number theory - How prove this $a^n-b^n$ always have prime factor $P$ and $P>n$

Let $p_{1},p_{2},p_{3}$ be different prime numbers, and let the positive integer $n$, be defined by
$$n=p_{1}p_{2}p_{3}.$$

Show that:

For any two positive integer $a,b$ ,then $a^n-b^n$ always has a prime factor $P$ satisfying $P>n$

This problem is from a maths exam a few days ago.
We only need to prove that $$a^{p_{1}p_{2}p_{3}}-b^{p_{1}p_{2}p_{3}}$$ has a prime factor $P>n$.
But I can't. Thank you very much!

Answer

If $a=b$ then $a^n-b^n=0$ is divisible by any prime, so we are done since there are infinitely many primes.

Otherwise we may WLOG assume $a>b$. Let $d=\gcd(a, b)$. By Zsigmondy's theorem, $\left(\frac{a}{d}\right)^n-\left(\frac{b}{d}\right)^n$ has a prime factor $p$ that does not divide $\left(\frac{a}{d}\right)^k-\left(\frac{b}{d}\right)^k$ for any positive integer $k

We cannot have $p \mid \frac{a}{d}$, for otherwise $p \mid \left(\frac{a}{d}\right)^n-\left(\frac{b}{d}\right)^n$ implies $p \mid \frac{b}{d}$ as well, so $p \mid \frac{a}{d}-\frac{b}{d}$ and $1

The order of $\left(\frac{a}{d}\right)\left(\frac{b}{d}\right)^{-1} \pmod{p}$ is thus $n$. Since $p \nmid \left(\frac{a}{d}\right)\left(\frac{b}{d}\right)^{-1}$, we have by Fermat's little theorem that $\left(\left(\frac{a}{d}\right)\left(\frac{b}{d}\right)^{-1}\right)^{p-1} \equiv 1\pmod{p}$. Thus $n \mid p-1$. This gives (since clearly $p>1$) that $n \leq p-1$ so $p>n$.

Finally since $a^n-b^n=d^n\left(\left(\frac{a}{d}\right)^n-\left(\frac{b}{d}\right)^n\right)$ we have $p \mid a^n-b^n$ and $p>n$, so we are done.

A simpler proof which does not appeal to Zsigmondy's theorem might be possible, as we only require a special case. (Here $n=p_1p_2p_3$) and may not require the full machinery used to prove Zsigmondy's theorem (Cyclotomic polynomials)