First, we prove that the Diophantine system f(z)=f(x)+f(y)=f(u)-f(v)=f(p)f(q)has infinitely many integer solutions for f(X) = X(X+ a) with nonzero integers a≡0,1,4(mod5). Second, we show that the above Diophantine system has an integer parametric solution for f(X) = X(X+ a) with nonzero integers a, if there are integers m,n,k such that {(n2-m2)(4mnk(k+a+1)+a(m2+2mn-n2))≡0(mod(m2+n2)2),(m2+2mn-n2)((m2-2mn-n2)k(k+a+1)-2amn)≡0(mod(m2+n2)2),where k≡0(mod4) when a is even, and k≡2(mod4) when a is odd. Third, we get that the Diophantine system f...
