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

Jonathan Post jvospost3 at gmail.com
Tue Aug 14 00:41:29 CEST 2007

```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. Note that 2^8 + n^8 can be prime for composite n
beginning 21, 55, 69, 77,
87, 117.
%F A132216 Primes in A132215. {A001016(A000040(i)) +
A001016(A000040(j)) for i > j and are elements of A000040}.
%e A132216 a(1) = 2^8 + 13^8 = 256 + 815730721 =
815730977, which is prime.
a(2) = 2^8 + 137^8 = 124097929967680577, which is
prime.
a(3) = 2^8 + 223^8 = 6115597639891380737, which is
prime.
a(4) = 2^8 + 331^8 = 144086718355753024097, which is
prime.
a(5) = 2^8 + 389^8 = 524320466699664691937, which is
prime.
a(6) = 2^8 + 491^8 = 3377940044732998170977, which is
prime.
a(7) = 2^8 + 563^8 = 10094089678769799935777, which is
prime.
%t A132216 *** extended with a(5) through a(7) since previous
submission, a few minutes ago ***
%Y A132216 Cf. A000040, A001016, A050997, A120398,
A122616, A130873, A130555, A132214, A132215.
%O A132216 1
%K A132216 ,easy,more,nonn,
%A A132216 Jonathan Vos Post (jvospost2 at yahoo.com),
Aug 13 2007

Then after I resubmitted, I found:

a(8) = 2^8 + 647^8 = 30706 777728 209453 204417, which is prime.

a(9) = 2^8 + 701^8 = 58310 148000 746221 725857 is prime.

a(10) = 2^8 + 773^8 = 127478 208443 761827 917537 is prime.

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?

What are the common characteristics of the index sequence:

13, 137, 223, 331, 398, 491, 563, 647, 701, 773, ...?

The next such sequence is based on 16th powers...

```