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Find all integer solutions for $a_1(a_1+k)(a_1+2k)...(a_1+(2k+1)d)+1 = n^2$ [closed]

+1
−3

Closed as not constructive by beloh‭ on Dec 22, 2022 at 10:47

This question cannot be answered in a way that is helpful to anyone. It's not possible to learn something from possible answers, except for the solution for the specific problem of the asker.

This question was closed; new answers can no longer be added. Users with the reopen privilege may vote to reopen this question if it has been improved or closed incorrectly.

Find all integer solution for the equation below: $$a_1(a_1+k)(a_1+2k)...(a_1+(2k+1)d)+1 = n^2$$

=$$ 1+\prod_{k=0}^{2k+1} (a_1 + kd) = n^2 $$

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4 comment threads

Is there supposed to be +1 in the title? (1 comment)
The product as you've written it now is not well defined, and I think not what you wanted anyway. It ... (1 comment)
Video (1 comment)
Counterexample? (3 comments)

1 answer

+3
−0

The initial observation is explained by the identity $$(x^2 + 3x + 1)^2 = x(x+1)(x+2)(x+3) + 1$$

The generalisation is messed up in the question as it currently stands, but the only smallish similar identity which I can find is the uninteresting $$(x+1)^2 = x(x+2) + 1$$

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