A358742 First of three consecutive primes p,q,r such that p^3 + q^3 - r^3 is prime.

13, 29, 89, 97, 127, 137, 151, 163, 199, 223, 241, 277, 313, 349, 367, 389, 419, 431, 457, 463, 521, 577, 613, 691, 823, 827, 829, 859, 877, 883, 911, 953, 971, 1049, 1087, 1097, 1129, 1151, 1163, 1217, 1409, 1489, 1499, 1579, 1699, 1723, 1867, 1879, 1993, 2089, 2111, 2141, 2293, 2339, 2399, 2411

Offset: 1

Author

J. M. Bergot and Robert Israel, Nov 29 2022

Keywords

• nonn

Comments

Note: for x >= 3275, there is a prime between x and x(1 + 1/(2 log^2 x)) < 1.01x (Dusart 1998). Together with finite checking this shows that for p > 19, p^3 + q^3 - r^3 > 0. - Charles R Greathouse IV, Nov 29 2022

Examples

			a(3) = 89 is a term because 89, 97, 101 are consecutive primes and 89^3 + 97^3 - 101^3 = 587341 is prime.
		

Links

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

Crossrefs

Cf. A255581, A358743, A358744.

Programs

Maple

R:= NULL: count:= 0: q:= 2: r:= 3:
while count < 100 do
  p:= q; q:= r; r:=nextprime(r);
  if isprime(p^3+q^3-r^3) then count:= count+1; R:= R,p; fi;
od:
R;

Mathematica

Select[Partition[Prime[Range[360]], 3, 1], (s = #[[1]]^3 + #[[2]]^3 - #[[3]]^3) > 0 && PrimeQ[s] &][[;; , 1]] (* Amiram Eldar, Nov 29 2022 *)

PARI

a358742(upto) = {my(p1=2, p2=3); forprime(p3=5, upto, if (isprime (p1^3+p2^3-p3^3), print1(p1,", ")); p1=p2; p2=p3)};
a358742(2500) \\ Hugo Pfoertner, Nov 29 2022

Python

from itertools import islice
from sympy import isprime, nextprime
def agen():
    p, q, r = 2, 3, 5
    while True:
        if isprime(p**3 + q**3 - r**3): yield p
        p, q, r = q, r, nextprime(r)
print(list(islice(agen(), 56))) # Michael S. Branicky, Nov 29 2022