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Number Theory Level 4

Let $S$ be a set of (distinct!) prime numbers, and let $\langle S\rangle_G$ be the geometric mean of $S$ :

$\langle S\rangle_G = \sqrt[\#S]{\prod S}.$

Suppose there exist a prime number $p$ and a positive integer $n$ such that $\langle S \rangle_G^n = p$ . What is the greatest possible number of elements in $S$ ?

3 There is no maximum size for

S

. 2 Such a set does not exist. 1 37

1 solution

Arjen Vreugdenhil
Feb 16, 2016

Let $m = \#S$ be the count of prime numbers in $S$ , and $N = \prod S$ their product. Then we have $N^{n/m} = p\ \ \therefore\ \ N^n = p^m.$ Unique prime factorization guarantees that $N$ is a power of the prime number $p$ . However, since $N$ is the product of distinct prime numbers, we must have $N = p$ . This means that $S = \{p\}$ has only one element.

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