Here are the first odd prime numbers....

3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97, 101, 103, ....

By definition, an odd number is that it is an integer of the form n = 2k + 1, where k is an integer.

So, 3 = 2*1 + 1, 5 = 2*2 + 1, 7 = 2*3 + 1, 11 = 2*5 + 1, 13 = 2*6 + 1, 17 = 2*8 + 1, etc.

Notice that....

3 = 2^2 - 1^2

5 = 3^2 - 2^2

7 = 4^2 - 3^2

11 = 6^2 - 5^2

13 = 7^2 - 6^2

17 = 9^2 - 8^2

19 = 10^2 - 9^2

23 = 12^2 - 11^2

etc.

This shows that a prime number p > 2, can be expressed as the difference of two squares, of two specific consecutive numbers.

This is true because 2k + 1 = (k + 1)^2 - k^2

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