MathDB

P(x)

and

Q(x)

are two polynomials with integer coefficients such that

P(x)|Q(x)^2+1

.
a) Prove that there exists polynomials

A(x)

and

B(x)

with rational coefficients and a rational number

c

such that

P(x)=c(A(x)^2+B(x)^2)

.
b) If

P(x)

is a monic polynomial with integer coefficients, Prove that there exists two polynomials

A(x)

and

B(x)

with integer coefficients such that

P(x)

can be written in the form of

A(x)^2+B(x)^2

.
Proposed by Mohammad Gharakhani

Problem Statement