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Index
Abel, Niels Henrik 25
AM-GM inequality 109, 144, 151,
155,160,162,163,167,232
for weighted means 162
algebra 23
algorithm
division 28
Euclidean 178, 181
alternating sums 15
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
general 165
binomial coefficients 2, 75
generalized 276
binomial theorem 2, 8, 175, 263
Bezout's equality 179
Bezout's theorem 28
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
in one variable 201
linear 202
of higher degree 211
quadratic 216
conjugate 65
conjecture
Goldbach 173
Shimura-Taniyama 173
twin prime 173
coprime 180
Cramer's rule 46
cubic polynomial 24
342
Index
cubic (continued)
de Moivre's theorem 75, 77, 78, 84
decimal expansion 261
degree
of a polynomial 24, 38
of a power mean 167
digit patterns 265
digit sum 261, 267
digits 258
final 261
Diophantine equations 217
cardinality of solution set 245
linear 217
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
in three variables 43
in two variables 39
elimination method 54
equivalent transformations 94
estimation method 101
Euclidean algorithm 178, 181
Euler's cp-function 194
Euler's criterion 216
Euler's theorem 198, 262, 264, 266
factorial 1
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
homogeneous inequality 105
homogeneous polynomial 38
Holder's inequality 166
implication method 61
incomplete quotient 176
induction principle 135, 136
finite 134
inequality 89
AM-GM 109, 144, 151, 155, 160,
162, 163, 167, 232
AM-GM, for weighted means 162
Bernoulli's 109, 141, 166, 168,
233
Bernoulli's, general 165
between power means 167
Cauchy's 127, 131, 133, 149, 166
Chebyshev's 145, 148, 149, 150,
159
geometric-power mean 167
homogeneous 105
Holder's 166
Jensen's 105, 169
Minkowski's 130, 166
symmetric 104
triangle 129
weak 91
Young's 163
infinite product 140
infinitude of primes
in arithmetic progressions 187
of the form 3k + 2 187
of the form 4k + 3 187
integer part 251, 267
integer-valued polynomials 276
interpolation 23
Index
irrational equations 60
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
substitution 66
symmetric polynomials 55
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
polynomials 23, 84, 274
cubic 24
degree of 24, 38
elementary symmetric 32, 39, 43
homogeneous 38
in several variables 38
integer-valued 276
irreducible 280
linear 24
343
Taylor 23
with integer coefficients 274
polynomial division 27
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
equations of higher degree 54
irrational equations 73
linear congruences in one variable
206
linear equations 46
linear equations with parameters
48
Taylor polynomials 23
theorem
binomial 2, 8, 175, 263
Bezout's 28
Chebyshev's 187
Chinese remainder 211
de Moivre's 75, 77, 78, 84
Dirichlet's 187
division 176
Euler's 198, 262, 264, 266
344
Index
theorem (continued)
Fermat's 197
Fermat's last 173
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
multiple 28
simple 28
zero polynomial 24, 38
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