# Close powers of integers

#### Loren

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... ?

#### DarnItJimImAnEngineer

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?

2 people

#### Loren

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
Trivially when $b=d=1$. What are your constrains such as the greater than three and the differences greater than two?

2 people

#### Loren

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?