A073608

[Jim Nastos]

This sequence (A073608) contains a conjecture. It's proof is below,
and I ask a question regarding it:

ID Number: A073608
Sequence:  1,3,5,8,10
Name:      a(1) = 1, a(n) = smallest number such that a(n)-a(n-k) is a 
prime or a prime power for all k.
Comments:  Differences |a(i)-a(j)| are primes or prime powers for all i,j.
              Conjecture: sequence is bounded.
Example:   a(5) = 10 as 10-8,10-5,10-3, 10-1 or 2,5,7,9 are either prime 
or prime powers.

  I assume "sequence is bounded" means "sequence is finite," as a(n) >= 
a(n-2) if it exists. For the proof of this, I require the sixth term, 
which is 12. (The differences are 11,9,7,4,2, all primes/prime powers.)
  Assume some k was the 7th term. Then {k-1, k-3, k-5} must contain a 
multiple of 3 as does the set {k-8, k-10, k-12}, and all these numbers 
must be primes or prime powers. k>12, so there must be a power of 3 in 
each set, and they are distinct. Their difference can be at most 11, which 
is impossible.
  My question: Is there a more fundamental proof that doesn't use the 
knowledge of the 6th term? If there didn't exist a 6th term (or if it was 
extremely difficult to find) would there still be an elementary proof?
 Thanks.
-- 
Jim Nastos, B.Math, B.Ed          |     Office: 117 Athabasca Hall
MSc Candidate                     |     Office Phone: (780) 492-5046
University Of Alberta             |     Edmonton, Alberta
Department of Computing Science   |     T6G 2H1
nastos at cs.ualberta.ca             |
http://www.cs.ualberta.ca/people/grad/nastos.html