Let f(x)∈Z[x]; for each integer α it is interesting to
consider the number of iterates nα​, if possible, needed to satisfy
fnα​(α)=α. The sets {α,f(α),…,fnα​−1(α),α} generated by the iterates of f are
called cycles. For Z[x] it is known that cycles of length 1 and 2
occur, and no others. While much is known for extensions to number fields, we
concentrate on extending Z by adjoining reciprocals of primes. Let
Z[1/p1​,…,1/pn​] denote Z extended by adding in
the reciprocals of the n primes p1​,…,pn​ and all their products and
powers with each other and the elements of Z.
Interestingly, cycles of length 4, called 4-cycles, emerge for polynomials in
Z[1/p1​,…,1/pn​][x] under the appropriate
conditions. The problem of finding criteria under which 4-cycles emerge is
equivalent to determining how often a sum of four terms is zero, where the
terms are ±1 times a product of elements from the list of n primes. We
investigate conditions on sets of primes under which 4-cycles emerge. We
characterize when 4-cycles emerge if the set has one or two primes, and
(assuming a generalization of the ABC conjecture) find conditions on sets of
primes guaranteed not to cause 4-cycles to emerge.Comment: 14 pages, 1 figur