Concept

Primorial

The product of all the primes up to a given prime, written p#, as 2 × 3 × 5 = 30 for p = 5. Euclid's proof adds one to it to make a number no listed prime divides, and its logarithm is close to p by the prime number theorem.

Named by 2 essays across one field — each of them below, with the objects they name alongside it.

Euclid's construction on 2, 3, 5, 7. The product of the listed primes plus one, divided by each of them in turn; every division leaves one over.

There is no last prime

Euclid's argument is often described as producing a new prime from any finite list. It does not, and the number it builds is frequently composite — which makes the proof more interesting rather than less.

number · Infinitude of primes

The primes p for which Euclid's number p# ± 1 is itself prime. p up to 1100: p# + 1 prime for 2, 3, 5, 7, 11, 31, 379, 1019, 1021; p# − 1 prime for 3, 5, 11, 13, 41, 89, 317, 337, 991.

When Euclid's number is itself prime

Multiply the primes up to p and add one. Euclid needed only that the result has a prime factor beyond p; it is itself prime for p = 2, 3, 5, 7 and 11, and then not again until 31, and then not until 379. Up to 1,100 there are nine such primes of each sign, against thirteen that a careful estimate expects. The estimate predicts infinitely many and grows by about 1.8 every time x is multiplied by e; nobody can prove there is even one more.

number · Infinitude of primes

Named alongside it

The objects these essays reach for when they reach for this one.

Primes Composite Counting argument Divisibility Euclid Existence proof Factorial Harmonic series Heuristic Open problem Prime number theorem Proof by contradiction

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