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What is the maximum possible value of the greatest common divisor of two consecutive terms of the sequence a_n = n^2 - 1, where n is greater than 0?

 Nov 26, 2020
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The sequence computed to 100 terms:

 

listfor(n, 1, 100, gcd((n^2 - 1),(n+1)^2 - 1))=(3, 1, 1, 3, 1, 1, 3, 1, 1, 3, 1, 1, 3, 1, 1, 3, 1, 1, 3, 1, 1, 3, 1, 1, 3, 1, 1, 3, 1, 1, 3, 1, 1, 3, 1, 1, 3, 1, 1, 3, 1, 1, 3, 1, 1, 3, 1, 1, 3, 1, 1, 3, 1, 1, 3, 1, 1, 3, 1, 1, 3, 1, 1, 3, 1, 1, 3, 1, 1, 3, 1, 1, 3, 1, 1, 3, 1, 1, 3, 1, 1, 3, 1, 1, 3, 1, 1, 3, 1, 1, 3, 1, 1, 3, 1, 1, 3, 1, 1, 3).

 

As you can see, the maximum GCD is 3.

 Nov 26, 2020

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