Feit-Thompson conjecture

Feit-Thompson conjecture

In mathematics, the Feit-Thompson conjecture is a conjecture in number theory, suggested by harvs|txt=yes|first=Walter|last= Feit|authorlink=Walter Feit| first2=John G. |last2=Thompson|author2-link=John G. Thompson|year=1962. The conjecture states that there are no distinct prime numbers "p" and "q" for which

:frac{p^{q} - 1}{p - 1} mbox{ divides } frac{q^{p} - 1}{q - 1}.

If the conjecture were true, it would greatly simplify the final chapter of the proof harv|Feit|Thompson|1963 of the Feit-Thompson theorem that every finite group of odd order is solvable. A stronger (and rather implausible) conjecture that the two numbers are always coprime was disproved by harvtxt|Stephens|1971 with the counterexample "p" = 17 and "q" = 3313 with common factor 2"pq" + 1 = 112643.

Informal probability 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.

ee also

*Cyclotomic polynomials
*Goormaghtigh conjecture

References

* citation
last= Feit|first=Walter|last2=Thompson|first2= John G.
title = A solvability criterion for finite groups and some consequences
journal = Proc. Nat. Acad. Sci. U.S.A.
volume = 48
year = 1962
pages = 968–970
url= http://links.jstor.org/sici?sici=0027-8424%2819620615%2948%3A6%3C968%3AASCFFG%3E2.0.CO%3B2-Q MathSciNet|id=0143802
* citation
last= Feit|first=Walter|last2=Thompson|first2= John G.
title = Solvability of groups of odd order
journal = Pacific J. Math.
volume = 13
year = 1963
pages = 775–1029
issn = 0030-8730 MathSciNet|id=0166261
* citation
last = Stephens
first = Nelson M.
title = On the Feit-Thompson conjecture
journal = Math. Comp.
volume = 25
year = 1971
pages = 625
issn = 0025-5718
url=http://links.jstor.org/sici?sici=0025-5718%28197107%2925%3A115%3C625%3AOTFC%3E2.0.CO%3B2-H MathSciNet|id=0297686

External links

* (This article confuses the Feit-Thompson conjecture with the stronger disproved conjecture mentioned above.)


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