Today I had a math test, and there was one problem I couldn’t solve. Please help!
Prove that the following expressions are not prime numbers (there always exist a shorter beautiful solution):
a) $2^{10}+5^{12}$
b) $65^{64}+64$
c) $989\cdot 1001\cdot 1007+320$
For (a) and (b), we can use an Aurifeuillean factorization: $$ a^{4}+4b^{4}=(a^{2}-2ab+2b^{2})(a^{2}+2ab+2b^{2}) $$ Take $a=5^3$ and $b=2^2$. Then $$ 2^{10}+5^{12} = 4b^{4}+a^{4} = a^{4}+4b^{4} = (a^{2}-2ab+2b^{2})(a^{2}+2ab+2b^{2}) = 14657 \cdot 16657 $$
Now take $a=65^{16}$ and $b=2$. Then $$ 65^{64}+64 = a^{4}+4b^{4} = (a^{2}-2ab+2b^{2})(a^{2}+2ab+2b^{2}) = \small 10309258098174834118790766041058622855698420907745361328133 \quad\cdot 10309258098174834118790766041870898989955232237091064453133 $$
(c) is easier: $$ x(x+12)(x+18)+320 = (x + 2) (x + 8) (x + 20) $$ Now take $x=989$ and get $$ 989\cdot 1001\cdot 1007+320 = 991 \cdot 997 \cdot 1009 $$ (I was lucky that the first form I tried, $x(x+12)(x+18)+320$, factored so nicely!)