# Lemoine's conjecture

In number theory, Lemoine's conjecture, named after Émile Lemoine, also known as Levy's conjecture, after Hyman Levy, states that all odd integers greater than 5 can be represented as the sum of an odd prime number and an even semiprime.

## History

The conjecture was posed by Émile Lemoine in 1895, but was erroneously attributed by MathWorld to Hyman Levy who pondered it in the 1960s.

A similar conjecture by Sun in 2008 states that all odd integers greater than 3 can be represented as the sum of prime number and the product of two consecutive positive integers ( p+x(x+1) ).

## Formal definition

To put it algebraically, 2n + 1 = p + 2q always has a solution in primes p and q (not necessarily distinct) for n > 2. The Lemoine conjecture is similar to but stronger than Goldbach's weak conjecture.

## Example

For example, 47 = 13 + 2 × 17 = 37 + 2 × 5 = 41 + 2 × 3 = 43 + 2 × 2. (sequence A046927 in the OEIS) counts how many different ways 2n + 1 can be represented as p + 2q.

## Evidence

According to MathWorld, the conjecture has been verified by Corbitt up to 109.[citation needed] A blog post in June of 2019 additionally claimed to have verified the conjecture up to 1010.