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| Herman J., Simsa J., Kucera R. — Equations and Inequalities: Elementary Problems and Theorems in Algebra and Number Theory |
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| Предметный указатель |
Abel, Niels Henrik 25
Algebra 23
Algorithm, division 28
Algorithm, Euclidean 178 181
Alternating sums 15
AM-GM inequality 109 144 151 155 160 162 163 167 232
AM-GM inequality for weighted means 162
Arithmetic mean 33 109 150
Arithmetic progression 5
Base of the natural logarithm 113
Base representations 258
Bernoulli numbers 13
Bernoulli’s formula 12
Bernoulli’s inequality 109 141 166 168 233
Bernoulli’s inequality, general 165
Bezout’s equality 179
Bezout’s Theorem 28
Binomial coefficients 2 75
Binomial coefficients, generalized 276
Binomial theorem 2 8 175 263
Cauchy’s inequality 127 131 133 149 166
Chebyshev’s inequality 145 148 149 150 159
Chebyshev’s theorem 187
Chinese remainder theorem 211
Combinatorial identities 6
Combinatorial numbers 1
Common divisor 178 180
Common multiple 178 180
Completing the square 113
Complex number 75
Composite number 183
Congruences 189
Congruences in one variable 201
Congruences, linear 202
Congruences, of higher degree 211
Congruences, quadratic 216
Conjecture, Goldbach 173
Conjecture, Shimura — Taniyama 173
Conjecture, twin prime 173
conjugate 65
Coprime 180
Cramer’s Rule 46
Cubic polynomial 24
De Moivre’s Theorem 75 77 78 84
Decimal expansion 261
Degree of a polynomial 24 38
Degree of a power mean 167
Digit patterns 265
Digit sum 261 267
DIGITS 258
Digits, final 261
Diophantine equations 217
Diophantine equations, cardinality of solution set 245
Diophantine equations, linear 217
Diophantine equations, linear in some variable 220
Diophantus of Alexandria 217
Dirichlet’s principle 268
Dirichlet’s Theorem 187
Discriminant 124
Divisibility of numbers 174
Division algorithm 28
Division theorem 176
Eisenstein’s irreducibility criterion 280
Elementary symmetric polynomials 32 39
Elementary symmetric polynomials in three variables 43
Elementary symmetric polynomials in two variables 39
Elimination method 54
Equivalent transformations 94
Estimation method 101
Euclidean algorithm 178 181
Euler’s  -function 194
Euler’s criterion 216
Euler’s theorem 198 262 264 266
Factorial 1
Factorial, prime decomposition of 255
False roots 61
Fermat’s Last Theorem 173
Fermat’s theorem 197
Finite induction 136
Finite sums 5
Fractional part 251
Fundamental mean property 165
Fundamental theorem of algebra 29
Gauss, Carl Friedrich 189
Gaussian elimination 46
Generating functions 9
Geometric mean 151 167
Geometric progression 6
Geometric series 110
Geometric-power mean inequalities 167
Goldbach conjecture 173
Greatest common divisor 178 180
Harmonic mean 156
Hoelder’s inequality 166
Homogeneous inequality 105
Homogeneous polynomial 38
Implication method 61
Incomplete quotient 176
Induction principle 135 136
Induction principle, finite 134
Inequality 89
Inequality, AM-GM 109 144 151 155 160 162 163 167 232
Inequality, AM-GM, for weighted means 162
Inequality, Bernoulli’s 109 141 166 168 233
Inequality, Bernoulli’s, general 165
Inequality, between power means 167
Inequality, Cauchy’s 127 131 133 149 166
Inequality, Chebyshev’s 145 148 149 150 159
Inequality, geometric-power mean 167
Inequality, Hoelder’s 166
Inequality, homogeneous 105
Inequality, Jensen’s 105 169
Inequality, Minkowski’s 130 166
| Inequality, symmetric 104
Inequality, triangle 129
Inequality, weak 91
Inequality, Young’s 163
Infinite product 140
Infinitude of primes in arithmetic progressions 187
Infinitude of primes of the form 3k+2 187
Infinitude of primes of the form 4k+3 187
Integer part 251 267
Integer-valued polynomials 276
Interpolation 23
Irrational equations 60
Irrational equations, with a parameter 71
Irrational numbers 169
Irreversible transformations 96
Jensen’s Inequality 105 169
Least common multiple 178 180
Length of a vector 130
Linear congruences 202
Linear diophantine equations 217
Linear polynomial 24
Lower bound 101
Mathematical induction 5 135
Mean of degree zero 167
Mean value 150
Method of squares 113
Method of Substitution 66
Method of symmetric polynomials 55
Method of undetermined coefficients 19
Minkowski’s inequality 130 166
Monomial 38
Multiple 174
Multiple root 28
Multiple zero 28
Multiplicity of a zero 28
Nth roots 60
Number theory 173
Ordering of an n-tuple 146
Pairwise relatively prime 180
Partial fraction decomposition 21
Partial summation 16
Pigeonhole Principle 268
Polynomial division 27
Polynomials 23 84 274
Polynomials in several variables 38
Polynomials, cubic 24
Polynomials, degree of 24 38
Polynomials, elementary symmetric 32 39 43
Polynomials, homogeneous 38
Polynomials, integer-valued 276
Polynomials, irreducible 280
Polynomials, linear 24
Polynomials, Taylor 23
Polynomials, with integer coefficients 274
Power mean 151 166 167
Prime number 183
Pythagorean equation 237
Quadratic congruences 216
Quadratic polynomial 24
Quotient 28 176
Rational zeros 35
Regular n-gon 80
Relatively prime 180
Remainder 28 176
Repunit 265
Root of a polynomial equation 25
Scalar product 128
Shimura — Taniyama conjecture 173
Sierpinski, W. 187
Simple zero 28
Strict inequalities 91
Sum of digits 261 267
Sums of powers 11 39
Symmetric inequality 104
Symmetric polynomial 38
Systems of equations 46
Systems of equations of higher degree 54
Systems of irrational equations 73
Systems of linear congruences in one variable 206
Systems of linear equations 46
Systems of linear equations with parameters 48
Taylor polynomials 23
Theorem, Bezout’s 28
Theorem, binomial 2 8 175 263
Theorem, Chebyshev’s 187
Theorem, Chinese remainder 211
Theorem, de Moivre’s 75 77 78 84
Theorem, Dirichlet’s 187
Theorem, division 176
Theorem, Euler’s 198 262 264 266
Theorem, Fermat’s 197
Theorem, Fermat’s last 173
Theorem, Wilson’s 204
Transitivity of inequalities 91
Triangle inequality 129
Trichotomy Law 91
Trigonometric functions 78
Twin Prime Conjecture 173
Unique factorization 185
Upper bound 101
Vieta’s relations 32 38 41 56 78 80 82 83 125 132
Vinogradov, I. M. 216
Weak inequalities 91
Weighted arithmetic mean 162
Weighted geometric mean 162
Weighted means 151 162
Wiles, Andrew 173
Wilson’s Theorem 204
Young’s inequality 163
Zero of a polynomial 23 25 275
Zero polynomial 24 38
Zero, multiple 28
Zero, simple 28
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