Close powers of integers

May 2015
508
32
Arlington, VA
Do there exist powers of integer pairs, both greater than three, whose differences are greater than two and singly sequential?

For differences zero to two: 1^N-1^N=0...3^2-2^3=1...3^3-5^2=2... ?
 
Jun 2019
493
262
USA
Can you define that a little more clearly?
You're looking for integers a and b: a>3, b>3, \(\displaystyle |a^b-b^a|>2\)
Singly sequential meaning what? |a-b|=1?
 
  • Like
Reactions: 2 people
May 2015
508
32
Arlington, VA
Can you define that a little more clearly?
You're looking for integers a and b: a>3, b>3, \(\displaystyle |a^b-b^a|>2\)
Singly sequential meaning what? |a-b|=1?
Are there nonzero integers a, b, c and d such that |a^b-c^d| includes all Z?
 

v8archie

Math Team
Dec 2013
7,709
2,677
Colombia
Trivially when $b=d=1$. What are your constrains such as the greater than three and the differences greater than two?
 
  • Like
Reactions: 2 people
May 2015
508
32
Arlington, VA
Trivially when $b=d=1$. What are your constrains such as the greater than three and the differences greater than two?
Do integers a and c, and those b and d >1, obey |a^b-c^d|=Z for all Z?

I guess these are the only constraints I need, noting the trivial case b=d=1.

I know 1^b-1^d=0, 3^2-2^3=1, and 3^3-5^2=2. Likewise negatives. Does this pattern extend to all Z?