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No cubic integer polynomial generates a Sidon sequence

In this paper we show that for no integer n0 and no polynomial f with integer coefficients and degree at most 3 the sequence of values {f(n):n=n0,n0+1,⋯} can be a Sidon sequence. This settles a corresponding conjecture of Ruzsa. To prove this result for each f(x)=ax3+bx2+cx+d∈Z[x] we construct infinitely many solutions of the Diophantine equation f(m)+f(n)=f(r)+f(s) in pairwise distinct positive integers m,n,r,s .