In Note M, Burnside (1955) states, "The contrast that these results shew between groups of odd and of even order suggests inevitably that simple groups of odd order
do not exist." Of course, simple groups of prime
order do exist, namely the groups 
for any prime 
. Therefore, Burnside conjectured that every finite simple group of non-prime order must have even order.
The conjecture was proven true by Feit and Thompson (1963).
Burnside's Conjecture
See also
Abelian Group, Feit-Thompson Conjecture, Feit-Thompson Theorem, Simple GroupThis entry contributed by Nicolas Bray
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References
Burnside, W. Theory of Groups of Finite Order, 2nd ed. New York: Dover, 1955.Feit, W. and Thompson, J. G. "Solvability of Groups of Odd Order." Pacific J. Math. 13, 775-1029, 1963.Referenced on Wolfram|Alpha
Burnside's ConjectureCite this as:
Bray, Nicolas. "Burnside's Conjecture." From MathWorld--A Wolfram Web Resource, created by Eric W. Weisstein. https://mathworld.wolfram.com/BurnsidesConjecture.html