perplexus.info :: Just Math : Power Equality Poser

Power Equality Poser (Posted on 2015-03-29)

Find all nonzero integer solutions of this system of equations:

A^A+B= B¹² and, B^A+B = A³

Prove that there are no others.

See The Solution Submitted by K Sengupta

Rating: 4.0000 (2 votes)

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Quadratically

| Comment 2 of 4 |

After step 4 in the previous post, we know that

A + B = 6 and A = +/- B^2

If A = B^2, then B^2 + B - 6 = 0, making B either -3 or 2.

If A = -B^2, then B^2 - B + 6 = 0, and this has no real solutions.

So there are only two solutions in addition to (1,1), and these are (9,-3) and (4,2).

Posted by Steve Herman on 2015-03-29 19:03:39

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