| Preface to the First Edition |
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v | |
| Preface to the Second Edition |
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vii | |
| Glossary of Symbols |
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xiii | |
| Introduction |
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1 | (2) |
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3 | (41) |
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Prime values of quadratic functions |
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4 | (3) |
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Primes connected with factorials |
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7 | (1) |
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Mersenne primes. Repunits. Fermat numbers. Primes of shape k . 2n + 2 |
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8 | (5) |
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13 | (2) |
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Arithmetic progressions of primes |
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15 | (2) |
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Consecutive primes in A.P. |
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17 | (1) |
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18 | (1) |
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Gaps between primes. Twin primes |
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19 | (4) |
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23 | (2) |
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25 | (1) |
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Increasing and decreasing gaps |
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26 | (1) |
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Pseudoprimes. Euler pseudoprimes. Strong pseudoprimes |
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26 | (4) |
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30 | (2) |
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``Good'' primes and the prime number graph |
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32 | (1) |
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Congruent products of consecutive numbers |
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33 | (1) |
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Gaussian primes. Eisenstein-Jacobi primes |
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33 | (3) |
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36 | (5) |
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The Erdos-Selfridge classification of primes |
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41 | (1) |
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Values of n making n -- 2k prime. Odd numbers not of the form ±pa ± 2b |
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42 | (2) |
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44 | (61) |
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44 | (1) |
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Almost perfect, quasi-perfect, pseudoperfect, harmonic, weird, multiperfect and hyperperfect numbers |
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45 | (8) |
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53 | (2) |
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55 | (4) |
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Quasi-amicable or betrothed numbers |
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59 | (1) |
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60 | (2) |
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Aliquot cycles or sociable numbers |
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62 | (1) |
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Unitary aliquot sequences |
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63 | (2) |
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65 | (1) |
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66 | (1) |
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Solutions of mσ(m) = nσ(n) |
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67 | (1) |
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67 | (1) |
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Solutions of σ(n) = σ(n + 1) |
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68 | (1) |
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69 | (1) |
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Solutions of σ(q) + σ(r) = σ(q + r) |
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69 | (1) |
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70 | (3) |
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Exponential-perfect numbers |
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73 | (1) |
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Solutions of d(n) = d(n + 1) |
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73 | (2) |
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(m,n + 1) and (m + 1,n) with same set of prime factors |
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75 | (2) |
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77 | (1) |
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k . 2n + 1 composite for all n |
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77 | (2) |
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Factorial n as the product of n large factors |
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79 | (1) |
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Equal products of factorials |
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79 | (1) |
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The largest set with no member dividing two others |
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80 | (1) |
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Equal sums of geometic progressions with prime ratios |
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81 | (1) |
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Densest set with no l pairwise coprime |
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81 | (1) |
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The number of prime factors of n + k which don't divide n + i, 0 ≤ i < k |
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82 | (1) |
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Consecutive numbers with distinct prime factors |
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83 | (1) |
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Is x determined by the prime divisors of x + 1, x + 2, ..., x + k? |
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83 | (1) |
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A small set whose product is square |
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84 | (1) |
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84 | (1) |
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85 | (2) |
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Largest divisor of a binomial coefficient |
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87 | (2) |
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If there's an i such that n - i divides (nk) |
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89 | (1) |
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Products of consecutive numbers with the same prime factors |
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89 | (1) |
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90 | (2) |
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Does &phis;(n) properly divide n -- 1? |
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92 | (1) |
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Solutions of &phis;(m) = σ(n) |
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93 | (1) |
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94 | (1) |
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95 | (1) |
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Iterations of &phis; and σ |
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96 | (3) |
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Behavior of &phis;(σ(n)) and σ(&phis;(n)) |
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99 | (1) |
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Alternating sums of factorials |
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99 | (1) |
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100 | (1) |
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101 | (1) |
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The largest prime factor of n |
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101 | (1) |
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When does 2a -- 2b divide na -- nb? |
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102 | (1) |
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Products taken over primes |
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102 | (1) |
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103 | (2) |
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105 | (34) |
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105 | (2) |
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Sums of consecutive primes |
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107 | (1) |
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108 | (1) |
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109 | (1) |
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Sums determining members of a set |
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110 | (1) |
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Addition chains. Brauer chains. Hansen chains |
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111 | (2) |
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The money-changing problem |
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113 | (1) |
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Sets with distinct sums of subsets |
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114 | (1) |
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115 | (3) |
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Modular difference sets and error correcting codes |
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118 | (3) |
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Three-subsets with distinct sums |
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121 | (2) |
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The postage stamp problem |
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123 | (4) |
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The corresponding modular covering problem. Harmonious labelling of graphs |
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127 | (1) |
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128 | (1) |
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Maximal zero-sum-free sets |
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129 | (2) |
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Nonaveraging sets. Nondividing sets |
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131 | (1) |
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The minimum overlap problem |
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132 | (1) |
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133 | (2) |
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Is a weakly independent sequence the finite union of strongly independent ones? |
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135 | (1) |
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136 | (3) |
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139 | (60) |
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Sums of like powers. Euler's conjecture |
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139 | (5) |
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144 | (2) |
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146 | (4) |
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150 | (1) |
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151 | (1) |
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An elementary solution of x2 = 2y4 -- 1 |
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152 | (1) |
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Sum of consecutive powers made a power |
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153 | (1) |
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A pyramidal diophantine equation |
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154 | (1) |
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155 | (2) |
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Exponential diophantine equations |
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157 | (1) |
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158 | (8) |
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166 | (2) |
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168 | (1) |
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169 | (1) |
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Numbers whose sums in pairs make squares |
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170 | (1) |
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Triples with the same sum and same product |
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171 | (1) |
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Product of blocks of consecutive integers not a power |
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172 | (1) |
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Is there a perfect cuboid? Four squares whose sums in pairs are square. Four squares whose differences are square |
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173 | (8) |
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Rational distances from the corners of a square |
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181 | (4) |
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Six general points at rational distances |
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185 | (3) |
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Triangles with integer sides, medians and area |
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188 | (2) |
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Simplexes with rational contents |
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190 | (1) |
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191 | (2) |
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193 | (1) |
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Equations involving factorial n |
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193 | (1) |
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Fibonacci numbers of various shapes |
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194 | (1) |
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195 | (2) |
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A reciprocal diophantine equation |
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197 | (2) |
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199 | (41) |
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A thin sequence with all numbers equal to a member plus a prime |
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199 | (1) |
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Density of a sequence with l.c.m. of each pair less than x |
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200 | (1) |
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Density of integers with two comparable divisors |
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201 | (1) |
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Sequence with no member dividing the product of r others |
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201 | (1) |
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Sequence with members divisible by at least one of a given set |
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202 | (1) |
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Sequence with sums of pairs not members of a given sequence |
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203 | (1) |
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A series and a sequence involving primes |
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203 | (1) |
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Sequence with no sum of a pair a square |
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203 | (1) |
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Partitioning the integers into classes with numerous sums of pairs |
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204 | (1) |
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Theorem of van der Waerden. Szemeredi's theorem. Partitioning the integers into classes; at least one contains an A.P. |
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204 | (5) |
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Schur's problem. Partitioning integers into sum-free classes |
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209 | (2) |
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The modular version of Schur's problem |
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211 | (2) |
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Partitioning into strongly sum-free classes |
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213 | (1) |
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Rado's generalizations of van der Waerden's and Schur's problems |
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213 | (1) |
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214 | (1) |
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215 | (3) |
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218 | (1) |
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219 | (1) |
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Are the integer parts of the powers of a fraction infinitely often prime? |
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220 | (1) |
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Davenport-Schinzel sequences |
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220 | (2) |
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222 | (2) |
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Cycles and sequences containing all permutations as subsequences |
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224 | (1) |
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Covering the integers with A.P.s |
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224 | (1) |
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225 | (1) |
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225 | (1) |
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Epstein's Put-or-Take-a-Square game |
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226 | (1) |
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227 | (1) |
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228 | (1) |
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Sequence with sums and products all in one of two classes |
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229 | (1) |
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MacMahon's prime numbers of measurement |
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230 | (1) |
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Three sequences of Hofstadter |
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231 | (1) |
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B2-sequences formed by the greedy algorithm |
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232 | (1) |
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Sequences containing no monotone A.P.s |
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233 | (1) |
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234 | (1) |
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235 | (2) |
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237 | (1) |
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237 | (1) |
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238 | (2) |
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240 | (28) |
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Gauß's lattice point problem |
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240 | (1) |
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Lattice points with distinct distances |
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241 | (1) |
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Lattice points, no four on a circle |
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241 | (1) |
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The no-three-in-line problem |
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242 | (2) |
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Quadratic residues. Schur's conjecture |
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244 | (1) |
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Patterns of quadratic residues |
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245 | (3) |
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A cubic analog of a Pell equation |
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248 | (1) |
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Quadratic residues whose differences are quadratic residues |
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248 | (1) |
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248 | (1) |
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Residues of powers of two |
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249 | (1) |
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Distribution of residues of factorials |
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250 | (1) |
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How often are a number and its inverse of opposite parity? |
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251 | (1) |
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Covering systems of congruences |
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251 | (2) |
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253 | (3) |
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256 | (1) |
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Products of small prime powers dividing n |
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256 | (1) |
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Series associated with the ζ-function |
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257 | (1) |
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Size of the set of sums and products of a set |
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258 | (1) |
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Partitions into distinct primes with maximum product |
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258 | (1) |
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259 | (1) |
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All partial quotients one or two |
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259 | (1) |
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Algebraic numbers with unbounded partial quotients |
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260 | (1) |
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Small differences between powers of 2 and 3 |
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261 | (1) |
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Squares with just two different decimal digits |
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262 | (1) |
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The persistence of a number |
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262 | (1) |
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Expressing numbers using just ones |
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263 | (1) |
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Mahler's generalization of Farey series |
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263 | (2) |
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A determinant of value one |
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265 | (1) |
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Two congruences, one of which is always solvable |
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266 | (1) |
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A polynomial whose sums of pairs of values are all distinct |
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266 | (1) |
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An unusual digital problem |
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266 | (2) |
| Index of Authors Cited |
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268 | (12) |
| General Index |
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280 | |
