Enormous product of primes

Number Theory Level pending

Let $p_1=2, p_2=3, p_3=5, p_4=7,...$ be the primes. Compute $(65537-p_1)(65537-p_2)...(65537-p_{10000}) \mod 65537$

The answer is 0.

1 solution

Alex Li
Jul 9, 2015

Note that $65537=2^{16}+1$ , which is a well-known Fermat prime. A quick search will reveal that there are $9592$ primes less than $100000$ , so one of the primes used will be $65537$ , which makes the whole product $\boxed{0}$ .

Moderator note:

There's a simpler way to prove that the 10000th prime number is larger than 65537.

Hint : Prime Number Theorem.

Enormous product of primes

The answer is 0.

1 solution

Moderator note:

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