JP Journal of Algebra, Number Theory and Applications
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Abstract: We introduce and discuss a generalized ultimate
periodicity conjecture for prime sequences in which every term is recursively defined as the
maximum element of the finite set where represents the greatest prime factor
of x, while and are fixed positive integers for A special case of the conjecture
follows from a general elementary result (making use of the existence of
arbitrarily large prime gaps) guaranteeing the ultimate periodicity of integer
sequences satisfying under certain conditions fulfilled
by the function f. The conditions are
not overly restrictive, among the functions satisfying them being the Euler’s
phi function.
Keywords and phrases: recurrences, greatest prime factor, periodicity, prime gaps.
