Mykhailo Papenko - cp's OEIS frontend

User: Mykhailo Papenko

Mykhailo Papenko has authored 2 sequences.

A375217 Primes p such that p^64 + 2^64 is prime.

37, 53, 181, 491, 547, 619, 661, 677, 911, 941, 1297, 1423, 1867, 2441, 2687, 3137, 3571, 5387, 5821, 5881, 6449, 6551, 6899, 8263, 8537, 8999, 9803, 9931, 10861, 11057, 11131, 11423, 12377, 12941, 13147, 14009, 14519, 14759, 14813, 15493, 16103, 16573, 19949

Offset: 1

Mykhailo Papenko, Oct 17 2024

nonn

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.

But it's proven that either p1 or p2 must be a 2.

Mykhailo Papenko, Table of n, a(n) for n = 1..8079
Mykhailo Papenko, Primes-Made-Up-of-Primes, Github.

6th row of A132260.

/* see link for code with instructions */

Select[Prime[Range[2255]],PrimeQ[#^64+2^64]&] (* James C. McMahon, Nov 19 2024 *)

p^64 + 2^64 ∈ A000040 (p^64 + 2^64 belongs to the set of the prime numbers)

A375215 Primes of the form p^16 + 2^16, where p is prime (see also A157764).

Original entry on oeis.org

15496731425178936435099327796097, 295216374856540727739668685343937, 4579937329576774398276408998557697, 19419444565344683427626434801775297, 643780251284828743866259724717471297, 1110832290554380967776058484990830657, 57196271293373441589892672200988689857, 75456166331666628614079195878996262017

Offset: 1

Mykhailo Papenko, Oct 17 2024

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

			a(1) = 89^16 + 2^16 = 15496731425178936435099327796097, which is prime.
a(2) = 107^16 + 2^16 = 295216374856540727739668685343937, which is prime.
a(3) = 127^16 + 2^16 = 4579937329576774398276408998557697, which is prime.
a(4) = 139^16 + 2^16 = 19419444565344683427626434801775297, which is prime.
a(5) = 173^16 + 2^16 = 643780251284828743866259724717471297, which is prime.
a(6) = 179^16 + 2^16 = 1110832290554380967776058484990830657, which is prime.
a(7) = 229^16 + 2^16 = 57196271293373441589892672200988689857, which is prime.
a(8) = 233^16 + 2^16 = 75456166331666628614079195878996262017, which is prime.
a(9) = 349^16 + 2^16 = 48440300802975619860301347588732379759937, which is prime.
a(10) = 421^16 + 2^16 = 973898133213875918230007677219773667320257, which is prime.

Mykhailo Papenko, Table of n, a(n) for n = 1..29908
Mykhailo Papenko, Primes-Made-Up-of-Primes, Github.

The corresponding primes p are in A157764.

/* see link for code with instructions */

Select[Table[Prime[p]^16+2^16,{p,60}],PrimeQ] (* James C. McMahon, Nov 18 2024 *)

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

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User: Mykhailo Papenko

A375217 Primes p such that p^64 + 2^64 is prime.

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