Flatters, Anthony; Ward, Thomas A polynomial Zsigmondy theorem. (English) Zbl 1257.11028 J. Algebra 343, No. 1, 138-142 (2011). Let \(k\) be a field, and let \((F_n)_{n\geq 1}\) be a sequence of elements of \(k[T]\). Note that \(k[T]\) is a unique factorization domain. An irreducible factor that divides a term in the sequence but does not divide an earlier term is called a primitive prime divisor. In particular, the authors consider the sequence \(F_n=f^n-g^n\), where \(f\) and \(g\) are nonzero coprime elements of \(k[T]\) that are not both units.The main result of this paper is the following. If \(\text{char}(k)=0,\) then each term of \((F_n)_{n\geq 1}\) beyond the second term has a primitive prime divisor. If \(\text{char}(k)=p>0,\) let \(F'\) be the sequence obtained from \((F_n)_{n\geq 1}\) by deleting the terms with \(p\mid n.\) Then each term of \(F'\) beyond the second term has a primitive prime divisor. This is an analogue of work of A. S. Bang [Zeuthen Tidskr. (5) 4, 70–80, 130–137 (1886; JFM 19.0168.02)] and K. Zsigmondy [“Zur Theorie der Potenzreste”, Monatsh. Math. 3, 265–284 (1892; JFM 24.0167.02)] on the sequence \((a^n-b^n)_{n\geq 1},\) where \(a>b>0\) are coprime integers. (In that case, every term beyond the sixth has a primitive prime divisor.) Reviewer: Elizabeth Sell (Millersville) Cited in 1 ReviewCited in 4 Documents MSC: 11C08 Polynomials in number theory 11B83 Special sequences and polynomials Keywords:Zsigmondy theorem; polynomial ring; primitive divisor Citations:JFM 19.0168.02; JFM 24.0167.02 PDFBibTeX XMLCite \textit{A. Flatters} and \textit{T. Ward}, J. Algebra 343, No. 1, 138--142 (2011; Zbl 1257.11028) Full Text: DOI arXiv References: [1] Bang, A. S., Taltheoretiske undersølgelser, Tidskrifft Math., 5, 70-80 (1886), 130-137 [2] Everest, G.; van der Poorten, A.; Shparlinski, I.; Ward, T., Recurrence Sequences, Math. Surveys Monogr., vol. 104 (2003), American Mathematical Society: American Mathematical Society Providence, RI · Zbl 1033.11006 [3] Ingram, P.; Mahé, V.; Silverman, J. H.; Stange, K.; Streng, M., Algebraic divisibility sequences over function fields (2011) [4] Silverman, J. H., The Arithmetic of Dynamical Systems, Grad. Texts in Math., vol. 241 (2007), Springer: Springer New York · Zbl 1130.37001 [5] Washington, L. C., Introduction to Cyclotomic Fields, Grad. Texts in Math., vol. 83 (1982), Springer-Verlag: Springer-Verlag New York · Zbl 0484.12001 [6] Zsigmondy, K., Zur Theorie der Potenzreste, Monatsh. Math., 3, 265-284 (1892) · JFM 24.0176.02 This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.