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Question
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!

Answer
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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Janet Yang

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I can answer questions in Algebra, Basic Math, Calculus, Differential Equations, Geometry, Number Theory, and Word Problems. I would not feel comfortable answering questions in Probability and Statistics or Topology because I have not studied these in depth.

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I tutor students (fifth through twelfth grades) and am a Top Contributor on Yahoo!Answers with over 24,000 math solutions.

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