The Sum Of The Squares Of Tow Consecutive Positive Integers Is 340. Find The Integers?

Let n and (n+1) be the integers.
n² + (n+1)² = 340
n² + n² + 2n + 1 = 340
2n² + 2n = 339    (subtract 1 from both sides of the equation, collect terms)
2(n²+n) = 339    (factor out 2 for clarity)

This equation has no integer solution for n. If n is an integer, n² is an integer and (n²+n) is an integer. No integer exists such that multiplying it by 2 will give 339.

There are no such integers.

Find four consecutive even
integers whose sum is 340.
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