Find all triples of natural numbers (a, b, c) satisfying the system of equations

9514 1404 393

Step-by-step explanation:

Solution of the system by the usual means yields the parametric equations ...

  (a², b², c²) = (41 +t, t -7, 2t)

Since we want t-7 to be a perfect square, we can let ...

  t -7 = n²

  t = n² +7

Then the solutions are ...

  (a, b, c) = (±√(n²+48), n, ±√(2(n² +7)))

The only integer solutions are for n=±1 and n=±11. Then the 16 possible triples are ...

  (±7, ±1, ±4) and (±13, ±11, ±16) . . . where the signs can have any combination

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Step-by-step explanation:

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