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### A Prime Number?

(Text as at 04/06/2020 23:48:29)

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• Question: Prove that 210 + 512 cannot be a prime number
• To my shame, I didn’t give this trivial problem a proper go, having stopped when the solution should have been obvious. Anyway, none of the other friends I sent it to seem to have given it a proper go either.
• I did write a little Access VBA program (Jacks_Non_Prime) to check that it was indeed non-prime, and the number is 244,141,649 = 14,657 x 16657.
• Well, the proof is simple …..
• Proof:-
1. First complete the square, ie. (a + b)2 = a2 + 2ab + b2, so
2. a2 + b2 = (a + b)2 – 2ab, with obvious substitutions, a = 25 and b = 56
3. So, if 2ab is a square, we would then have the difference of two squares, which we could factorise in the form c2 – d2 = (c + d) x (c – d).
4. Which, if so, would be a composite number, hence not prime.
5. Now, of course, the 2ab term is 2 x 25 x 56, which is
6. 26 x 56, which, of course is
7. (23 x 53)2
• So, there we are … I’d done everything apart from the critical step of spotting the difference of two squares. Going senile.

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