An equation P(x)=Q(y) is called Interesting if P and Q are polynomials with degree at least one and integer coefficients and the equations has an infinite number of answers in N.
An interesting equation P(x)=Q(y) yields in interesting equation F(x)=G(y) if there exists polynomial R(x)∈Q[x] such that F(x)≡R(P(x)) and G(x)≡R(Q(x)).
(a) Suppose that S is an infinite subset of N×N.S is an answer of interesting equation P(x)=Q(y) if each element of S is an answer of this equation. Prove that for each S there's an interesting equation P0(x)=Q0(y) such that if there exists any interesting equation that S is an answer of it, P0(x)=Q0(y) yields in that equation.
(b) Define the degree of an interesting equation P(x)=Q(y) by max{deg(P),deg(Q)}. An interesting equation is called primary if there's no other interesting equation with lower degree that yields in it.
Prove that if P(x)=Q(y) is a primary interesting equation and P and Q are monic then (deg(P),deg(Q))=1.Time allowed for this question was 2 hours. algebrapolynomialalgebra unsolved