number theory Number theory is a branch of pure mathematics devoted primarily to the study of the integers and arithmetic functions. Number theorists study prime numbers as well as the properties of mathematical objects constructed from integers (for example ...

, 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, mo ...

, also known as Levy's conjecture, after Hyman Levy, 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 divisible by 2, and odd if it is not.. For example, −4, 0, and 82 are even numbers, while −3, 5, 23, and 69 are odd numbers. The ...

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 (mathematics), 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 ...

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 n ...

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 ...

to Hyman Levy 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 prime number, prime ...

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 . Lemoine's conjecture is that this sequence contains no zeros after the first three.

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. A proof was claimed in 2017 by Agama and Gensel, but this was later found to be flawed.

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

External links

Levy's Conjecture
by Jay Warendorff,

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