Goormaghtigh conjecture
Encyclopedia
In mathematics
, the Goormaghtigh conjecture is a conjecture
in number theory
named for the Belgian
mathematician
René Goormaghtigh
. The conjecture is that the only non-trivial integer
solutions of the exponential Diophantine equation
satisfying x, y > 1 and n, m > 2 are
This may be expressed as saying that 31 and 8191 are the only two numbers that are repunit
s with at least 3 digits in two different bases.
Balasubramanian
and Shorey have proved that there are only finitely many possible solutions to the equations in (x,y,m,n) with prime divisors of x and y lying in a given finite set and that they may be effectively computed
.
Mathematics
Mathematics is the study of quantity, space, structure, and change. Mathematicians seek out patterns and formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proofs, which are arguments sufficient to convince other mathematicians of their validity...
, the Goormaghtigh conjecture is a conjecture
Conjecture
A conjecture is a proposition that is unproven but is thought to be true and has not been disproven. Karl Popper pioneered the use of the term "conjecture" in scientific philosophy. Conjecture is contrasted by hypothesis , which is a testable statement based on accepted grounds...
in number theory
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...
named for the Belgian
Belgium
Belgium , officially the Kingdom of Belgium, is a federal state in Western Europe. It is a founding member of the European Union and hosts the EU's headquarters, and those of several other major international organisations such as NATO.Belgium is also a member of, or affiliated to, many...
mathematician
Mathematician
A mathematician is a person whose primary area of study is the field of mathematics. Mathematicians are concerned with quantity, structure, space, and change....
René Goormaghtigh
René Goormaghtigh
René Goormaghtigh was a Belgian engineer, after whom the Goormaghtigh Conjecture is named....
. The conjecture is that the only non-trivial integer
Integer
The integers are formed by the natural numbers together with the negatives of the non-zero natural numbers .They are known as Positive and Negative Integers respectively...
solutions of the exponential Diophantine equation
satisfying x, y > 1 and n, m > 2 are
- (x, y, m, n) = (5, 2, 3, 5); and
- (x, y, m, n) = (90, 2, 3, 13).
This may be expressed as saying that 31 and 8191 are the only two numbers that are repunit
Repunit
In recreational mathematics, a repunit is a number like 11, 111, or 1111 that contains only the digit 1. The term stands for repeated unit and was coined in 1966 by Albert H. Beiler...
s with at least 3 digits in two different bases.
Balasubramanian
Ramachandran Balasubramanian
Ramachandran Balasubramanian is an Indian mathematician and is currently the Director of the Institute of Mathematical Sciences in Chennai, India...
and Shorey have proved that there are only finitely many possible solutions to the equations in (x,y,m,n) with prime divisors of x and y lying in a given finite set and that they may be effectively computed
Effective results in number theory
For historical reasons and in order to have application to the solution of Diophantine equations, results in number theory have been scrutinised more than in other branches of mathematics to see if their content is effectively computable...
.