Feit-Thompson Conjecture

The Feit-Thompson conjecture asserts that there are no primes  and  for which  and  have a common factor.

Parker noticed that if this were true, it would greatly simplify the lengthy proof of the Feit-Thompson theorem that every group of odd order is solvable. (Guy 1994, p. 81). However, the counterexample (, ) with a common factor  was subsequently found by Stephens (1971), demonstrating that the conjecture is, in fact, false.

No other such pairs exist with both values less than .

REFERENCES:

Apostol, T. M. "The Resultant of the Cyclotomic Polynomials  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. Unsolved Problems in Number Theory, 2nd ed. New York: Springer-Verlag, p. 81, 1994.

Stephens, N. M. "On the Feit-Thompson Conjecture." Math. Comput. 25, 625, 1971.

Wells, D. G. The Penguin Dictionary of Curious and Interesting Numbers. London: Penguin, p. 17, 1986.