# Levy’s conjecture

Conjecture (Émile Lemoine). All odd integers greater than 5 can be represented as the sum of an odd prime and an even semiprime. In other words, $2n+1=p+2q$ always has a solution in primes $p$ and $q$ (not necessarily distinct) for $n>2$.

For example, $47=13+2\times 17=37+2\times 5=41+2\times 3=43+2\times 2$. A046927 in Sloane’s OEIS counts how many different ways $2n+1$ can be represented as $p+2q$.

The conjecture was first stated by Émile Lemoine in 1894. In 1963, Hyman Levy published a paper mentioning this conjecture in relation to Goldbach’s conjecture.

## References

• 1 L. E. Dickson, History of the Theory of Numbers Vol. I. Providence, Rhode Island: American Mathematical Society & Chelsea Publications (1999): 424
• 2 R. K. Guy, Unsolved Problems in Number Theory New York: Springer-Verlag 2004: C1
• 3 L. Hodges, “A lesser-known Goldbach conjecture”, Math. Mag., 66 (1993): 45 - 47.
• 4 É. Lemoine, “title” L’intermediaire des mathematiques 179 3 (1896): 151
• 5 H. Levy, “On Goldbach’s Conjecture”, Math. Gaz. 47 (1963): 274
Title Levy’s conjecture LevysConjecture 2013-03-22 17:26:32 2013-03-22 17:26:32 PrimeFan (13766) PrimeFan (13766) 6 PrimeFan (13766) Conjecture msc 11P32 Levy conjecture Lemoine’s conjecture