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Cover......Page 1
Advisory Board......Page 2
Andrzej Schinzel......Page 3
Title......Page 4
ISBN 978-3-03719-038-8......Page 5
Preface......Page 6
Contents......Page 10
Part A: Diophantine equations and integral forms......Page 16
Commentary on A: Diophantine equations and integral forms......Page 18
Sur les nombres de Mersennequi sont triangulaires......Page 26
Sur quelques propriétés des nombres 3/n et 4/n,où n est un nombre impair......Page 28
Sur l’existence d’un cercle passant par un nombre donnéde points aux coordonnées entières......Page 32
Sur les sommes de trois carrés......Page 33
On the Diophantine equation......Page 37
Polynomials of certain special types......Page 42
An improvement of Runge’s theoremon Diophantine equations......Page 51
On the equation y\'\'\' = P(x)......Page 56
Zeta functionsand the equivalence of integral forms......Page 62
Quadratic Diophantine equations with parameters......Page 69
Selmer’s conjecture and families of elliptic curves......Page 77
Families of curves having each an integer point......Page 82
Hasse’s principlefor systems of ternary quadratic formsand for one biquadratic form......Page 102
On Runge’s theoremabout Diophantine equations......Page 108
On sums of three unit fractionswith polynomial denominators......Page 131
On equations y^2 = x\'\' + k in a finite field......Page 139
Part B: Continued fractions......Page 142
Commentary on B: Continued fractions......Page 144
On some problems of the arithmetical theoryof continued fractions......Page 146
On some problems of the arithmetical theoryof continued fractions II......Page 164
On two conjectures of P. Chowla and S. Chowlaconcerning continued fractions......Page 176
Part C: Algebraic number theory......Page 182
Commentary on C: Algebraic numbers......Page 184
A refinement of two theorems of Kronecker......Page 190
On a theorem of Bauerand some of its applications......Page 194
An extension of the theorem of Bauerand polynomials of certain special types......Page 205
On sums of roots of unity(Solution of two problems of R. M. Robinson)......Page 212
On a theorem of Bauerand some of its applications II......Page 225
On the product of the conjugatesoutside the unit circle of an algebraic number......Page 236
On linear dependence of roots......Page 253
1. Introduction......Page 268
3. Real quadratic fields......Page 269
4. Imaginary quadratic fields......Page 276
A class of algebraic numbers......Page 279
On values of the Mahler measurein a quadratic field(solution of a problem of Dixon and Dubickas)......Page 287
Part D: Polynomials in one variable......Page 296
Commentary on D: Polynomials in one variable......Page 298
Solution d’un problème de K. Zarankiewiczsur les suites de puissances consécutivesde nombres irrationnels......Page 310
On the reducibility of polynomialsand in particular of trinomials......Page 316
Reducibility of polynomialsand covering systems of congruences......Page 348
Reducibility of lacunary polynomials I......Page 359
Reducibility of lacunary polynomials II......Page 396
A note on the paper“Reducibility of lacunary polynomials I”......Page 418
Reducibility of lacunary polynomials III......Page 424
Reducibility of lacunary polynomials IV......Page 462
On the number of terms of a power of a polynomial......Page 465
Introduction and the statement of results......Page 481
1. Auxiliary results from the theory of algebraic functions......Page 490
2. Determination of the range of Tables 1 and 2 (Lemmas 3–27)......Page 491
3. Determination of the content of Table 1 (Lemmas 28 to 40)......Page 515
4. Determination of the content of Table 2 (Lemmas 41 to 48)......Page 523
5. Proof of Theorems 1, 2 and 3......Page 533
6. Proof of Theorems 4 and 5......Page 536
7. Proof of Theorem 6 and of the subsequent remarks......Page 540
8. Deduction of Consequences 1–3 from Conjecture......Page 544
9. Proof of Theorems 7 and 8......Page 545
10. Proof of Theorem 9 and of Corollary 1......Page 547
11. Proof of Theorem 10 and of Corollary 2......Page 554
1. Introduction......Page 564
2. Proofs......Page 566
Reducibility of lacunary polynomials XII......Page 578
1. Introduction......Page 595
2. 16 lemmas to Theorems 1–2......Page 596
3. Proof of Theorems 1 and 2......Page 612
4. Two lemmas to Theorem 3......Page 613
5. Proof of Theorem 3......Page 616
6. Addendum to the paper [5](2)......Page 619
On reducible trinomials III......Page 620
On the greatest common divisorof two univariate polynomials I......Page 647
On the greatest common divisorof two univariate polynomials II......Page 661
On the reduced length of a polynomialwith real coefficients......Page 673
Part E: Polynomials in several variables......Page 708
Commentary on E: Polynomials in several variables......Page 710
Some unsolved problems on polynomials......Page 718
Reducibility of polynomials in several variables......Page 724
Reducibility of polynomials of the form f (x) − g(y)......Page 730
Reducibility of quadrinomials......Page 735
A general irreducibility criterion......Page 754
1. Statement of results......Page 762
2. Proof of Theorem 1......Page 764
4. Proof of Theorem 3......Page 767
6. Proof of Theorem 5......Page 768
On difference polynomialsand hereditarily irreducible polynomials......Page 770
On a decomposition of polynomialsin several variables......Page 775
Introduction......Page 794
1. Lemmas on PGL2(K)......Page 796
2. Determination of all binary formswith a given group of weak automorphs......Page 804
3. Upper bounds for |Aut(f,K)|......Page 821
5. The case of an algebraically closed field......Page 836
Reducibility of symmetric polynomials......Page 843
Part F: Hilbert’s Irreducibility Theorem......Page 850
Commentary on F:Hilbert’s Irreducibility Theorem......Page 852
On Hilbert’s Irreducibility Theorem......Page 854
A class of polynomials......Page 861
The least admissible value of the parameterin Hilbert’s Irreducibility Theorem......Page 864
Part G: Arithmetic functions......Page 884
Commentary on G: Arithmetic functions......Page 886
On functions ϕ(n) and σ(n)......Page 891
Sur l’équation ϕ(x) = m......Page 896
Sur un problème concernant la fonction ϕ(n)......Page 900
Distributions of the valuesof some arithmetical functions......Page 902
On the functions ϕ(n) and σ(n)......Page 915
On integers not of the form n − ϕ(n)......Page 920
Part H: Divisibility and congruences......Page 924
Commentary on H: Divisibility and congruences......Page 926
Sur un problème de P. Erdos......Page 928
On the congruence ax ≡ b (modp)......Page 934
On the composite integers of the form c(ak + b)! ± 1......Page 937
On power residues and exponential congruences......Page 940
Abelian binomials,power residues and exponential congruences......Page 964
An extension ofWilson’s theorem......Page 996
Systems of exponential congruences......Page 1000
On a problem in elementary number theory......Page 1012
On exponential congruences......Page 1021
I. Introduction......Page 1026
II. Résultats......Page 1027
III. Lemmes préliminaires......Page 1028
IV. Démonstrations des résultats......Page 1031
V. Exemples de polynômes......Page 1035
On power residues......Page 1037
Part I: Primitive divisors......Page 1056
Commentary on I: Primitive divisors......Page 1058
On primitive prime factors of an − bn......Page 1061
On primitive prime factors of Lehmer numbers I......Page 1071
On primitive prime factors of Lehmer numbers II......Page 1084
On primitive prime factors of Lehmer numbers III......Page 1091
Primitive divisors of the expression An − Bnin algebraic number fields......Page 1115
An extension of the theorem on primitive divisorsin algebraic number fields......Page 1123
Part J: Prime numbers......Page 1128
Commentary on J: Prime numbers......Page 1130
Sur certaines hypothèsesconcernant les nombres premiers......Page 1138
Remarks on the paper“Sur certaines hypothèsesconcernant les nombres premiers”......Page 1159
A remark on a paper of Bateman and Horn......Page 1167
5. The greatest prime factorof a quadratic or cubic polynomial......Page 1170
On the relation between two conjectureson polynomials......Page 1179
Part K: Analytic number theory......Page 1218
Commentary on K: Analytic number theory......Page 1220
On Siegel’s zero......Page 1224
Multiplicative properties of the partition function......Page 1236
On an analytic problemconsidered by Sierpi ´ nski and Ramanujan......Page 1242
1. Introduction......Page 1249
2. Dirichlet characters......Page 1251
3. Character sums in terms of Bernoulli numbers......Page 1259
4. Results......Page 1261
5. Tables......Page 1264
Part L: Geometry of numbers......Page 1270
Commentary on L: Geometry of numbers......Page 1272
A decomposition of integer vectors II......Page 1274
A decomposition of integer vectors IV......Page 1284
A property of polynomialswith an application to Siegel’s lemma......Page 1299
On vectors whose span containsa given linear subspace......Page 1313
Part M: Other papers......Page 1328
Commentary on M: Other papers......Page 1330
The influence of the Davenport–Schinzel paperin discrete and computational geometry......Page 1336
Sur l’équation fonctionnellef [x + y · f (x)] = f (x) · f (y)......Page 1339
A combinatorial problemconnected with differential equations......Page 1352
An analogue of Harnack’s inequalityfor discrete superharmonic functions......Page 1363
An inequality for determinants with real entries......Page 1372
1. Introduction......Page 1375
2. Assertions (i), (ii) of Theorem 1......Page 1376
3. Assertion (iii) of Theorem 1......Page 1378
5. The upper bound B  7/4......Page 1380
7. The upper bound 1.7373......Page 1384
Unsolved problems and unproved conjectures......Page 1390
Unsolved problems and unproved conjecturesproposed by Andrzej Schinzelin the years 1956–2006arranged chronologically......Page 1392
Publication list of Andrzej Schinzel......Page 1400
A. Research papers......Page 1402
B.2. Other papers......Page 1415
C. Books......Page 1418
Back Cover......Page 1419