Proof that $2^{10}+5^{12}$ is a composite number

Question

A hint or full answer will help a lot, because I have no idea what to do.

http://www.wolframalpha.com/input/?i=factorize+2%5E10%2B5%5E12 — Wolfram, Jan 29 '17 at 13:38
It is a number of the form $a^4+4b^4$ and $$ a^4+4b^4 = (a^2-2ab+2b^2)(a^2+2ab+2b^2)$$ — Jack D'Aurizio, Jan 29 '17 at 13:41

score 3 · Accepted Answer · answered Jan 29 '17 at 13:40

3

$$5^{12}+4\cdot 2^{8} = (5^6-2\cdot 5^3 \cdot 2^2+2\cdot 2^4)(5^6+2\cdot 5^3 \cdot 2^2+2\cdot 2^4)$$

Using $a^4+4b^4 =(a^2-2ab+2b^2)(a^2+2ab+2b^2)$

answered Jan 29 '17 at 13:40

juantheron

score -1 · Answer 2 · answered Jan 29 '17 at 13:39

-1

Hint: This can be expressed as the difference of two squares in a reasonably easy way.

answered Jan 29 '17 at 13:39

Mark Bennet

1

Note that factorising by using the difference of two squares is a respectable technique. – Mark Bennet Jan 29 '17 at 13:44

2 Answers2