MathDB

An infinite table whose rows and columns are numbered with positive integers, is given. For a sequence of functions

f_1(x), f_2(x), \ldots

let us place the number

f_i(j)

into the cell

(i,j)

of the table (for all

i, j\in \mathbb{N}

). A sequence

f_1(x), f_2(x), \ldots

is said to be {\it nice}, if all the numbers in the table are positive integers, and each positive integer appears exactly once. Determine if there exists a nice sequence of functions

f_1(x), f_2(x), \ldots

, such that each

f_i(x)

is a polynomial of degree 101 with integer coefficients and its leading coefficient equals to 1.

Problem Statement