Lemoine's conjecture
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
. 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
.
For example, 47 = 13 + 2 × 17 = 37 + 2 × 5 = 41 + 2 × 3 = 43 + 2 × 2. counts how many different ways 2n + 1 can be represented as p + 2q.
According to MathWorld
, the conjecture has been verified by Corbitt up to 109.
The conjecture was posed by Émile Lemoine
in 1895, but in more recent years came to be attributed to Hyman Levy
who pondered it in the 1960s.
Number theory
Number theory is a branch of pure mathematics devoted primarily to the study of the integers. Number theorists study prime numbers as well...
, Lemoine's conjecture, named after Émile Lemoine
Émile Lemoine
Émile Michel Hyacinthe Lemoine 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, most notably, the École Polytechnique...
, also known as Levy's conjecture, after Hyman Levy
Hyman Levy
Hyman Levy was a Scottish philosopher, mathematician and political activist.The son of Minna Cohen and Marcus Levy, a Jewish art dealer in Edinburgh, Hyman was the third oldest of eight children. He went to school at George Heriot's School, and was the School Dux. Thanks to scholarships, he was...
, states that all odd integers greater than 5 can be represented as the sum of an odd prime number
Prime number
A prime number is a natural number greater than 1 that has no positive divisors other than 1 and itself. A natural number greater than 1 that is not a prime number is called a composite number. For example 5 is prime, as only 1 and 5 divide it, whereas 6 is composite, since it has the divisors 2...
and an even semiprime
Semiprime
In mathematics, a semiprime is a natural number that is the product of two prime numbers. The first few semiprimes are 4, 6, 9, 10, 14, 15, 21, 22, 25, 26, ... ....
. 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
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:...
.
For example, 47 = 13 + 2 × 17 = 37 + 2 × 5 = 41 + 2 × 3 = 43 + 2 × 2. counts how many different ways 2n + 1 can be represented as p + 2q.
According to MathWorld
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 Digital Library grant to the University of Illinois at...
, the conjecture has been verified by Corbitt up to 109.
The conjecture was posed by Émile Lemoine
Émile Lemoine
Émile Michel Hyacinthe Lemoine 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, most notably, the École Polytechnique...
in 1895, but in more recent years came to be attributed to Hyman Levy
Hyman Levy
Hyman Levy was a Scottish philosopher, mathematician and political activist.The son of Minna Cohen and Marcus Levy, a Jewish art dealer in Edinburgh, Hyman was the third oldest of eight children. He went to school at George Heriot's School, and was the School Dux. Thanks to scholarships, he was...
who pondered it in the 1960s.
See also
- Lemoine's conjecture and extensions
External links
- Levy's Conjecture by Jay Warendorff, Wolfram Demonstrations ProjectWolfram Demonstrations ProjectThe Wolfram Demonstrations Project is hosted by Wolfram Research, whose stated goal is to bring computational exploration to the widest possible audience. It consists of an organized, open-source collection of small interactive programs called Demonstrations, which are meant to visually and...
.