Primes that are sums of eighth powers of two distinct primes.

[Max Alekseyev]

On 8/13/07, Jonathan Post <jvospost3 at gmail.com> wrote:
> PRE-NUMBERED NEW SEQUENCE A132216 FROM Jonathan Vos
> Post
>
> %I A132216
> %S A132216 815730977, 124097929967680577, 6115597639891380737,
> 144086718355753024097, 524320466699664691937, 3377940044732998170977,
> 10094089678769799935777
> %N A132216 Primes that are sums of eighth powers of two distinct primes.
> %C A132216 These primes exist because the polynomial x^8 + y^8 is
> irreducible over Z.

This is a strange comment.
The value a(1)=815730977 alone implies that "these primes exist".
It seems that you wanted to make a comment about the infiniteness of
such primes. But it is a long-standing conjecture that the irreducible
over Z polynomial takes an infinite number of prime values.

[...]

> It doesn't need to be stated that x^8 + y^8 is prime for integers x,
> y, means that either x or y is 2, does it?

I would better state that. "Primes of the form 2^8 + n^8" sounds more
natural and straightforward to me than "primes of the form x^8+y^8".

> What are the common characteristics of the index sequence:
>
> 13, 137, 223, 331, 398, 491, 563, 647, 701, 773, ...?

We can make some sieve-based claims like:
these numbers are odd;
these numbers equal 0, 1, 2, 4, 8, 9, 13, 15, or 16 modulo 17;
etc.

Max