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Feit–Thompson conjecture
In
mathematics , the Feit–Thompson conjecture is a
conjecture in
number theory , suggested by
Walter Feit and
John G. Thompson (
1962 ). The conjecture states that there are no distinct
prime numbers p and q such that
p
q
−
1
p
−
1
{\displaystyle {\frac {p^{q}-1}{p-1}}}
divides
q
p
−
1
q
−
1
{\displaystyle {\frac {q^{p}-1}{q-1}}}
.
If the conjecture were true, it would greatly simplify the final chapter of the
proof (
Feit & Thompson 1963 ) of the
Feit–Thompson theorem that every
finite group of odd
order is
solvable . A stronger conjecture that the two numbers are always
coprime was disproved by
Stephens (1971) with the
counterexample p = 17 and q = 3313 with common
factor 2pq + 1 = 112643.
It is known that the conjecture is true for q = 2 (
Stephens 1971 ) and q = 3 (
Le 2012 ).
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.
See also
References
Feit, Walter; Thompson, John G. (1962), "A solvability criterion for finite groups and some consequences", Proc. Natl. Acad. Sci. U.S.A. , 48 (6): 968–970,
Bibcode :
1962PNAS...48..968F ,
doi :
10.1073/pnas.48.6.968 ,
JSTOR
71265 ,
PMC
220889 ,
PMID
16590960
MR
0143802
Feit, Walter; Thompson, John G. (1963),
"Solvability of groups of odd order" (PDF) , Pacific J. Math. , 13 : 775–1029,
doi :
10.2140/pjm.1963.13.775 ,
ISSN
0030-8730 ,
MR
0166261
Le, Mao Hua (2012), "A divisibility problem concerning group theory", Pure Appl. Math. Q. , 8 (3): 689–691,
doi :
10.4310/PAMQ.2012.v8.n3.a5 ,
ISSN
1558-8599 ,
MR
2900154
Stephens, Nelson M. (1971), "On the Feit–Thompson conjecture", Math. Comp. , 25 (115): 625,
doi :
10.2307/2005226 ,
JSTOR
2005226 ,
MR
0297686
External links