GENERALIZED BINOMIAL EXPANSIONS AND BERNOULLI POLYNOMIALS

Kapitel in diesem Buch

1. Frontmatter I
2. PREFACE TO VOLUME 13 OF INTEGERS V
3. CONTENTS VII
4. SIMPLE ARITHMETICAL CRITERIA FOR IRREDUCIBILITY OF POLYNOMIALS WITH INTEGER COEFFICIENTS 1
5. A SHORT PROOF OF A RESULT OF GICA AND LUCA 22
6. COUNTING HERON TRIANGLES WITH CONSTRAINTS 24
7. RAMSEY TYPE RESULTS ON THE SOLVABILITY OF CERTAIN EQUATIONS IN Z_M 41
8. A COMBINATORIAL PROOF OF S-ADICITY FOR SEQUENCES WITH LINEAR COMPLEXITY 50
9. ON INTERPOLATING POWER SERIES 69
10. CAPTURED-REVERSIBLE MOVES AND STAR DECOMPOSITION DOMINATION IN HEX 75
11. ON BOUNDS FOR TWO DAVENPORT-TYPE CONSTANTS 89
12. ON THE DIOPHANTINE EQUATION x² − kxy + y² + lx = 0 92
13. ON A SEQUENCE OF POLYNOMIALS WITH HYPOTHETICALLY INTEGER COEFFICIENTS 100
14. ON DIVISIBILITY PROPERTIES OF SOME DIFFERENCES OF THE CENTRAL BINOMIAL COEFFICIENTS AND CATALAN NUMBERS 129
15. GENERALIZED BINOMIAL EXPANSIONS AND BERNOULLI POLYNOMIALS 149
16. THE DISTRIBUTION OF SOLUTIONS TO xy = n (mod a) WITH AN APPLICATION TO FACTORING INTEGERS 162
17. DISTRIBUTION LAWS OF PAIRS OF DIVISORS 182
18. SHARP THRESHOLD ASYMPTOTICS FOR THE EMERGENCE OF ADDITIVE BASES 195
19. ON AN ANALOGUE TO THE LUCAS-LEHMER-RIESEL TEST USING ELLIPTIC CURVES 212
20. RESTRICTED INTEGER PARTITION FUNCTIONS 228
21. BRUN MEETS SELMER 237
22. THICK SUBSETS THAT DO NOT CONTAIN ARITHMETIC PROGRESSIONS 249
23. COFINITE INDUCED SUBGRAPHS OF IMPARTIAL COMBINATORIAL GAMES: AN ANALYSIS OF CIS-NIM 259
24. PIRATES AND TREASURE 286
25. TAKING THE CONVOLUTED OUT OF BERNOULLI CONVOLUTIONS: A DISCRETE APPROACH 304
26. SOME RESULTS ON BALANCING, COBALANCING, (a, b)-TYPE BALANCING, AND (a, b)-TYPE COBALANCING NUMBERS 316
27. VALUATIONS AND COMBINATORICS OF TRUNCATED EXPONENTIAL SUMS 330
28. STRICT SCHUR NUMBERS 346
29. ON q-ANALOG OF WOLSTENHOLME TYPE CONGRUENCES FOR MULTIPLE HARMONIC SUMS 358
30. ON THE FOURIER TRANSFORM OF THE GREATEST COMMON DIVISOR 369
31. WEIGHTED LONESUM MATRICES AND THEIR GENERATING FUNCTION 385
32. ON MULTIPLICATIVE SIDON SETS 392
33. SOME WEIGHTED SUMS OF PRODUCTS OF LUCAS SEQUENCES 402
34. ADDITIVE ENERGY AND THE FALCONER DISTANCE PROBLEM IN FINITE FIELDS 411
35. A COMBINATORIAL INTERPRETATION OF THE CATALAN AND BELL NUMBER DIFFERENCE TABLES 421
36. PISOT NUMBERS AND CHROMATIC ZEROS 440
37. COORDINATE SUM AND DIFFERENCE SETS OF d-DIMENSIONAL MODULAR HYPERBOLAS 447
38. AN INFINITE FAMILY OF RECURSIVE FORMULAS GENERATING POWER MOMENTS OF KLOOSTERMAN SUMS WITH TRACE ONE ARGUMENTS: O(2n + 1, 2r) CASE 463
39. THE DIOPHANTINE EQUATION F^y_n + F^x_n+1 = F^x_m 479
40. A NOTE ON THE MULTIPLICATIVE STRUCTURE OF AN ADDITIVELY SHIFTED PRODUCT SET AA + 1 496
41. REPRESENTATIONS OF SQUARES BY CERTAIN SEPTENARY QUADRATIC FORMS 504
42. CATALAN NUMBERS MODULO A PRIME POWER 522
43. GONI: PRIMES REPRESENTED BY BINARY QUADRATIC FORMS 526
44. A COMBINATORIAL PROOF OF TWO EQUIVALENT IDENTITIES BY FREE 2-MOTZKIN PATHS 544
45. IMPROVING THE CHEN AND CHEN RESULT FOR ODD PERFECT NUMBERS 552
46. FINITE SUMS THAT INVOLVE RECIPROCALS OF PRODUCTS OF GENERALIZED FIBONACCI NUMBERS 560
47. COMPOSITIONS CONSTRAINED BY GRAPH LAPLACIAN MINORS 571
48. EDGE GROWTH IN GRAPH SQUARES 593
49. NEW BOUNDS AND COMPUTATIONS ON PRIME-INDEXED PRIMES 613
50. GAP DISTRIBUTION OF FAREY FRACTIONS UNDER SOME DIVISIBILITY CONSTRAINTS 634
51. ONE FINITE SUMS AND INTEGRAL REPRESENTATIONS 649
52. FINE-WILF GRAPHS AND THE GENERALIZED FINE-WILF THEOREM 661
53. ON THE LEAST SIGNIFICANT 2-ADIC AND TERNARY DIGITS OF CERTAIN STIRLING NUMBERS 679
54. ON THE DENSITY OF HAPPY NUMBERS 689
55. A REMARK ON RELATIVELY PRIME SETS 714
56. INTERSECTING RATIONAL BEATTY SEQUENCES 728
57. NEW PROOFS FOR THE p, q-ANALOGUE OF CHU-VANDERMONDE’S IDENTITY 741
58. REMARKS ON TINY ZERO-SUM SEQUENCES 752
59. A NOTE ON IRREGULARITIES OF DISTRIBUTION 760
60. GENERALIZING DELANNOY NUMBERS VIA COUNTING WEIGHTED LATTICE PATHS 764
61. ON THE LEAST PRIMITIVE ROOT EXPRESSIBLE AS A SUM OF TWO SQUARES 797
62. STABILIZATION OF COEFFICIENTS FOR PARTITION POLYNOMIALS 804
63. ON SETS WITH MORE RESTRICTED SUMS THAN DIFFERENCES 819
64. ON AN INCOMPLETE ARGUMENT OF ERDȌS ON THE IRRATIONALITY OF LAMBERT SERIES 843
65. GENERALIZATION OF UNIVERSAL PARTITION AND BIPARTITION THEOREMS 849
66. SOME WEIGHTED SUMS OF POWERS OF FIBONACCI POLYNOMIALS 860
67. CONGRUENCES OF CONCAVE COMPOSITION FUNCTIONS 879
68. DISJUNCTIVE RADO NUMBERS FOR ax₁ + x₂ = x₃ AND bx₁ + x₂ = x₃ 886
69. HOW TO WRITE A PERMUTATION AS A PRODUCT OF INVOLUTIONS (AND WHY YOU MIGHT CARE) 897
70. ON THE HIGHER-DIMENSIONAL GENERALIZATION OF A PROBLEM OF ROTH 917
71. INDEPENDENT DIVISIBILITY PAIRS ON THE SET OF INTEGERS FROM 1 TO N 924
72. CYCLES IN REPEATED EXPONENTIATION MODULO pn 937
73. COMPUTATION OF AN IMPROVED LOWER BOUND TO GIUGA’S PRIMALITY CONJECTURE 944
74. CONVOLUTION SUMS INVOLVING LEGENDRE-JACOBI SYMBOL AND DIVISOR FUNCTIONS 958
75. OPTIMAL PRIMITIVE SETS WITH RESTRICTED PRIMES 972
76. A ZERO-SUM THEOREM OVER Z 982
77. ON THE DIVISIBILITY OF aⁿ ± bⁿ BY POWERS OF n 993
78. LOWER BOUNDS FOR SUMSETS OF MULTISETS IN Z2 999
79. ON SEQUENCES WITHOUT GEOMETRIC PROGRESSIONS 1016
80. INCLUSION-EXCLUSION POLYNOMIALS WITH LARGE COEFFICIENTS 1021
81. AN EXPLICIT FORMULA FOR HIGHER ORDER BERNOULLI POLYNOMIALS OF THE SECOND KIND 1024
82. A NOTE ON A CONJECTURE OF ERDȌS, GRAHAM, AND SPENCER 1031
83. SQUARES AND DIFFERENCE SETS IN FINITE FIELDS 1037
84. A FAMILY OF FIBONACCI-LIKE CONDITIONAL SEQUENCES 1042
85. ON THE RESIDUE CLASSES OF π(n) MODULO t 1056
86. S-DIOPHANTINE QUADRUPLES WITH TWO PRIMES CONGRUENT TO 3 MODULO 4 1066
87. THE AVERAGE LARGEST PRIME FACTOR 1075