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**The work of Andrzej Schinzel in number theory.**
*(English)*
Zbl 0935.11001

Győry, Kálmán (ed.) et al., Number theory in progress. Proceedings of the international conference organized by the Stefan Banach International Mathematical Center in honor of the 60th birthday of Andrzej Schinzel, Zakopane, Poland, June 30-July 9, 1997. Volume 1: Diophantine problems and polynomials. Berlin: de Gruyter. 341-357 (1999).

This paper presents a survey of the work of Andrzej Schinzel in number theory. In this review I cannot give a complete description of the content of this survey: Andrzej Schinzel has published more than two hundred papers and the survey of W. Narkiewicz contains 13 sections. I shall content myself to enumerate the points which seem to me the most important. Schinzel worked first under the guidance of W. Sierpiński, and his first papers deal with elementary number theory, they contain studies about the Euler \(\varphi \)-function, some others deal with certain diophantine equations.

In 1958, in a joint work with Sierpiński, appeared the famous conjecture \(\mathbb{H}\) about polynomials with integer coefficients: if \(f_1,\dots,f_s\in\mathbb{Z}[X]\) have positive leading coefficients and if moreover the product \(\prod_{i=1}^s f_i\) has no fixed divisor \({}>1\), then for infinitely many \(n\) the values \(f_1(n)\), …, \(f_s(n)\) are prime numbers. From this time the consequences of this hypothesis were intensively studied (the proof seems to be presently out of reach), and they are extremely striking.

The habilitation thesis of Schinzel dealt with primitive prime divisors of certain recurrent sequences. Schinzel obtained very nice results on this problem, the first appeared in 1962. In 1977, C. L. Stewart, using Baker’s theory, could prove that every Lehmer number \(P_n(\alpha,\beta)\), with \(\alpha\) and \(\beta\) coprime, has a primitive prime divisor for \(n\geq 7\), \(n\notin\{8,10,12\}\), except for finitely many exceptions which could be (in principle) determined. It seems that the effective determination of all these exceptions has been done very recently by Bilu, Hanrot and Voutier.

Certainly, the main theme of Schinzel researches is the study of questions of irreducibility of polynomials. In 1982 he published the book Selected topics on polynomials devoted to this study. Many details are given in Narkiewicz’ survey.

Another subject for which Schinzel obtained very important results is Lehmer’s problem about what is called now Mahler’s measure.

I cannot omit the more recent work of Schinzel about diophantine equations, more precisely on Runge’s theorem and on exponential diophantine equations.

For the entire collection see [Zbl 0911.00025].

In 1958, in a joint work with Sierpiński, appeared the famous conjecture \(\mathbb{H}\) about polynomials with integer coefficients: if \(f_1,\dots,f_s\in\mathbb{Z}[X]\) have positive leading coefficients and if moreover the product \(\prod_{i=1}^s f_i\) has no fixed divisor \({}>1\), then for infinitely many \(n\) the values \(f_1(n)\), …, \(f_s(n)\) are prime numbers. From this time the consequences of this hypothesis were intensively studied (the proof seems to be presently out of reach), and they are extremely striking.

The habilitation thesis of Schinzel dealt with primitive prime divisors of certain recurrent sequences. Schinzel obtained very nice results on this problem, the first appeared in 1962. In 1977, C. L. Stewart, using Baker’s theory, could prove that every Lehmer number \(P_n(\alpha,\beta)\), with \(\alpha\) and \(\beta\) coprime, has a primitive prime divisor for \(n\geq 7\), \(n\notin\{8,10,12\}\), except for finitely many exceptions which could be (in principle) determined. It seems that the effective determination of all these exceptions has been done very recently by Bilu, Hanrot and Voutier.

Certainly, the main theme of Schinzel researches is the study of questions of irreducibility of polynomials. In 1982 he published the book Selected topics on polynomials devoted to this study. Many details are given in Narkiewicz’ survey.

Another subject for which Schinzel obtained very important results is Lehmer’s problem about what is called now Mahler’s measure.

I cannot omit the more recent work of Schinzel about diophantine equations, more precisely on Runge’s theorem and on exponential diophantine equations.

For the entire collection see [Zbl 0911.00025].

Reviewer: Maurice Mignotte (Strasbourg)

### MSC:

11-03 | History of number theory |

01A70 | Biographies, obituaries, personalia, bibliographies |

11-02 | Research exposition (monographs, survey articles) pertaining to number theory |

### Keywords:

Euler phi-function; diophantine equations; conjecture \(\mathbb{H}\); Lehmer number; irreducibility of polynomials; Lehmer problem about Mahler’s measure; Runge’s theorem; exponential diophantine equations### Biographic References:

Schinzel, A.
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\textit{W. Narkiewicz}, in: Number theory in progress. Proceedings of the international conference organized by the Stefan Banach International Mathematical Center in honor of the 60th birthday of Andrzej Schinzel, Zakopane, Poland, June 30--July 9, 1997. Volume 1: Diophantine problems and polynomials. Berlin: de Gruyter. 341--357 (1999; Zbl 0935.11001)