Integers (eBook)

PREFACE TO VOLUME 13 OF INTEGERS 5
CONTENTS 7
SIMPLE ARITHMETICAL CRITERIA FOR IRREDUCIBILITY OF POLYNOMIALS WITH INTEGER COEFFICIENTS 13
A SHORT PROOF OF A RESULT OF GICA AND LUCA 34
COUNTING HERON TRIANGLES WITH CONSTRAINTS 36
RAMSEY TYPE RESULTS ON THE SOLVABILITY OF CERTAIN EQUATIONS IN ZM 53
A COMBINATORIAL PROOF OF S-ADICITY FOR SEQUENCES WITH LINEAR COMPLEXITY 62
ON INTERPOLATING POWER SERIES 81
CAPTURED-REVERSIBLE MOVES AND STAR DECOMPOSITION DOMINATION IN HEX 87
ON BOUNDS FOR TWO DAVENPORT-TYPE CONSTANTS 101
ON THE DIOPHANTINE EQUATION x² - kxy + y² + lx = 0 104
ON A SEQUENCE OF POLYNOMIALS WITH HYPOTHETICALLY INTEGER COEFFICIENTS 112
ON DIVISIBILITY PROPERTIES OF SOME DIFFERENCES OF THE CENTRAL BINOMIAL COEFFICIENTS AND CATALAN NUMBERS 141
GENERALIZED BINOMIAL EXPANSIONS AND BERNOULLI POLYNOMIALS 161
THE DISTRIBUTION OF SOLUTIONS TO xy = n (mod a) WITH AN APPLICATION TO FACTORING INTEGERS 174
DISTRIBUTION LAWS OF PAIRS OF DIVISORS 194
SHARP THRESHOLD ASYMPTOTICS FOR THE EMERGENCE OF ADDITIVE BASES 207
ON AN ANALOGUE TO THE LUCAS-LEHMER-RIESEL TEST USING ELLIPTIC CURVES 224
RESTRICTED INTEGER PARTITION FUNCTIONS 240
BRUN MEETS SELMER 249
THICK SUBSETS THAT DO NOT CONTAIN ARITHMETIC PROGRESSIONS 261
COFINITE INDUCED SUBGRAPHS OF IMPARTIAL COMBINATORIAL GAMES: AN ANALYSIS OF CIS-NIM 271
PIRATES AND TREASURE 298
TAKING THE CONVOLUTED OUT OF BERNOULLI CONVOLUTIONS: A DISCRETE APPROACH 316
SOME RESULTS ON BALANCING, COBALANCING, (a, b)-TYPE BALANCING, AND (a, b)-TYPE COBALANCING NUMBERS 328
VALUATIONS AND COMBINATORICS OF TRUNCATED EXPONENTIAL SUMS 342
STRICT SCHUR NUMBERS 358
ON q-ANALOG OF WOLSTENHOLME TYPE CONGRUENCES FOR MULTIPLE HARMONIC SUMS 370
ON THE FOURIER TRANSFORM OF THE GREATEST COMMON DIVISOR 381
WEIGHTED LONESUM MATRICES AND THEIR GENERATING FUNCTION 397
ON MULTIPLICATIVE SIDON SETS 404
SOME WEIGHTED SUMS OF PRODUCTS OF LUCAS SEQUENCES 414
ADDITIVE ENERGY AND THE FALCONER DISTANCE PROBLEM IN FINITE FIELDS 423
A COMBINATORIAL INTERPRETATION OF THE CATALAN AND BELL NUMBER DIFFERENCE TABLES 433
PISOT NUMBERS AND CHROMATIC ZEROS 452
COORDINATE SUM AND DIFFERENCE SETS OF d-DIMENSIONAL MODULAR HYPERBOLAS 459
AN INFINITE FAMILY OF RECURSIVE FORMULAS GENERATING POWER MOMENTS OF KLOOSTERMAN SUMS WITH TRACE ONE ARGUMENTS: O(2n + 1, 2r) CASE 475
THE DIOPHANTINE EQUATION Fyn + Fxn+1 = Fxm 491
A NOTE ON THE MULTIPLICATIVE STRUCTURE OF AN ADDITIVELY SHIFTED PRODUCT SET AA + 1 508
REPRESENTATIONS OF SQUARES BY CERTAIN SEPTENARY QUADRATIC FORMS 516
CATALAN NUMBERS MODULO A PRIME POWER 534
GONI: PRIMES REPRESENTED BY BINARY QUADRATIC FORMS 538
A COMBINATORIAL PROOF OF TWO EQUIVALENT IDENTITIES BY FREE 2-MOTZKIN PATHS 556
IMPROVING THE CHEN AND CHEN RESULT FOR ODD PERFECT NUMBERS 564
FINITE SUMS THAT INVOLVE RECIPROCALS OF PRODUCTS OF GENERALIZED FIBONACCI NUMBERS 572
COMPOSITIONS CONSTRAINED BY GRAPH LAPLACIAN MINORS 583
EDGE GROWTH IN GRAPH SQUARES 605
NEW BOUNDS AND COMPUTATIONS ON PRIME-INDEXED PRIMES 625
GAP DISTRIBUTION OF FAREY FRACTIONS UNDER SOME DIVISIBILITY CONSTRAINTS 646
ONE FINITE SUMS AND INTEGRAL REPRESENTATIONS 661
FINE-WILF GRAPHS AND THE GENERALIZED FINE-WILF THEOREM 673
ON THE LEAST SIGNIFICANT 2-ADIC AND TERNARY DIGITS OF CERTAIN STIRLING NUMBERS 691
ON THE DENSITY OF HAPPY NUMBERS 701
A REMARK ON RELATIVELY PRIME SETS 726
INTERSECTING RATIONAL BEATTY SEQUENCES 740
NEW PROOFS FOR THE p, q-ANALOGUE OF CHU-VANDERMONDE’S IDENTITY 753
REMARKS ON TINY ZERO-SUM SEQUENCES 764
A NOTE ON IRREGULARITIES OF DISTRIBUTION 772
GENERALIZING DELANNOY NUMBERS VIA COUNTING WEIGHTED LATTICE PATHS 776
ON THE LEAST PRIMITIVE ROOT EXPRESSIBLE AS A SUM OF TWO SQUARES 809
STABILIZATION OF COEFFICIENTS FOR PARTITION POLYNOMIALS 816
ON SETS WITH MORE RESTRICTED SUMS THAN DIFFERENCES 831
ON AN INCOMPLETE ARGUMENT OF ERDO 
855 
GENERALIZATION OF UNIVERSAL PARTITION AND BIPARTITION THEOREMS 861
SOME WEIGHTED SUMS OF POWERS OF FIBONACCI POLYNOMIALS 872
CONGRUENCES OF CONCAVE COMPOSITION FUNCTIONS 891
DISJUNCTIVE RADO NUMBERS FOR ax1 + x2 = x3 AND bx1 + x2 = x3 
898 
HOW TO WRITE A PERMUTATION AS A PRODUCT OF INVOLUTIONS (AND WHY YOU MIGHT CARE) 909
ON THE HIGHER-DIMENSIONAL GENERALIZATION OF A PROBLEM OF ROTH 929
INDEPENDENT DIVISIBILITY PAIRS ON THE SET OF INTEGERS FROM 1 TO N 936
CYCLES IN REPEATED EXPONENTIATION MODULO pn 949
COMPUTATION OF AN IMPROVED LOWER BOUND TO GIUGA’S PRIMALITY CONJECTURE 956
CONVOLUTION SUMS INVOLVING LEGENDRE-JACOBI SYMBOL AND DIVISOR FUNCTIONS 970
OPTIMAL PRIMITIVE SETS WITH RESTRICTED PRIMES 984
A ZERO-SUM THEOREM OVER Z 994
ON THE DIVISIBILITY OF an ± bn 
1005 
LOWER BOUNDS FOR SUMSETS OF MULTISETS IN Z2p 
1011 
ON SEQUENCES WITHOUT GEOMETRIC PROGRESSIONS 1028
INCLUSION-EXCLUSION POLYNOMIALS WITH LARGE COEFFICIENTS 1033
AN EXPLICIT FORMULA FOR HIGHER ORDER BERNOULLI POLYNOMIALS OF THE SECOND KIND 1036
A NOTE ON A CONJECTURE OF ERDO 
1043 
SQUARES AND DIFFERENCE SETS IN FINITE FIELDS 1049
A FAMILY OF FIBONACCI-LIKE CONDITIONAL SEQUENCES 1054
ON THE RESIDUE CLASSES OF p 
1068 
S-DIOPHANTINE QUADRUPLES WITH TWO PRIMES CONGRUENT TO 3 MODULO 4 1078
THE AVERAGE LARGEST PRIME FACTOR 1087