# 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

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