Abstract
The 3 x + 1 semigroup is the multiplicative semigroup S of positive rational numbers generated by { 2 k + 1 3 k + 2 : k ⩾ 0 } together with { 2 } . This semigroup encodes backwards iteration under the 3 x + 1 map, and the 3 x + 1 conjecture implies that it contains every positive integer. This semigroup is proved to be the set of positive rationals a b in lowest terms with b ≢ 0 ( mod 3 ) , and so contains all positive integers.
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