In
mathematics, the Feit–Thompson conjecture is a
conjecture
In mathematics, a conjecture is a conclusion or a proposition that is proffered on a tentative basis without proof. Some conjectures, such as the Riemann hypothesis (still a conjecture) or Fermat's Last Theorem (a conjecture until proven in 1 ...
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 integer-valued functions. German mathematician Carl Friedrich Gauss (1777–1855) said, "Math ...
, suggested by . The conjecture states that there are no distinct
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 ...
s ''p'' and ''q'' such that
:
divides
.
If the conjecture were true, it would greatly simplify the final chapter of the
proof
Proof most often refers to:
* Proof (truth), argument or sufficient evidence for the truth of a proposition
* Alcohol proof, a measure of an alcoholic drink's strength
Proof may also refer to:
Mathematics and formal logic
* Formal proof, a con ...
of the
Feit–Thompson theorem In mathematics, the Feit–Thompson theorem, or odd order theorem, states that every finite group of odd order is solvable. It was proved by .
History
conjectured that every nonabelian finite simple group has even order. suggested using ...
that every
finite group of odd
order
Order, ORDER or Orders may refer to:
* Categorization, the process in which ideas and objects are recognized, differentiated, and understood
* Heterarchy, a system of organization wherein the elements have the potential to be ranked a number of d ...
is
solvable. A stronger conjecture that the two numbers are always
coprime
In mathematics, two integers and are coprime, relatively prime or mutually prime if the only positive integer that is a divisor of both of them is 1. Consequently, any prime number that divides does not divide , and vice versa. This is equival ...
was disproved by with the
counterexample
A counterexample is any exception to a generalization. In logic a counterexample disproves the generalization, and does so rigorously in the fields of mathematics and philosophy. For example, the fact that "John Smith is not a lazy student" is ...
''p'' = 17 and ''q'' = 3313 with common
factor
Factor, a Latin word meaning "who/which acts", may refer to:
Commerce
* Factor (agent), a person who acts for, notably a mercantile and colonial agent
* Factor (Scotland), a person or firm managing a Scottish estate
* Factors of production, ...
2''pq'' + 1 = 112643.
It is known that the conjecture is true for ''q'' = 3 .
Informal
probability
Probability is the branch of mathematics concerning numerical descriptions of how likely an Event (probability theory), event is to occur, or how likely it is that a proposition is true. The probability of an event is a number between 0 and ...
arguments suggest that the "expected" number of counterexamples to the Feit–Thompson conjecture is very close to 0, suggesting that the Feit–Thompson conjecture is likely to be true.
See also
*
Cyclotomic polynomials
*
Goormaghtigh conjecture
References
*
*
*
*
External links
* (This article confuses the Feit–Thompson conjecture with the stronger disproved conjecture mentioned above.)
{{DEFAULTSORT:Feit-Thompson conjecture
Conjectures
Unsolved problems in number theory