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

Jonathan Post jvospost3 at gmail.com
Tue Aug 14 02:16:47 CEST 2007

Max, what I meant, and the cross-references detail this, is that:

p^k + q^k can NOT be prime for k = 3, 5, 6, 7, 9, and those values of
k such that x^k + y^k is reducible over Z.  That is, k must NOT be a
power of 2.

The seive comment is useful, and I do thank you for your time and consideration.

On 8/13/07, Max Alekseyev <maxale at gmail.com> wrote:
> 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
>