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prime factors of f(a),f(a+1),...,f(a+n-1), ≥k in count

Source: Bulgaria 1985 P1

June 16, 2021
number theoryPolynomialsalgebra

Problem Statement

Let f(x)f(x) be a non-constant polynomial with integer coefficients and n,kn,k be natural numbers. Show that there exist nn consecutive natural numbers a,a+1,…,a+n−1a,a+1,\ldots,a+n-1 such that the numbers f(a),f(a+1),…,f(a+n−1)f(a),f(a+1),\ldots,f(a+n-1) all have at least kk prime factors. (We say that the number p1α1⋯psαsp_1^{\alpha_1}\cdots p_s^{\alpha_s} has α1+…+αs\alpha_1+\ldots+\alpha_s prime factors.)
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