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September 8th, 2019, 08:31 PM   #1
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Close powers of integers

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... ?
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September 9th, 2019, 05:19 AM   #2
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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?
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September 10th, 2019, 01:23 PM   #3
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Originally Posted by DarnItJimImAnEngineer View Post
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?
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September 11th, 2019, 06:44 AM   #4
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Trivially when $b=d=1$. What are your constrains such as the greater than three and the differences greater than two?
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September 11th, 2019, 02:09 PM   #5
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Originally Posted by v8archie View Post
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?
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