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A Quintic Problem (Posted on 2006-03-15) Difficulty: 3 of 5
Determine all possible integer solutions, whether positive or negative, to this equation:

5y4 + 2560y2 = x5 - 65536

See The Solution Submitted by K Sengupta    
Rating: 2.3333 (3 votes)

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Hints/Tips One solution, no proof | Comment 1 of 7

x = 16, y = 16    or    x = 16, y = -16  (x has to be positive, but y could go either way, of course)

I wish I had some formal mathematical proof for this, but I don't, I'll leave that to the smart people.  I sort of happened upon it by accident, noticing that 2560 = 10(16^2) and 65536 = 16^4.  I figured if x or y was 16 it would be easy to simplify...turns out it was. 

One of those I just "saw", can't really explain the inspiration.  Just figured I'd check out the "16" theme, normally this is the type of puzzle I won't even try - I'll wait for someone else to show me how it's done. :)

 


  Posted by tomarken on 2006-03-15 15:15:43
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