Question 197955
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Let <b><i>x</i></b> represent the first integer.  Then the next four consecutive integers are:


*[tex \LARGE  x + 1]


*[tex \LARGE  x + 2]


*[tex \LARGE  x + 3]


*[tex \LARGE  x + 4]


The sum of the squares of the first 3 is equal to the sum of the squares of the next two:


*[tex \LARGE \ \ \ \ \ \ \ \ \ \ x^2\ +\ (x + 1)^2\ +\ (x + 2)^2\ =\ (x + 3)^2\ +\ (x + 4)^2]


*[tex \LARGE \ \ \ \ \ \ \ \ \ \ x^2\ +\ x^2 + 2x + 1\ +\ x^2 + 4x + 4\ =\ x^2 + 6x + 9\ +\ x^2 + 8x + 16]


*[tex \LARGE \ \ \ \ \ \ \ \ \ \ 3x^2 + 6x + 5\ =\ 2x^2 + 14x + 25]


*[tex \LARGE \ \ \ \ \ \ \ \ \ \ x^2 - 8x - 20\ =\ 0]


Solve the quadratic (it factors) to determine the first integer.  Exclude the negative root because the problem asks for positive integers.


John
*[tex \LARGE e^{i\pi} + 1 = 0]
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