A375215 - OEIS

%I #28 Nov 19 2024 08:59:35

%S 15496731425178936435099327796097,295216374856540727739668685343937,

%T 4579937329576774398276408998557697,

%U 19419444565344683427626434801775297,643780251284828743866259724717471297,1110832290554380967776058484990830657,57196271293373441589892672200988689857,75456166331666628614079195878996262017

%N Primes of the form p^16 + 2^16, where p is prime (see also A157764).

%C It is conjectured that solutions for p1^n + p2^n = p3 (where p1, p2, and p3 are all primes and n is a natural number) exist only when n is itself a power of two (when n is a number in A000079); and would have infinitely many solutions.

%C But it's known that either p1 or p2 must be a 2.

%H Mykhailo Papenko, <a href="/A375215/b375215.txt">Table of n, a(n) for n = 1..29908</a>

%H Mykhailo Papenko, <a href="https://github.com/PiTBody/Primes-Made-Up-of-Primes">Primes-Made-Up-of-Primes</a>, Github.

%F a(n) = A157764(n)^16 + 2^16.

%e a(1) = 89^16 + 2^16 = 15496731425178936435099327796097, which is prime.

%e a(2) = 107^16 + 2^16 = 295216374856540727739668685343937, which is prime.

%e a(3) = 127^16 + 2^16 = 4579937329576774398276408998557697, which is prime.

%e a(4) = 139^16 + 2^16 = 19419444565344683427626434801775297, which is prime.

%e a(5) = 173^16 + 2^16 = 643780251284828743866259724717471297, which is prime.

%e a(6) = 179^16 + 2^16 = 1110832290554380967776058484990830657, which is prime.

%e a(7) = 229^16 + 2^16 = 57196271293373441589892672200988689857, which is prime.

%e a(8) = 233^16 + 2^16 = 75456166331666628614079195878996262017, which is prime.

%e a(9) = 349^16 + 2^16 = 48440300802975619860301347588732379759937, which is prime.

%e a(10) = 421^16 + 2^16 = 973898133213875918230007677219773667320257, which is prime.

%t Select[Table[Prime[p]^16+2^16,{p,60}],PrimeQ] (* _James C. McMahon_, Nov 18 2024 *)

%o (Java) /* see link for code with instructions */

%Y The corresponding primes p are in A157764.

%K nonn,less

%O 1,1

%A _Mykhailo Papenko_, Oct 17 2024