The Feit-Thompson conjecture asserts that, for distinct primes 
and 
,
the integer 
never divides 
.
Feit and Thompson (1962) noted that this would simplify the proof
of the Feit-Thompson theorem, which states
that every group of odd group order is a solvable
group. The conjecture remains unsolved (Hannusch
2024).
The stronger assertion that 
and 
are always coprime was
refuted by Stephens (1971). For 
and 
, their greatest
common divisor is 112643. This counterexample
does not refute the original nondivisibility conjecture.
Stephens's search considered odd primes with 
and 
,
testing possible common prime factors below 400000.
The pair above was the only one found in this restricted search (Stephens 1971).
and 
."
Math. Comput. 29, 1-6, 1975.Feit, W. and Thompson, J. G.
"A Solvability Criterion for Finite Groups and Some Consequences." Proc.
Nat. Acad. Sci. USA 48, 968-970, 1962.Feit, W. and Thompson,
J. G. "Solvability of Groups of Odd Order." Pacific J. Math. 13,
775-1029, 1963.Guy, R. K.