Infinitude of Primes - An Impossible Injection

This is a recent proof by Dustin J. Mixon. (I am grateful to Claus Ivo Doering for bringing it to my attention.)

Proof

Suppose there are a total of \(N\;\) primes, \(2=p_{1}\lt p_{2}\lt\ldots\lt p_{N}\), and pick \(K\;\) such that \(2^{K}\gt (K+1)^N\).

Consider the mapping \(f: \{1,\ldots,2^{K}\}\rightarrow\{0,\ldots,K\}^N\;\) defined by \(f(x):=(k_{1},\ldots,k_{N})\), where \(x=p_{1}^{k_1}\cdot\ldots\cdot p_{N}^{k_N}\;\) is the prime factorization of \(x\).

Here, the fact that \(f(x)\in\{0,\ldots,K\}^N\;\) follows from

\(\displaystyle K\ge\sum_{n=1}^{N}k_{n}\mbox{log}_{2}p_{n}\ge\sum_{n=1}^{N}k_{n}\ge \mbox{max}(k_{1},\ldots,k_{N}). \)

Then \(f\;\) is injective by the fundamental theorem of arithmetic, which contradicts the Pigeonhole Principle.

Note

The proof above appears to be based on ideas already used by A. Thue in 1897.

References

D. J. Mixon, Another Simple Proof of Infinitude of Primes, Am Math Monthly, 119 (2012), n 9, 811

Infinitude of Primes

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