MathDB

Let

p

be a prime number and

k

be a positive integer. Let

t=\sum_{i=0}^\infty\bigg\lfloor\frac{k}{p^i}\bigg\rfloor.

a) Let

f(x)

be a polynomial of degree

k

with integer coefficients such that its leading coefficient is

1

and its constant is divisible by

p.

prove that there exists

n\in\mathbb{N}

for which

p\mid f(n),

but

p^{t+1}\nmid f(n).

b) Prove that the statement above is sharp, i.e. there exists a polynomial

g(x)

of degree

k,

integer coefficients, leading coefficient

1

and constant divisible by

p

such that if

p\mid g(n)

is true for a certain

n\in\mathbb{N},

then

p^t\mid g(n)

also holds.
Proposed by Kristóf Szabó, Budapest

A. 813