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. To put it algebraically, 2"n" + 1 = "p" + 2"q" 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.

For example, 47 = 13 + 2 × 17 = 37 + 2 × 5 = 41 + 2 × 3 = 43 + 2 × 2. OEIS|id=A046927 counts how many different ways 2"n" + 1 can be represented as "p" + 2"q".

According to MathWorld, the conjecture has been checked for all odd positive integers less than 10⁹.

The conjecture was posed by Émile Lemoine in 1895, but in more recent years came to be attributed to Hyman Levy who pondered it in the 1960s.

ee also

* Lemoine's conjecture and extensions

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

External links

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* [http://demonstrations.wolfram.com/LevysConjecture/ Levy's Conjecture] by Jay Warendorff, The Wolfram Demonstrations Project.