Size Doesn't Matter

Number Theory Level 4

How many numbers in the sequence 70 , 71 , 78 , 79 , 86 , 87 , . . . , 2014 , 2015 70, 71, 78, 79, 86, 87, ... , 2014, 2015 can be expressed as the sum of the squares of two positive integers?

2 1 0 403 1007

Grant Bulaong by Grant Bulaong , 22, Philippines

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1 solution

Grant Bulaong
Jul 24, 2016

For any integer n n , n 2 ≡ 0 , 1 , 4 ( m o d 8 ) n^2 \equiv 0,1,4 \left(\mod 8\right) . Thus the sum of two squares can be 0 , 1 , 2 , 4 , 5 ( m o d 8 ) 0,1,2,4,5 \left(\mod 8\right) . All numbers in the given series are either 6 6 or 7 7 m o d 8 \mod 8 .

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