In
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 arithmetic function, integer-valued functions. German mathematician Carl Friedrich Gauss (1777â ...
, 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
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 ways ...
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 nu ...
.
History
The conjecture was posed by
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 ...
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 Dig ...
to
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 ...
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
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 Dig ...
, the conjecture has been verified by Corbitt up to 10
9.
A blog post in June of 2019 additionally claimed to have verified the conjecture up to 10
10.
See also
*
Lemoine's conjecture and extensions
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 Conjectureby 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