# Lemoine's conjecture

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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.[1]

A similar conjecture by Sun in 2008 states that all odd integers greater than 3 can be represented as the sum of an odd prime number and the product of two consecutive 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.

## References

• Emile Lemoine, L'intermédiare des mathématiciens, 1 (1894), 179; ibid 3 (1896), 151.
• H. Levy, "On Goldbach's Conjecture", Math. Gaz. 47 (1963): 274
• L. Hodges, "A lesser-known Goldbach conjecture", Math. Mag., 66 (1993): 45–47.
• John O. Kiltinen and Peter B. Young, "Goldbach, Lemoine, and a Know/Don't Know Problem", Mathematics Magazine, Vol. 58, No. 4 (Sep., 1985), pp. 195–203 (http://www.jstor.org/stable/2689513?seq=7)
• Richard K. Guy, Unsolved Problems in Number Theory New York: Springer-Verlag 2004: C1