[seqfan] Sequences with a^b +/- c^d

[Charles Greathouse]

Sequences A164063, A164064, A164065 are (p1, p2, p3) of the form p1 =
a^b - c^d, where a, b, c, d are primes and a + b + c + d = p2, where
p2 (A164064) is prime and conc(abcd) = p3 (concatenation of a, b, c,
d) is also prime.

Sequences A164077, A164078, A164079 are the analogous sequences for p1
= a^b + c^d.

1. Are A164064, A164065, A164078, and A164079 well-defined?
Conceivably there would be multiple solutions for a given p1 value. Or
does A164063(n) not necessarily correspond to A164064(n)?

2. Is A164063 correct?  In the absence of a constructive proof of
Pillai's conjecture that there are only finitely many nontrivial
powers within a fixed distance, can the non-membership of an element
in A164063 be proven?
I was not able to prove the members of A164063 to be correct, but
perhaps I'm overlooking something.  Failing that, can their membership
be verified under some well-known conjecture?  Pillai and abc are
ineffective.

3. Would someone care to extend these sequences (subject to #2)?  I submitted
%S A164077 3253,24517,78157,366103,548677,705097,1030429,1229257,5735467,6438391,
%T A164077 12221371,17498881,19618243,74084347,118370899,263374849,270840151,
%U A164077 286199371,410180599,418195621,418719781,529483321,565609411,698388391
but did not send in the corresponding entries for A164078 and A164079,
nor was I able to calculate enough entries for a respectable b-file.

4. These sequences, if updated, should get the ,base, keyword.

Charles Greathouse
Analyst/Programmer
Case Western Reserve University