number theory Number theory (or arithmetic or higher arithmetic in older usage) is a branch of pure mathematics devoted primarily to the study of the integers and integer-valued functions. German mathematician Carl Friedrich Gauss (1777–1855) said, "Math ...

, Lemoine's conjecture, named after

Émile Lemoine Émile Michel Hyacinthe Lemoine (; 22 November 1840 – 21 February 1912) was a French civil engineer and a mathematician, a geometer in particular. He was educated at a variety of institutions, including the Prytanée National Militaire and, mos ...

, also known as Levy's conjecture, after

Hyman Levy Prof Hyman Levy (1889–1975) was a Scottish-Jewish philosopher, Emeritus Professor of Imperial College London, mathematician, political activist and fellow of the Royal Society of Edinburgh.StafHyman LevyGazetteer for Scotland, Retrieved 10 Febr ...

, states that all

odd integer In mathematics, parity is the property of an integer of whether it is even or odd. An integer is even if it is a multiple of two, and odd if it is not.. For example, −4, 0, 82 are even because \begin -2 \cdot 2 &= -4 \\ 0 \cdot 2 &= 0 \\ 41 ...

s greater than 5 can be represented as the sum of an odd

prime number A prime number (or a prime) is a natural number greater than 1 that is not a product of two smaller natural numbers. A natural number greater than 1 that is not prime is called a composite number. For example, 5 is prime because the only way ...

and an even

semiprime In mathematics, a semiprime is a natural number that is the product of exactly two prime numbers. The two primes in the product may equal each other, so the semiprimes include the squares of prime numbers. Because there are infinitely many prime ...

History

The conjecture was posed by

in 1895, but was erroneously attributed by

MathWorld ''MathWorld'' is an online mathematics reference work, created and largely written by Eric W. Weisstein. It is sponsored by and licensed to Wolfram Research, Inc. and was partially funded by the National Science Foundation's National Science ...

who pondered it in the 1960s. 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) ).

Formal definition

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 In number theory, Goldbach's weak conjecture, also known as the odd Goldbach conjecture, the ternary Goldbach problem, or the 3-primes problem, states that : Every odd number greater than 5 can be expressed as the sum of three primes. (A prime ma ...

Example

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

Evidence

According to

, the conjecture has been verified by Corbitt up to 10⁹. A blog post in June of 2019 additionally claimed to have verified the conjecture up to 10¹⁰.

Notes

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'', 58(4) (Sep., 1985), pp. 195–203. . *

Richard K. Guy Richard Kenneth Guy (30 September 1916 – 9 March 2020) was a British mathematician. He was a professor in the Department of Mathematics at the University of Calgary. He is known for his work in number theory, geometry, recreational mathemat ...

, ''Unsolved Problems in Number Theory'' New York: Springer-Verlag 2004: C1

External links

Levy's Conjecture
by Jay Warendorff,

Wolfram Demonstrations Project The Wolfram Demonstrations Project is an organized, open-source collection of small (or medium-size) interactive programs called Demonstrations, which are meant to visually and interactively represent ideas from a range of fields. It is hos ...

. {{Prime number conjectures Additive number theory Conjectures about prime numbers Unsolved problems in number theory