MathDB

3

Part of 2002 Italy TST

Problems(2)

Finitely many x such that f(x)=1 functional equation

Source: Italy TST 2002

11/9/2010
Find all functions f:R+ā†’R+f:\mathbb{R}^+\rightarrow\mathbb{R}^+ which satisfy the following conditions: (i)(\text{i}) f(x+f(y))=f(x)f(y)f(x+f(y))=f(x)f(y) for all x,y>0;x,y>0; (ii)(\text{ii}) there are at most finitely many xx with f(x)=1f(x)=1.
functionalgebra unsolvedalgebra
x divides y^2+m

Source: Italy TST 2002

9/23/2008
Prove that for any positive integer m m there exist an infinite number of pairs of integers (x,y)(x,y) such that (i)(\text{i}) xx and yy are relatively prime; (ii)(\text{ii}) xx divides y2+m;y^2+m; (iii)(\text{iii}) yy divides x2+m.x^2+m.
algebrapolynomialVietanumber theoryrelatively primenumber theory unsolved