3

Part of 2002 Hungary-Israel Binational

Problems(2)

$r_n = p(r_{n+1} ) $, p is a polynomial

Source: 13-th Hungary-Israel Binational Mathematical Competition 2002

p(x)

be a polynomial with rational coefficients, of degree at least

2

. Suppose that a sequence

(r_{n})

of rational numbers satisﬁes

r_{n}= p(r_{n+1})

n\geq 1

. Prove that the sequence

(r_{n})

algebrapolynomialalgebra unsolved

neither of a^{p−1} - 1 & (a + 1)^{p−1}-1 is divisible p^2

Source: 13-th Hungary-Israel Binational Mathematical Competition 2002

p \geq 5

be a prime number. Prove that there exists a positive integer

a < p-1

such that neither of

a^{p-1}-1

(a+1)^{p-1}-1

is divisible by

p^{2}

number theory unsolvednumber theory