MathDB

Let

k

be an odd number that is greater than or equal to

3

. Prove that there exists a

k^{th}

-degree integer-valued polynomial with non-integer-coefficients that has the following properties: (1)

f(0)=0

and

f(1)=1

; and. (2) There exist infinitely many positive integers

n

so that if the following equation:

n= f(x_1)+\cdots+f(x_s),

has integer solutions

x_1, x_2, \dots, x_s

, then

s \geq 2^k-1

Problem Statement