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Prime Square Differences

Problem

Prove that all primes greater than 2 can be written as the difference of two squares.


Solution

By considering the difference of two consecutive square numbers,
(n + 1)2 minus n2 = (n2 + 2n + 1) minus n2 = 2n + 1

Hence all odd numbers greater than 1, which must include all primes greater than 2, can be written as the difference of two square numbers.

For example, 83 = 2times41 + 1 = 422 minus 412

Investigate primes that can be written as the difference of square numbers in general, including non-consecutive squares.

Problem ID: 94 (Dec 2002)     Difficulty: 2 Star

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