A358744 First of three consecutive primes p, q, r such that p + q - r, p^2 + q^2 - r^2 and p^3 + q^3 - r^3 are all prime.

13, 29, 137, 521, 577, 691, 823, 1879, 3469, 4799, 8783, 21569, 25453, 26263, 26591, 27529, 27919, 34607, 39509, 45631, 48869, 53653, 56099, 56633, 57641, 63313, 63809, 67733, 68819, 74381, 76031, 76421, 94781, 97187, 98873, 101279, 105683, 110291, 118967, 119569, 119849, 120577, 123737, 128951

Offset: 1

Author

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

Keywords

• nonn

Examples

			a(3) = 137 is a term because 137, 139, 149 are consecutive primes and
137^1 + 139^1 - 149^1 = 127,
137^2 + 139^2 - 149^2 = 15889,
and 137^3 + 139^3 - 149^3 = 1949023 are all prime.
		

Links

• Robert Israel, Table of n, a(n) for n = 1..2500

Crossrefs

Intersection of A358743, A255581 and A358742.

Programs

Maple

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

Mathematica

Select[Partition[Prime[Range[13000]], 3, 1], AllTrue[Table[#[[1]]^k + #[[2]]^k - #[[3]]^k, {k, 1, 3}], PrimeQ] &][[;; , 1]] (* Amiram Eldar, Nov 29 2022 *)

Python

from itertools import islice
from sympy import isprime, nextprime
def agen():
    p, q, r = 2, 3, 5
    while True:
        if all(isprime(t) for t in [p+q-r, p**2+q**2-r**2, p**3+q**3-r**3]):
            yield p
        p, q, r = q, r, nextprime(r)
print(list(islice(agen(), 44))) # Michael S. Branicky, Nov 29 2022