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The Sum of Fourth Powers (Posted on 2007-07-04) Difficulty: 3 of 5
Find all integer solutions to x4+y4=(x+y)(xy+1) with x≥y≥0.

  Submitted by Brian Smith    
Rating: 4.0000 (1 votes)
Solution: (Hide)
Assume that x >= 3, then:
x^4 > x^3 + x^3 + x + x
x^4 > x^2*y +x*y^2 + x + y
x^4 + y^4 > x^2*y +x*y^2 + x + y
This contradicts the given equation, therefore 3 > x.

The potential solutions are then limited to (0,0),(1,0),(2,0)(1,1),(2,1), and (2,2). Of these possibilites only (0,0) and (1,0) are solutions.

Comments: ( You must be logged in to post comments.)
  Subject Author Date
SolutionSolutionhoodat2007-07-04 19:16:36
SolutionSolutionK Sengupta2007-07-04 14:49:31
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