2

Following this post, I want to prove that $ \frac{\binom{n}{x_1,x_2...x_n}}{n}$ is an integer when $x_i\neq \{n,0 \}$ for $ 0 \leq i \leq n$ and prime $n$. The motivation is that I am trying to prove Fermat's little theorem.

For those interested , here is my proof so far:

To prove: $a^p - a $ is divisible by $p$

Proof:

$$ a^p -a = (1+1+1... \text{ a times})^p- a = \sum_{x_i \neq \{n,0 \} for 0 \leq i \leq n, } \binom{p}{x_1,x_2...,x_p}$$

Now.. if only if I could show that I could factor out a 'p' from the right most summation and still have an integer,I'd be done.

Bernard
  • 175,478
tryst with freedom
  • 11,538

1 Answers1

2

Since $p$ is a prime you can do it because $x_i<p$, and in the denominators of the multinomial coefficients there will not be a $p$.

Furbini
  • 388
  • Fair point on the factorability, but, how do I show it's an integer as well? – tryst with freedom Jun 13 '21 at 12:11
  • BY definition the multinomial coefficient is a natural number. @Fubini has shown clearly that except in the two extreme cases it is divisible by $p$. – ancient mathematician Jun 13 '21 at 12:18
  • @Buraian If you know that multinomial coefficients are integers, then this answer shows why you still have an integer after dividing by $p$. If you don't yet know that multinomial coefficients are integers, then you should learn about that before worrying about dividing it by $p$. – Andreas Blass Jun 13 '21 at 12:20
  • How do you mean it is so by definition? If you see the linked post, people had gone to elaborate schemes to show that it is an integer (however it could be made easy if you used recurrence definition,here I don't know of such) @ancientmathematician – tryst with freedom Jun 13 '21 at 12:22
  • Fair point, but the thing is that I tried some example values and I found it to evaluate to any integer only. That's how I hypothesized the idea in the question statement – tryst with freedom Jun 13 '21 at 12:22
  • 1
    The multinomial coefficient has a "concrete" definition: it is the number of ways to put $n$ objects in $m$ bowls such that the $i$th bowl contains $x_i$ objects. If you think a moment about this definition you will realize that it agrees with the numerical expression. From the "concrete" definition is pretty clear that multinomial coefficients are integers. @Buraian – Furbini Jun 13 '21 at 12:51
  • Is my proof of fermat's little theorem finished then? I felt my proof may have been a bit too simple lol – tryst with freedom Jun 14 '21 at 11:13
  • Yes I think that the proof is correct and that it is a very nice proof :) @Buraian – Furbini Jun 14 '21 at 11:53