(Харків : Фізико-технічний інститут низьких температур ім. Б. І. Вєркіна НАН України., 2022) Gefter, S. L.; Piven’, A. L.
Let {fn}∞ n=0 be a sequence of independent identically distributed integer valued random variables which are defined on a probability space (Ω, F, P). We assume that these variables have a non-degenerate distribution. Let a and b be integers, b 6= 0, ±1, and let a be not divisible by b. For every ω ∈ Ω, we consider the implicit first-order linear nonhomogeneous difference equa tion bxn+1 + axn = fn(ω), n = 0, 1, 2, . . .. It is proved that the probability that there exists an integer solution of this implicit difference equation is equal to zero. Hence, under the random choice of integers f0, f1, f2, . . ., the implicit linear difference equation bxn+1 + axn = fn, n = 0, 1, 2, . . ., has no solutions in integers. We also prove that if a and b are co-prime integers, then the solvability set for this difference equation is an uncountable dense meagre set in the space of all sequences of integers.
