MathDB

b_1 = a_0, b_{n+1} = P (b_n) , P (x) = a_dx^d + ... + a_2x^2 + a_0

Source: 2021 Saudi Arabia Training Lists p34 https://artofproblemsolving.com/community/c2758131_2021_saudi_arabia_training_tests

January 5, 2022

polynomialnumber theorydivides

Problem Statement

Let coefficients of the polynomial

P (x) = a_dx^d + ... + a_2x^2 + a_0

where

d \ge 2

, are positive integers. The sequences

(b_n)

is defined by

b_1 = a_0

and

b_{n+1} = P (b_n)

for

n \ge 1

. Prove that for any

n \ge 2

, there exists a prime number

p

such that

p|b_n

but it does not divide

b_1, b_2, ..., b_{n-1}