# The factors of a tetration plus an integer

There I was, just messing around with tetration, when I stumbled across this -

$$(x^x +1)/(x+1)$$ = integer (for odd integer values of x)

Playing some more with this it seems (not entirely sure as tetration quickly becomes to hard to compute) -

$$({{{^n}^+}^1}x + 1)$$/$$(^nx + 1)$$ --> Integer

(also for odd integer values of x)

I am aware of that this is to do with the factors of $$x^x +1$$ ,but can anyone give a full deeper visceral explanation to this? (If it is true - if not then tell me)

Thanks

• Do you have examples, where you can show it works? – Milten Oct 31 at 11:54
• sure, i'll get some up – Jakub Skop Oct 31 at 11:55
• (3^3^3 +1)/(3^3+1) = 272342767321 – Jakub Skop Oct 31 at 11:56
• (7^7+1)/(7+1) = 102943 – Jakub Skop Oct 31 at 11:57
• (11^11 + 1)/(11 +1) = 23775972551 – Jakub Skop Oct 31 at 11:57

We have the factorisation: $$x^{k}+1 = (x+1)(1-x+x^{2}-\cdots +x^{(k-1)})$$ for any odd natural number $$k$$.

$$^n x$$ divides $$^{n+1}x$$, since they are both just a bunch of $$x$$-factors. So we can write $$^{n+1}x = k\cdot{}^n x$$ for some odd $$k$$ (since $$^{n+1}x$$ is odd, $$k$$ has to be odd). Then we can factorise: $${}^{n+1}x + 1 = x^{{}^nx} + 1 = x^{k\cdot ({}^{n-1}x)} + 1 \\ = ({}^n x)^k+1 = ({}^n x+1)\left(1-({}^n x)^{({}^n x)}+({}^n x)^{2({}^n x)}-\cdots +({}^n x)^{(k-1)({}^n x)}\right)$$ which proves the result.

BONUS:

The factorisation we used is a variation of the following, which works for any $$k$$: $$x^{k}-1 = (x-1)(1+x+x^{2}+\cdots +x^{(k-1)})$$ Therefore we can conclude by the same method as above that $$({}^n x-1)$$ divides $$({}^{n+1}x - 1)$$ for any $$x$$.

• Where did you get the factorisation from - I've never seen it before. – Jakub Skop Oct 31 at 12:24
• Oops, I wrote it all wrong. I'll edit. – Milten Oct 31 at 12:48
• Have you edited it; I can't tell – Jakub Skop Oct 31 at 12:54
• Haha, yes I have. You can find it here under "Sum, odd exponent" with $E=x^m$, $F=1$. – Milten Oct 31 at 12:55
• @JakubSkop I finally proved it :) – Milten Oct 31 at 13:15