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a^2+1=b^3

Does the above sum have any solutions where both a and b are positive real integers? If yes, what are they? If not, what is the proof? Thank you!

David,

Please ignore any "duplicate question" notice you may receive—I hit the "reject duplicate" button accidentally.

I did mean to reject the question, but only because it stumped me.

a˛ + 1 = bł

Short answer: There are no solutions where a and b are both positive real integers.

I saw this question decades ago and know that the only integer solution is a = 0, b = 1.

Problem is, I cannot remember the proof!

It makes use of the fact that

a˛ + 1 = (a + i)(a - i)

and shows that (a + i) and (a - i) are relatively prime.

But that's all I can dig up.

If you find the answer, please let me know!

Janet

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