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

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**Question: Prove that 2**^{10}+ 5^{12}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 …..
- …. Think about it before reading on ….
**Proof**:-- First complete the square, ie. (a + b)
^{2}= a^{2}+ 2ab + b^{2}, so - a
^{2}+ b^{2}= (a + b)^{2}– 2ab, with obvious substitutions, a = 2^{5}and b = 5^{6} - So, if 2ab is a square, we would then have the difference of two squares, which we could factorise in the form c
^{2}– d^{2}= (c + d) x (c – d). - Which, if so, would be a composite number, hence not prime.
- Now, of course, the 2ab term is 2 x 2
^{5}x 5^{6}, which is - 2
^{6}x 5^{6}, which, of course is - (2
^{3}x 5^{3})^{2}

- First complete the square, ie. (a + b)
- 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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04/06/2020 23:48:29 | 1303 (A Prime Number?) | None |

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