how the hell are we supposed to somehow notice that 2438100000001 = x^5 + x^4 + 1 ? xD

Haha, exactly my point ;p

Actually I'm somewhat embarassed to say this. But I noticed I made an error in my argument so I'll be breaking my head over this one too. However my argument works for other large numbers. So I'll make a number myself, and change the puzzle ;p

About the original puzzle:

Wolframalpha gives the prime-factorisation

\(\displaystyle 73\times829\times1237\times32569\)

I find it somewhat an odd puzzle, cause I don't think any normal human being is going to prove that the number is composite without knowing any of the prime-factors beforehand.

The answer however provided in Wilmer's link is not fallacious. I found it somewhat a clever trick. But with a inhuman observation that the number can be written as \(\displaystyle f(x)=x^5+x^4+1\) with \(\displaystyle x=300\).

I don't fully understand the proof, but the simpel observation that \(\displaystyle f(x)=(x^2+x+1)(x^3-x+1)\) seems to me enough.

then \(\displaystyle 300^2+300+1 = 90301\), and \(\displaystyle 300^3-300+1= 26999701\) are both factors, so the number is not prime.

Can you guys prove that

**139801079 **is composite without using the help of a computer?