Lemoine's conjecture

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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 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, 47 = 13 + 2 × 17 = 37 + 2 × 5 = 41 + 2 × 3 = 43 + 2 × 2. (sequence A046927 in the OEIS) counts how many different ways 2n + 1 can be represented as p + 2q.

Evidence[edit]

According to MathWorld, the conjecture has been verified by Corbitt up to 109.[citation needed] A blog post in June of 2019 additionally claimed to have verified the conjecture up to 1010.[3]

See also[edit]

Notes[edit]

  1. ^ Weisstein, Eric W. "Levy's Conjecture". MathWorld.
  2. ^ Sun, Zhi-Wei. "On sums of primes and triangular numbers." arXiv preprint arXiv:0803.3737 (2008).
  3. ^ "Lemoine's Conjecture Verified to 10^10". June 19, 2019. Retrieved June 19, 2019.

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]