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

1,1

Links

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

Example

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.

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 *)

Prog

(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

Crossrefs

Intersection of A358743, A255581 and A358742.

Sequence in context: A209989 A269515 A166272 * A317897 A195878 A013545

Adjacent sequences: A358741 A358742 A358743 * A358745 A358746 A358747

Keyword

nonn

Author

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

Status

approved