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No cubic integer polynomial generates a Sidon sequence

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  • Artūras Dubickas
  • Aivaras Novikas

Abstract

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.

Suggested Citation

  • Artūras Dubickas & Aivaras Novikas, 2021. "No cubic integer polynomial generates a Sidon sequence," Mathematische Nachrichten, Wiley Blackwell, vol. 294(10), pages 1859-1865, October.
    • Handle: RePEc:bla:mathna:v:294:y:2021:i:10:p:1859-1865
    DOI: 10.1002/mana.202000334
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