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 September 8th, 2019, 08:31 PM #1 Senior Member   Joined: May 2015 From: Arlington, VA Posts: 458 Thanks: 29 Math Focus: Number theory 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... ?
 September 9th, 2019, 05:19 AM #2 Senior Member   Joined: Jun 2019 From: USA Posts: 213 Thanks: 90 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? Thanks from topsquark and Loren
September 10th, 2019, 01:23 PM   #3
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 Originally Posted by 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?
Are there nonzero integers a, b, c and d such that |a^b-c^d| includes all Z?

 September 11th, 2019, 06:44 AM #4 Math Team   Joined: Dec 2013 From: Colombia Posts: 7,683 Thanks: 2664 Math Focus: Mainly analysis and algebra Trivially when $b=d=1$. What are your constrains such as the greater than three and the differences greater than two? Thanks from topsquark and Loren
September 11th, 2019, 02:09 PM   #5
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 Originally Posted by v8archie 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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