A358745 a(n) is the least prime p that is the first of three consecutive primes p, q, r such that p^i + q^i - r^i is prime for i from 1 to n but not n+1.

2, 7, 41, 13, 4799, 45631, 332576273, 157108359787, 4001045161

Offset

0,1

Comments

For any prime x, if p == r (mod x) and q <> x, or q == r (mod x) and p <> x, p^i + q^i - r^i is not divisible by x. Thus there is no modular obstruction to the sequence being infinite.

If a(9) exists, then it exceeds 8*10^12. - Lucas A. Brown, Mar 08 2024

Links

Table of n, a(n) for n=0..8.

Example

a(3) = 13 because 13, 17, 19 are consecutive primes with 13 + 17 - 19 = 11, 13^2 + 17^2 - 19^2 = 97 and 13^3 + 17^3 - 19^3 = 251 are prime but 13^4 + 17^4 - 19^4 = -18239 is not, and no prime less than 13 works.

Maple

V:= Array(0..5):
q:= 2: r:= 3: count:= 0:
while count < 6 do
  p:= q; q:= r; r:= nextprime(r);
  for i from 1 while isprime(p^i+q^i-r^i) do od:
  if V[i-1] = 0 then V[i-1]:= p; count:= count+1 fi;
od:
convert(V, list);

Prog

(PARI) a(n) = my(p=2, q=nextprime(p+1)); forprime(r=nextprime(q+1), oo, my(c=0); for(k=1, oo, if(isprime(p^k + q^k - r^k), c+=1, break)); if(c==n, return(p)); p = q; q = r); \\ Daniel Suteu, Jan 04 2023

Crossrefs

Cf. A255581, A358742, A358743, A358744.

Sequence in context: A317723 A340005 A325061 * A215207 A106871 A107376

Adjacent sequences: A358742 A358743 A358744 * A358746 A358747 A358748

Keyword

nonn,more,hard

Author

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

Extensions

a(6) from Michael S. Branicky, Nov 29 2022

a(7) from Daniel Suteu, Jan 04 2023

Status

approved