**Number Theory**

The number-theoretic theorem of Abouabdillah is about antichains in the partially ordered set *E _{N}* consisting of the positive integers in the interval, partially ordered by divisibility. With this partial order, an antichain is a set of integers within this interval, such that no member of this set is a divisor of any other member. It possible to prove using ideas related to Dilworth's theorem that the maximum number of elements in an antichain of

*E*

_{2n}is exactly

*n*: there exists an antichain of this size consisting of all the numbers in the subinterval, so the maximum size of an antichain is at least

*n*. However, there are only

*n*odd numbers within the interval, for each odd number

*c*in this interval at most one number of the form 2

*k*

*c*may belong to any antichain, and every number in the interval has this form for some

*c*, so the maximum size of an antichain is also at most

*n*.

Abouabdillah's theorem characterizes more precisely the numbers that may belong to an antichain of maximum size in *E*_{2n}. Specifically, if *x* is any integer in the interval, decompose *x* as the product of a power of two and an odd number: *x* = 2*k**c*, where *c* is odd. Then, according to Abouabdillah's theorem, there exists an antichain of cardinality *n* in *E*_{2n} that contains *x* if and only if 2*n* < 3*k* + 1*c*.

The smallest value in any maximum antichain of *E*_{2n} is at least 2*k*, where 3*k* + 1 is the first power of three that is greater than 2*n*, as had been posed as a problem by Paul Erdős (1937) and solved by Emma Lehmer (1939). Lehmer's solution immediately implies the special case of Abouabdillah's theorem for *c* = 1. Abouabdillah's theorem generalizes this solution to all values within the given interval.

Read more about this topic: Abouabdillah's Theorem

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