2002 IMO Shortlist S08

Let $p_1,p_2,\ldots,p_n$ be distinct primes greater than $3$ . Show that $2^{p_1p_2\cdots p_n}+1$ has at least $4^n$ divisors.

令 $p_1,p_2,\ldots,p_n$ 为大于 $3$ 的不同素数。证明 $2^{p_1p_2\cdots p_n}+1$ 至少有 $4^n$ 个除数。