A309594 Smallest members of prime triples, the sum of which results in a perfect square.

13, 37, 277, 613, 12157, 14557, 23053, 55213, 81013, 203317, 331333, 393853, 824773, 867253, 1008037, 2038573, 3026053, 3322213, 5198197, 5497237, 5793517, 5984053, 9107173, 17246413, 20850757, 20871853, 21327997, 25363573, 25678573, 27258613, 29134597, 30153037, 33313333

Offset

1,1

Comments

A prime triple is a set of three prime numbers of the form (p, p+2, p+6) or (p, p+4, p+6).

The smallest prime of the first form of these triples is not part of this sequence because p + (p+2) + (p+6) = 3p +8 and a number of this form is never a square.

PROOF:

From Bernard Schott, Aug 09 2019: (Start)

If a == 0 (mod 3) ==> a^2 == 0 (mod 3),

If a == 1 (mod 3) ==> a^2 == 1 (mod 3),

If a == 2 (mod 3) ==> a^2 == 4 == 1 (mod 3).

Hence, a square is always == 0 or == 1 (mod 3)

As p + (p+2) + (p+6) = 3*p+8, and 3*p+8 == 2 (mod 3), there is no prime triple of the form (p, p+2, p+6) whose sum 3*p + 8 can be a square. (End)

Links

Robert Israel, Table of n, a(n) for n = 1..10000

Example

Let p = 277 (prime), q = p+4 = 281 (prime), r = p+6 = 283 (prime). We now have a prime triple. p+q+r = 841 = 29^2, a perfect square.

Maple

Res:= NULL: count:= 0:
for k from 0 while count < 100 do
  for x in [6*k+1, 6*k+5] do
    p:= (x^2-10)/3;
    if isprime(p) and isprime(p+4) and isprime(p+6) then
      count:= count+1;
      Res:= Res, p
    fi
od od:
Res; # Robert Israel, Aug 13 2019

Mathematica

ok[p_] := If[AllTrue[{p, p+4, p+6}, PrimeQ], Sow@p]; Reap[Do[ok[3 y^2 + 2 y - 3]; ok[3 y^2 + 4 y - 2], {y, 4000}]][[2, 1]] (* Giovanni Resta, Aug 09 2019 *)

Prog

(PARI) issq(p) = issquare(3*p+10);
istriple(p) = isprime(p+4) && isprime(p+6);
isok(p) = isprime(p) && istriple(p) && issq(p); \\ Michel Marcus, Aug 10 2019

Crossrefs

Cf. A130621.

Intersection of A022005 and A206279.

Sequence in context: A078952 A206279 A130621 * A098265 A195540 A262475

Adjacent sequences: A309591 A309592 A309593 * A309595 A309596 A309597

Keyword

nonn

Author

Philip Mizzi, Aug 09 2019

Extensions

More terms from Michel Marcus, Aug 09 2019

Status

approved