Euler prime

From Wikipedia, the free encyclopedia

In number theory, Euler primes or symmetric primes are primes that are the same distance from a given integer. ^{[citation needed]} For example 3 and 13 are both 5 units from the number 8, hence are symmetric primes. All twin primes, cousin primes, and sexy primes are symmetric primes.

Euler/symmetric primes constitute a Goldbach partition, which is defined as a pair of primes p,q that sum to an even integer. It would appear that all three terms are synonymous, the latter one seemingly most used in the literature.

[edit] Every natural number ≥ 2 has related symmetric primes

Goldbach's conjecture implies that there is at least one (pair of, not necessarily distinct) symmetric primes for every natural number n ≥ 2. Assuming then that symmetric primes p,q may have distance 0 (ie, p = q = n), this conjecture might be formally expressed as:

Let n be a natural number ≥ 2 and p,q primes. If p + q = 2n, then p,q are symmetric primes over n. Such p,q exist for all n.

The first part of this conjecture is trivially true, it is the last sentence which has yet to be proven/disproven.

Clearly the lower bound of n should be increased if one were to insist that symmetric primes be distinct, ie, have a minimal distance of 1 (q - n = n - p ≥ 1) — a judgement call which we leave to the reader.