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A Prime Number?

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

• 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 …..
• …. Think about it before reading on ….
• 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.