Divisibility for the Win - Problem 3

A polynomial $f(x)\in\mathbb{Z}[x]$ has the property that $2^{n-1}-1\mid f\left(2^{n}-1\right)$ for all integers $n>1$ . Show that $f(x)=0$ has an integer root.

#Proofathon

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`2 \times 3`	$2 \times 3$
`2^{34}`	$2^{34}$
`a_{i-1}`	$a_{i-1}$
`\frac{2}{3}`	$\frac{2}{3}$
`\sqrt{2}`	$\sqrt{2}$
`\sum_{i=1}^3`	$\sum_{i=1}^3$
`\sin \theta`	$\sin \theta$
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Comments

Is there a more general statement that can be made here?

On a related note, show that if $n \mid f(n)$ for infinitely many integers $n$ , then $f(0) = 0$ .

Calvin Lin Staff - 7 years, 2 months ago

The general statement, I believe, would be that if $f(m) \mid f(g(m))$ for infinitely many $m \in \mathbb{Z}$ for some $f, g \in \mathbb{Z}[x],$ then $f(g(0))=0,$ although there are probably other ways generalize this.

Note that $f(n) \equiv f(0) \pmod{n},$ so $n \mid f(0)$ for infinitely many $n \in \mathbb{Z},$ which is possible if and only if $f(0)=0,$ since $0$ is the only integer with infinitely many divisors.

Sreejato Bhattacharya - 7 years, 2 months ago

We have the condition $A\mid f(2A+1)$ where $A$ is a power of two minus one. Assuming that $n$ is big enough, let $p$ be the biggest prime number dividing $A$ : we have $p\mid f(2A+1)$ , or just $p\mid f(1)$ . Since the set of prime numbers dividing a number of the form $2^m-1$ is infinite (*), we have that $f(1)$ is divisible by an infinite number of primes - this gives $f(1)=0$ .

(*) Assuming that every $2^n-1$ can be factored in terms of $p_1,\ldots,p_k$ , we have a contradiction, since: $\forall i\in[1,k],\qquad 2^{1+\prod_{j=1}^k(p_j-1)}-1\equiv 2-1\equiv 1\not\equiv 0\pmod{p_i}.$

Jack D'Aurizio - 7 years, 2 months ago

Nice way to show that the set of primes which divide $2^n -1$ is infinite.

Note that $p$ doesn't need to be the biggest prime number that divides $A$ . (doesn't impact your proof)

Calvin Lin Staff - 7 years, 2 months ago

Minor typo: this gives $f(1)=0.$

Sreejato Bhattacharya - 7 years, 2 months ago

Math	Appears as
Remember to wrap math in `\(` ... `\)` or `\[` ... `\]` to ensure proper formatting.
`2 \times 3`	$2 \times 3$
`2^{34}`	$2^{34}$
`a_{i-1}`	$a_{i-1}$
`\frac{2}{3}`	$\frac{2}{3}$
`\sqrt{2}`	$\sqrt{2}$
`\sum_{i=1}^3`	$\sum_{i=1}^3$
`\sin \theta`	$\sin \theta$
`\boxed{123}`	$\boxed{123}$