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.

History

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

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

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

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]

Notes

^ ^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

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