Feit-Thompson conjecture: Difference between revisions
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==Statement== | ==Statement== | ||
For any two distinct primes <math>p,q</math>, the | For any two distinct primes <math>p,q</math>, the number <math>\Phi_p(q) = (q^p - 1)/(q-1)</math> does not divide <math>\Phi_q(p) = (p^q - 1)/(p-1)</math> are relatively prime. | ||
A stronger form of the conjecture is that the numbers <math>\Phi_p(q)</math> and <math>\Phi_q(p)</math> are relatively prime, and a counterexample to this stronger form is provided by <math>p = 17, q = 3313</math>. This is the only counterexample for <math>p,q < 400000</math>. | |||
==Related facts== | ==Related facts and conjectures== | ||
* [[Groupprops:Odd-order implies solvable|Feit-Thompson theorem]]: The Feit-Thompson theorem states that any [[groupprops:odd-order group of odd order]] is [[groupprops:solvable group]]. The proof of this theorem would be considerably simplified if the Feit-Thompson conjecture were true. | * [[Groupprops:Odd-order implies solvable|Feit-Thompson theorem]]: The Feit-Thompson theorem states that any [[groupprops:odd-order group of odd order]] is [[groupprops:solvable group]]. The proof of this theorem would be considerably simplified if the Feit-Thompson conjecture were true. | ||
* [[Goormaghtigh conjecture]]: This conjecture states that the equation: | |||
<math>\frac{x^m - 1}{x - 1} = \frac{y^n - 1}{y - 1}</math> | |||
has only two solution pairs. | |||
==External links== | |||
===Other subject wikis=== | |||
* [[Groupprops:Feit-Thompson conjecture]] | |||
Latest revision as of 13:25, 28 April 2009
Statement
For any two distinct primes , the number does not divide are relatively prime.
A stronger form of the conjecture is that the numbers and are relatively prime, and a counterexample to this stronger form is provided by . This is the only counterexample for .
Related facts and conjectures
- Feit-Thompson theorem: The Feit-Thompson theorem states that any groupprops:odd-order group of odd order is groupprops:solvable group. The proof of this theorem would be considerably simplified if the Feit-Thompson conjecture were true.
- Goormaghtigh conjecture: This conjecture states that the equation:
has only two solution pairs.