Note, however that those limits allow for solutions where q=2, and therefore two solutions when p and q are prime.)

a(n) is a subsequence of the Pythagorean Primes (A002144), which are of the form 4k+1.

The density of a(n) values among the primes declines with increasing n. For example, a(n) is about 22% of the first 1000 primes, and drops to about 15% of "incremental" primes around prime(10000). The density continues to fall among even larger primes. Twice those percentages apply as a portion of A002144.

All values of q also belong to A002144. It appears the set of q values "intends to" fully comprise A002144. This is notable because p values comprise an increasingly sparse subsequence within A002144, and each p value has just one q value.

The ability to fully comprise A002144 with q values is further challenged by the fact that for any given q value (i.e., any term of A002144) multiple values of p > q can be found such that p*q-1 is square. Thus q values are "promiscuous", and apparently without bounds on the number of p values they can serve.

Contrast this with primes p and q such that p*q+1 is square. The result are the Twin Primes (A001359 and A006512), arranged in a simple one-to-one correspondence, with p = q+2.

Examples

			13 is in this sequence because 13*5 - 1 = 64, which is square, with 5 < 13.
		

Links

• Chai Wah Wu, Table of n, a(n) for n = 1..10471

Crossrefs

Cf. A002144.

Programs

Mathematica

result = {}; Do[p = Prime[i]; Do[q = Prime[j]; r = p*q - 1;
  If[Mod[r, 8] == 1 || Mod[r, 8] == 0 || Mod[r, 8] == 4,
   If[IntegerQ[Sqrt[r]], AppendTo[result, p]]], {j, 2, i - 1}],
{i, 3, 1000}]; result

PARI

lista(nn) = {forprime(p=3, nn, ok = 0; forprime (q=3, p-1, if (issquare(p*q-1), ok = 1; break);); if (ok, print1(p, ", ")););} \\ Michel Marcus, Apr 06 2016

Python

from gmpy2 import is_prime, is_square
for p in range(3, 10 ** 4, 2):
    flag = 0
    if is_prime(p):
        for q in range(3, p, 2):
            if is_square(p * q - 1) and is_prime(q):
                flag = 1
                break
    if flag:
        print(p, end=", ")
# Soumil Mandal, Apr 07 2016