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January 8th, 2017, 05:42 PM   #1
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Integer solutions to (x^2+x) + (y^2+y) = (z^2+z)

Is there any integer solution to the equation, (x^2+x) + (y^2+y) = (z^2+z) ?
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January 8th, 2017, 08:12 PM   #2
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$\displaystyle (3^2+3)+(5^2+5)=(6^2+6)$
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January 9th, 2017, 12:26 AM   #3
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http://www.artofproblemsolving.com/c...3046h1046687__
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January 9th, 2017, 11:24 AM   #4
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(x, y, z) = (0, t, t), where t is an integer is the easiest solution.

A better question would be to find all the solutions in positive integers.

Let t be a natural number, then some (not necessarily distinct) solutions in positive integers are given below.

(x, y, z) = (3t, 4t + 1, 5t + 1)

(x, y, z) = (3t + 2, 4t + 2, 5t + 3)

(x, y, z) = (t + 2, (t + 1)(t + 4)/2, (t + 2)(t + 3)/2)

(x, y, z) = (3t + 5, (3t + 2)(t + 3)/2, (3t² + 11t + 12)/2)

One might speculate that every solution with x < y can be given in a similar way.
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January 9th, 2017, 12:03 PM   #5
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z = {-1 + SQRT[1 + 4(x^2 + x + y^2 + y)]} / 2

Kinda interesting to "look at"!!
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