# Infinitude of Primes Via *-Sets

A *-set is a finite set of positive integers {a_{1}, ..., a_{n}} such that

(1)

a_{i} - a_{j} divides a_{i}

for all distinct i and j. |X| denotes for the size of set X, e.g. _{1}, ..., a_{n}}| = n.

### Lemma 1

For all n ≥ 2, there is a *-set of size n.

### Proof

The proof is by induction.

Suppose {a_{1}, ..., a_{n}} is a *-set. Define _{0} = a_{1}·...·a_{n},_{k}'s, _{k} = b_{0} + a_{k}, _{k}: k = 0, ..., n}

### Lemma 2

Suppose {a_{1}, ..., a_{n}} is a *-set of size n. For _{k} = 2^{ak} + 1._{1}, ..., f_{n}

### Proof

Assume there are f_{k} and f_{m}, f_{k} > f_{m}, divisible by the same odd prime p. Then p divides 2^{am}(2^{ak - am} - 1).

Since p is odd, it ought to divide 2^{ak - am} - 1. Now, if s|t then ^{s} - 1 | 2^{t} - 1^{ak - am} - 1^{ak} - 1.^{ak} - 1.^{ak} + 1.

### Theorem

There exists an infinite number of primes.

### Proof

By lemmas 1 and 2, for any N, there is a set of N mutually prime terms and, therefore, the same number of distinct primes. Thus the assumption that the number of primes is finite would lead to a contradiction.

## Reference

- M. Gilchrist,
__A Proof That There Are an Infinite Number of Rational Primes__,*Am Math Monthly*, 114 (August-September 2007), p. 622

|Contact| |Front page| |Contents| |Algebra| |Up|

Copyright © 1996-2018 Alexander Bogomolny