Ernst Kummer

Ernst Eduard Kummer (29 January 1810 in Sorau, Brandenburg, Prussia - 14 May 1893 in Berlin, Germany) was a German mathematician. Highly skilled in applied mathematics, Kummer trained German army officers in ballistics; afterwards, he taught for 10 years in a Gymnasium (the German equivalent of high school), where he inspired the mathematical career of Leopold Kronecker. He retired from teaching and from mathematics in 1890.

Proof of an infinite number of primes

In 1878 Kummer gave this proof that there are an infinite number of primes.

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Assume the number of primes is finite: p1, p2, ..., pk. Let n be the product of all of these primes. Now n?1 is not prime because it is greater than pk. If n?1 is not prime then there is some i such that pi divides n?1. But since pi also divides n, it divides their difference, n ? (n?1) = 1. This is impossible, so there are an infinite number of primes.

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