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# 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

Synonym:

Levy conjecture, Lemoine's conjecture

Type of Math Object:

Conjecture

Major Section:

Reference

## Mathematics Subject Classification

11P32*no label found*

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## Comments

## Levy's conjecture is NOT due to Levy

Actually, the so-called "Levy conjecture" was already stated in the 19th century, see Dickson's book "History of the Theory of Numbers", vol. I. I have made a copy of that page but forgot it in the library.

## Re: Levy's conjecture is NOT due to Levy

Can you at least remember the chapter number or chapter title, or the general neighborhood the conjecture could be found in Dickson's book?

## Levy's conjecture (2n+1=p+2q) should be called Lemoine's con...

Here is the last paragraph on page 424 of the following book

L.E. Dickson, History of the Theory of Numbers, Vol. I,

Amer. Math. Soc., Chelsea Publ., Providence RI, 1999.

"E. Lemoine [L'intermediaire des math., 1(1894), 179; 3(1896), 151]

stated empirically that every odd number >3 is a sum of a prime p and the double of a prime q, and is also of the forms p-2q and 2q'-p'. " [I think 3 should be a typo for 5.]

Since Lemoine's paper appeared much earlier than Levy's paper in 1963, surely we should call the conjecture 2n+1=p+2q Lemoine's conjecture, not Levy's conjecture.