A347531 - OEIS

A347531

First of three consecutive primes p,q,r such that r^2-p^2+p, r^2-p^2+q and r^2-p^2+r are consecutive primes.

5, 19, 31, 97, 107, 1429, 1597, 1997, 2441, 2917, 3203, 3581, 3659, 3691, 3847, 7481, 8237, 8377, 8737, 9697, 11467, 12107, 12251, 14071, 17317, 17659, 18481, 18583, 21383, 21491, 23189, 24799, 30931, 33563, 34337, 37507, 38317, 38639, 38707, 39397, 40483, 41221, 43577, 43649, 45121, 46411, 51059

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OFFSET

1,1

COMMENTS

Dickson's conjecture implies that there are infinitely many terms with, for example, the first three consecutive primes p, p+2, p+6 and the second three 13*p+36, 13*p+38 and 13*p+42.

LINKS

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

EXAMPLE

a(3) = 31 is a term because 31, 37, 41 are consecutive primes and 41^2-31^2+31 = 751, 41^2-31^2+37 = 757, 41^2-31^2+41 = 761 are consecutive primes.

MAPLE

R:= NULL: count:= 0:

q:= 2: r:= 3:

while count < 50 do

p:= q; q:= r; r:= nextprime(r);

d:= r^2 - p^2;

if isprime(p+d) and nextprime(p+d)=q+d and nextprime(q+d)=r+d then

count:= count+1; R:= R, p