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Levy's conjecture (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.

Bibliography

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



"Levy's conjecture" is owned by PrimeFan.
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Other names:  Levy conjecture, Lemoine's conjecture
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Cross-references: Goldbach's conjecture, relation, Hyman Levy, OEIS, solution, semiprime, even, prime, odd, sum, odd integers, Émile Lemoine, conjecture
There are 2 references to this entry.

This is version 3 of Levy's conjecture, born on 2007-07-31, modified 2008-06-18.
Object id is 9822, canonical name is LevysConjecture.
Accessed 1067 times total.

Classification:
AMS MSC: 11P32 (Number theory :: Additive number theory; partitions :: Goldbach-type theorems; other additive questions involving primes)
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Discussion
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Levy's conjecture (2n+1=p+2q) should be called Lemoine's conjecture by RST on 2008-06-04 21:31:21

 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.
[ reply | up ]
Levy's conjecture is NOT due to Levy by RST on 2008-06-04 06:37:41
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.
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