Lemoine's conjecture
From Wikipedia the free encyclopedia
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[edit]
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 a prime number and the product of two consecutive positive integers ( p+x(x+1) ).[2]
Formal definition[edit]
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[edit]
For example, the odd integer 47 can be expressed as the sum of a prime and a semiprime in four different ways:
- 47 = 13 + 2×17 = 37 + 2×5 = 41 + 2×3 = 43 + 2×2.
The number of ways this can be done is given by OEIS sequence A046927 (Number of ways to express 2n+1 as p+2q where p and q are primes). Lemoine's conjecture is that this sequence contains no zeros after the first three.
Evidence[edit]
According to MathWorld, the conjecture has been verified by Corbitt up to 109.[1] A blog post in June of 2019 additionally claimed to have verified the conjecture up to 1010.[3]
A proof was claimed in 2017 by Agama and Gensel, but this was later found to be flawed.[4]
See also[edit]
Notes[edit]
- ^ a b Weisstein, Eric W. "Levy's Conjecture". MathWorld.
- ^ Sun, Zhi-Wei. "On sums of primes and triangular numbers." arXiv preprint arXiv:0803.3737 (2008).
- ^ "Lemoine's Conjecture Verified to 10^10". 19 June 2019. Retrieved 19 June 2019.
- ^ Agama, Theophilus; Gensel, Berndt (21 March 2021). "A Proof of Lemoine's Conjecture by Circles of Partition". arXiv:1709.05335v6 [math.NT].
References[edit]
- 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. doi:10.2307/2690477. JSTOR 2690477
- John O. Kiltinen and Peter B. Young, "Goldbach, Lemoine, and a Know/Don't Know Problem", Mathematics Magazine, 58(4) (Sep., 1985), pp. 195–203. doi:10.2307/2689513. JSTOR 2689513
- Richard K. Guy, Unsolved Problems in Number Theory New York: Springer-Verlag 2004: C1
External links[edit]
- Levy's Conjecture by Jay Warendorff, Wolfram Demonstrations Project.