Question

欧拉如何用因式分解证明费马数F5是合数

Answer 1

证明设a=2^7,b=5,则a-b^3=128-125=3,

2^(2^5)+1=2^32+1=(2a)^4+1

=16a^4+1=(1+3×5)a^4+1

=(1+(a-b^3)b)a^4+1

=(1+ab-b^4)a^4+1

=(1+ab)a^4-a^4×b^4+1

=(1+ab)a^4-(a^2×b^2+1)(a^2×b^2-1)

=(1+ab)a^4-(a^2×b^2+1)(ab+1)(ab-1)

=(ab+1)(a^4-(a^2×b^2+1)(ab-1))

=641×6700417