SOLUTION: determine whether it is possible for the sum of the squares of three consecutive numbers to ever be a square number

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Question 430309: determine whether it is possible for the sum of the squares of three consecutive numbers to ever be a square number
Answer by richard1234(7193) About Me  (Show Source):
You can put this solution on YOUR website!
If we let the integers be x, x+1, and x+2, then

x%5E2+%2B+%28x%2B1%29%5E2+%2B+%28x%2B2%29%5E2+=+3x%5E2+%2B+6x+%2B+5. This number is congruent to 2 modulo 3, since 3x%5E2+%2B+6x is divisible by 3 and 5 is 2 modulo 3. Note that the perfect squares are either 0 or 1 modulo 3. This means that 3x%5E2+%2B+6x+%2B+5 can never be equal to a perfect square.