Marketing Fragment 6 X 10.Long.T65

Cambridge University Press 0521615240 - Elementary Number Theory in Nine Chapters, Second Edition James J. Tattersall Index More information

Index

Abel, Niels Henrik, 18 Apian, Peter, 33 absolute value function, 56 Apollonius of Perga, 274 Abu Bakr Ahmad, 232 Apuleius of Madaura, 10 Abu Kamil ibn Aslam, 188 Archimedes, 4, 274, 275 abundancy of a number, 156 cattle problem, 274–275 Adalbolt, Bishop of Utrecht, 13 Archytas of Tarentum, 136 Adleman, Leonard M., 252, 271 Argenti, Matteo, 239 Aeneas the Tactician, 227 Aristagorus, 227 Ahmad, Abu Bakr, 232 Aristotle, 60 Aiyer, V.Ramaswami, 130 arithmetic function, 94 Alaoglu, Leon, 157 arithmetic mean of the divisors of a number, Alberti, Leon Battista, 235, 239 158 Alcuin of York, 137, 188, 189 Armengaud, Joel, 142 Al-Duraihim, 233 Artin, Emil, 210, 218 Alford, William Robert, 155 conjecture, 218 Alfred the Great, 11 Aryabhata, 12, 22, 42, 77, 283 aliquot cycle, 99 Asadulla, Syed, 146 aliquot parts, 99 asymptotically equivalent, 117 algorithms Avicenna, 171 Collatz, 34 Ayrton, Hertha, 192–193 Crelle, 216–217 division, 58 Babbage, Charles, 242 Euclidean, 70–71 Bachet, Claude-Gaspard, 16–19, 21, 42, 51, Gauss’s Easter, 167 64, 70, 82, 137, 144, 189, 262, 266, 270, Kaprekhar, 35 279, 280 Moessner, 129–130 theorem, 182 reverse–subtract–reverse–add, 38 equation, 279–280 Saunderson, 73 Bacon, Roger, 234 Al-Hasan, 187 Baker, Alan, 280 Al-Karaji, 15, 32 Barbeau, Edward, 49 Al-Khalil, 226 Barlow, P., 183 Al-Khwarizimi, 34, 169 Barrow, Isaac, 95 Al-Qulgashandi, 232, 233 Battle of Hastings, 47 amicable pairs, 147 Baudhayana, 276 amicable triple, 150 Bear, Robert, 154 Amthov, Carl, 275 Beaufort, Sir Francis, 242 Andrews, George, 321 tableau, 241–242 Annapurna, U., 99 Belaso, Giovan Batista, 238, 239

421

422 Index

Bellerophon, 227 Charles I, 240, 242 Bernoulli, Daniel, 19 Charles II, 240 Bernoulli, Jakob, 42, 48 Charles IX of France, 240 Bernoulli, Johann, 19, 307 Chaucer, Geodfrey, 11, 234 Bernoulli, Nicolas, 30 Chebyshev, Pafnuty Lvovich, 116–117, 199 Bertrand, Joseph Louis François, 116–117 Chernick, J., 155 postulate, 116 Chinese Remainder Theorem, 185 Bhaskara, 187, 189, 276, 283, 289, 297 Chowla, Sarvadaman, 100, 111 Bhaskara–Brouncker theorm, 297 function, 100 Bianchini, Giovanni, 16 ciphers binary-square technique, 165–166 affine, 230 binary quadratic form, 280 Argenti, 240 Binet, Jacques-Philippe-Marie, 30 auto key, 240–242 formula, 30–31 Belaso, 238–239 Bode, Johann, 25 block, 247 law, 24–26 Caesar, 228 Boethius, 10–13, 137 character, 228 Boleyn, Ann, 45 disk cipher, 236–37 Bombelli, Rafael, 16–18, 76, 283, 296 exponential, 250 Book of Genesis, 147 knapsack, 245 Bourlet, Carlo, 144 monoalphabetic, 226 Bouton, Charles, 62 polyalphabetic, 236 Bouvellus, see Charles de Bovilles Polybius, 227 Bovilles, Charles de, 91, 138 polygraphic, 247 Brahmagupta, 189, 276 shift, 229 Brancker, Thomas, 109 Trithemius, 237–238 Brent, R.P., 144 Vigene`re, 240 Briggs, Henry, 33 Wheatstone, 242 Broscius, see Jan Brozek wheel cipher, 243 Brouncker, William (Lord), 18, 276, 297 ciphertext, 226 Brown, Alan, 151 Clark, William, 243 Brozek, Jan, 138 Clarkson, Roland, 142 Brunn, Viggo, 120 Clemens, Samuel, 15 constant, 120 Clement, Paul, 120 Bruno, Giordano, 239 formula, 120 clique, 157 Caesar, Augustus (Gaius Julius Caesar Cotes, Roger, 72, 294 Octavianus), 228 Cohen, Graeme Laurence, 144 Caesar, Julius, 228 Cohen, Henri, 9 Caldwell, Chris K., 125 Cohn, J.H.E., 29 Cameron, Michael, 142 Cole, Frank Nelson, 141 Capella, Martianus, 11 Collatz, Lothar, 34 Cardano, Girolamo, 94–95, 101, 239 algorithm, 34 Carmichael, Robert Daniel, 174 Colquitt, Walter, N., 142 lambda function, 191 Colson, John, 95 Carroll, Lewis see Charles Dodgson Columbus, Christopher, 15 casting out nines, 171 complete residue system, 163 Catalan, Euge`ne Charles, 274 congruence, 161 Cataldi, Pietro, 109, 138, 139, 144, 145, 283 Constantine the Great, 166 Catherine the Great, 197 Conway, John, 23, 48, 282 Cauchy, Augustin-Louis, 18, 122, 178, 184 Cooper, C., 64 sequence, 301 Copernicus, Nicholas, 237 ceiling function, 135 coprime, 67 central trinomial coefficient, 23 Cotes, Roger, 72, 294 Ceres, 25 Council of Nicaea, 166 Cesa`ro, Ernesto, 118, 126, 130 Crelle, August Leopold, 109, 216 Charlemagne, 137 algorithm, 216–217

Index 423

Cromwell, Oliver, 274 Dyson, F.J., 324 cross bones check, 172 Cunningham, Alan, Easter, 166–167 chain, 128 Easton, Belle, 132 Curie, Marie, 87 Edict of Nantes, 30 cycling digits property, 85 Eisenstein, Ferdinand, 207 cyclotonic integer, 271 Electric Frontier Foundation, 142 Cyril, Bishop of Alexandria, 14 elephantine triple, 82 Elizabeth I, 11 Darius, 227 Elkies, Noam J., 270 Dase, Zacharias, 109 El-Sedy, Esam, 26 Davenport, Harold, 166 equivalence relation, 162 coverings, 166 Eratosthenes, 87–88, 123 Davis, Martin, 181 sieve, 87–88 day of the week, 47 Erdo¨s, Paul, 118, 157, 166 de Bessy, Bernard Frenicle, 77–78, 97, 101, Ernst, Thomas, 237 111, 139, 176, 278 Escott, E.B., 123, 148 de Billy, Jacques, 18 Essennell, Emma, 63 Dedekind, Richard, 271 Euclid, 5, 14, 58, 60, 67, 70, 76, 88–90, 115, deficiency of a number, 156 136, 233 Degenin, C.F., 278 lemma, 89 degree of a number, 100 theorem, 115 de la Valleé Poussin, Charles Jean Nicholas Euclidean Algorithm, 70–71 Baron, 118, 132–133, 173 Euler, Johannes Albert, 268 de Me´re´, Chevalier, 31 Euler, Leonard, 17, 19, 20, 60, 98, 105, 111, DeMoivre, Abraham, 30 116, 118, 120, 123, 125, 126, 139, 140, DeMorgan, Augustus, 40, 47, 188 144, 145, 148, 154, 161, 176, 185, 189, de Nemore, Jordanus, 32, 138 190, 197, 201, 202, 204, 207, 215, 216, de Polignac, Alphonse Armand Charles 260, 262, 263, 267, 270, 274, 276, 279, Marie, 120, 124 280, 283, 304, 305, 307, 308, 314, 318 Descartes, Rene´, 95, 97, 139, 150, 151, 158, criterion, 203 265 parity law, 310 DeTemple, Duane, 29 pentagonal number theorem, 317 Dickson, Leonard Eugene, 95, 148 phi-function, 173 Diffie, W., 252, 253 product, 119 Digby, Sir Kenelem, 100 theorem, 315, 318 digital root, 35 Euler–Fermat theorem, 176 digital sum, 68 Euler–Maclaurin theorem, 95 Diophantine equation, 181 Euler–Mascheroni constant, 95, 132–133 Diophantus, 14–17, 45, 78, 181, 258, 260, extended digital sum, 68 265, 266, 270, 272, 275 polynomial, 280 Fagot, Henry, see Bruno Dirichlet, Johann Peter Gustav Lejeune, 95, Faltings, Gert, 271 99, 115, 116, 117, 123, 180, 265, 270, Faraday, Michael, 45 271, 283 Farey, John, 178 principle, 190 complementary fractions, 178 product, 105 fractions, 178 distance function, 299 interval, 181 division algorithm, 58 pairs, 178 Dixon, John, 72 Faulhaber, Johann, 48 Dodgson, Charles, 1, 241, 272 Fechner, Gustav, 30 Donne, John, 196 Felkel, Antonio, 109 Doyle, Sir Arthur Conan, 258 Fermat, Cle´ment-Samuel, 18 Dubner, Harvey, 122, 125 Fermat, Pierre de, 17, 18, 20, 22, 77, Duncan, R.L., 99 100–101, 110, 111, 139, 140, 145, 148, Durfee, William Pitt, 312, 313 150, 151, 176, 204, 260, 265, 267, 269, Durfee–Sylvester theorem, 313 270, 271, 274, 276, 279, 280

424 Index

formula, 22 problem, 260 Last Theorem, 18, 270 Girand–Euler theorem, 263 Little Theorem, 176 Glaisher, James Whitbread Lee, 97, 131 method of descent, 7 Goldbach, Christian, 123, 125, 262 Ferrers, Norman M., 311, 313 conjecture, 125 conjugate diagram, 311 golden ratio, 29–31 diagram, 311 golden rectangle, 30 theorem, 313 golden triangle, 29 Fibonacci, 3, 15, 27, 41, 42, 109, 138, 169, Goodwyn, Henry, 178 187, 189, 258, 260, 283 Guo Shoujing, 7 identity, 260 Granville, Andrew, 155 ﬁeld, 192–193 greatest common divisor, 64–65 Findley, Josh, 142 greatest integer function, 112 ﬁnite difference method, 8–10 Gregory, James, 8 Flaccus Albinus, see Alciun group, 192 Fortune, Reo, 122 Abelian, 192 Fouvrey, E., 271 cyclic, 223 Franklin, Fabian, 315, 316, 319 Gunpowder Plot, 1, 100 theorem, 316 Gupta, H., 174 Franqui, Benito, 151 Guy, Richard K., 48, 95, 100, 321 Frederick II, 3 Frenecle, see de Bessy Hadamard, Jacques, 118 Frey, Gerhard, 271 Haggard, Paul, 21 Friedman, William, 241 Haggis, Peter, 157 Fundamental Theorem of Arithmetic, 90 Hajratwala, Nayan, 142 Halley, Edmund, 164 Gage, Paul, 142 comet, 164 Galbreath, Norman, 124 Hammer, Monica Suzanne, 93 Gale, Kate, 82 Hardy, Godfrey Harold, 96, 119, 125, 268, Galileo Galilei, 1, 23, 78, 110, 237 323 Galois, Evariste, 18, 107, 296 harmonic mean of the divisors of a number, Garcia, Mariano, 148, 151 158 Gardiner, Vera, 124 Haros, C.H., 178, 180 Gauss, Karl Friedrich, 17, 25, 71, 90, 117, Harriot, Thomas, 1, 8 122, 146, 161, 167, 173, 175, 184, 185, Harrison, John, 33 196, 198, 201, 205, 207, 216, 218, 220, Haselgrove, C.B., 100 265, 266, 270, 271, 280, 281, 283 Heath-Brown, D.R., 271 lemma, 205 Heidel, Wolfgang Ernest, 227 method for calculating the date of Easter, Heilbronn, Hans Arnold, 111 167 Hellman, Martin, 250, 252, 253 quadratic reciprocity law, 207 Henriettta, Maria, 242 Theorem, 175 Henry IVof France, 16, 240 Gelin, A., Henry VIII of England, 47 Gelin–Cesa`ro identity, 54 Hensel, Kurt, 299 geometric mean of the divisors of a number, Heron‘s formula, 83 158 Herodotus, 227 Ge´radin, A., 35, 101 Hilbert, David, 92, 119, 181, 210, 269 Gerbert of Aurillac, 13, 234 primes, 92 problem, 23, Hildegard von Bingen, 234 Germain, Sophie, 122, 270 Hill, John, 43 prime, 122 Hill, Lester, 247 German, R.A., 275 cipher system, 247 Gergonne, Joseph Diez, 170 Histaeus, 227 Gillies, Don, 141 Holtzman, William, 16, 17 GIMPS (Great Internet Mersenne Prime Homer, 227 Search), 142 Horadam, A.F., 84 Girand, Albert, 27, 260 House of Wisdom, 10, 34

Index 425

Hrotsvita, 138 theorem, 215 Hurwitz, Alexander, 141, 269, 292 Lame´, Gabriel, 30, 71–72, 270, 271 Huygens, Christiaan, 283, 288 formula, 30 Hypatia, 14 Lander, L.J., 270 Hypsicles, 4, 14 Landry, Fortune, 140 Laplace, Pierre-Simon, 202 Iamblicus of Chalis, 10, 82, 137, 147 Larison, Sidney, 26–27 Illiad, 227 lattice point, 113 index of a number modulo a prime, 220 least common multiple, 68 Index librorum prohibitorum, 237 least residue system, 163 irrationality of e,60 Le Blanc, Monsieur, see Sophie Germain irrationality of , 60, 272 Legendre, Adrien Marie, 18, 112–113, 116, Iwaniec, Hendrik, 125 117, 122, 123, 161, 201, 202, 206, 207, 265, 266, 270 Jacobi, Carl Gustav Jacob, 97, 117, 220, 263, conjecture, 117 267, 319 symbol, 202 conjecture, 97 theorem, 113 symbol, 210 Lehman, R.S., 100 triple product identity, 319 Lehmer, Derrick Henry, 80, 110, 140, 150, Jefferson, 226, 243 155, 174 wheel cipher, 243 Lebesque, V.A., 184 Jia Xian, 31 Leibniz, Gottfried, 16, 45, 154, 176, Jing-Run, Chen, 125 198, 307 Jordan, Barbara, 55 Lenthe´ric, P., 81 Jumeau, Andre´, 151 Leonardo of Pisa, see Fibonnaci Juzuk, Dov, 49 Lerch, Mathias, 130 formula, 130 Kaisiki, F.W., 241 Lessing, Gotthold, 274 Kamasutra, 232 Levy, Paul, 129 Kanold, Hans-Joachim, 142, 157 Lewis, Meriweather, 243 Kaprekhar, D.R., 35, 39 Leybourn, Judy, 121 algorithm, 35 Library of Alexandria, 14, 87 constant, 35 Lindemann, Carl, 283 Kay, David C., 53 Liouville, Joseph, 107, 173, 191, 271 Kennedy, R.E., 64 formula, 102 Kepler, Johnannes, 237 lambda function, 107 Kersey, John, 95 Littlewood, John Edensor, 125, 269 Khaldun, Ibn, 147, 232 Lord Chesterﬁeld, (Philip Dormer Stanhope), Khayyam, Omar, 32 75 Korselt, Alwin Reinhold, 155 Lucas, Edouard, 21, 27–28, 34, 43, 48, 139, Kramp, Christian, 8 140, 141, 155, 177 Kronecker, Leopold, 97 Lucas–Lehmer test, 140–141 Kulik, J.P., 109 Luther, Martin, 14 Kumar, V.Sasi, 35 Kummer, Ernst Eduard, 271 MacMahon, Percy, 52, 297, 318, 320 Kuratowsi, Scott, 142 theorem, 320 Magna Carta, 47 Ladd-Franklin, Christine, 222, 315 Mahler, Kurt, 270 Lagrange, Joseph-Louis, 17, 27, 116, 124, Maillet, E., 269 140, 161, 196, 197, 198, 201, 202, 267, Malo, E., 155 277, 281, 294 Manhattan Project, 36 identity, 16, 42 Marcellus, 4 theorem, 198, 267, 277 Marks, Sarah, see Hertha Ayrton Lambda Triangle, 53 Mathematical Tripos, 96 Lambert, Johann Heinrich, 60, 109, 215, Matiasevich, Yuri, 181 291, 305 Matsuoka, Y., 26 quadrilateral, 60 Maurolico, Francesco, 138

426 Index

Maurolycus, Franciscius, see Francesco degree, 100 Maurolico digital root, 35 Maximilian I, Emperor, 16 digital sum, 64 Mazur, Barry, 271 Euler phi-, 173 McDaniel, Wayne, 93 extended digital sum, 64 Mead, Margaret, 122 kth powers of the divisors, 98 Mengoli, Pietro, 45 Mo¨bius, 106 Merkle, R.C., 252 multiplicative, 103 Mersenne, Marin, 22, 102, 110, 139, 140, number of distinct prime factors, 99 150, 151, 176, 265 number of divisors, 94 prime, 140 strongly additive, 103 Mertens, Frantz, 106 sum of divisors, 97 conjecture, 106 sum of the aliquot parts, 99 Metric, 299 numbers, Metrodorus, 45 abundant, 137 Meyer, Margaret, 39 almost perfect, 156 Miller, J.C.P., 220 amicable pair, 147 Minkowski, Hermann, 267 amicable triple, 150 Minoli, Daniel, 154 appropriate, 111 Mo¨bius, A.F., 105 arithmetic, 158 function, 105 Armstrong, 44 inversion formuala, 106 automorphic, 43 Moessner, Alfred, 129 balanced, 193 Mohanty, S.P., 20 Bantu, 51 Monica set, 93 Bernoulli, 48 Montcula, Jean Etienne, 22 betrothed pair, 150 Montmort, Pierre Re´mond de, 33 bicomposite, 133 Moore, Eliakim Hastings, 62 biprime, 133 Mordell, Louis J., 271, 279 b-Nivern, see b-riven conjecture, 271 b-riven, 86 Morehead, J.C., 146 Brunn’s constant, 120 Moss, T., 184 canonical representation, 90 Motzkin, T.S., 93 Carmichael, 155, 177 mth power residue, 213 cheap, cheaper, cheapest, 51–52 Mu¨ller, Johannes, see Regiomontanus Columbian, 39 Muqaddimah, 147 composite, 87 convenient, 11 Naude´, Philipp, 307 copperbach, 128 Nelson, Harry, 141 coprime, 67 Nemorarius, see Jordanus de Nemore cubic happy, 26 Neugebauer, Otto, 76 Cullen, 135 Neuve´glise, Charles de, 144 Cunningham chain, 126 Neville, Edward Hope, 96 curious, 44 Newton, Isaac, 8, 17, 95, 230, 294 curvaceous, 51 Niccolo´of Brescia, see Tartaglia Curzon, 85 Nickel, Laura, 141 deacci, 53 Nicomachus, 5–7, 10, 14, 19, 20, 21, 51, 88, deﬁcient, 137 137, 138, 139, 184 eban, 51 Nim, 62 elephantine triple, 82 Niven, Ivan, 64 emirp, 134 Noll, Curt, 141 evil, 84 Novarra, Dominico, 16 extraordinary, 44 number theoretic function, 94 Euclidean perfect, 136 additive, 103 Euler–Mascheroni constant, 95, 132–133 Chowla, 100 Fermat, 145 completely additive, 103 Fibonacci, 27 completely multiplicative, 105 Fibonacci-type, 28

Index 427

fortunate, 112 pheptagonal, 51 figurate, 13 practical, 155 friendly, 157 polite, 83, 131 h-fold perfect, 158 polygonal, 2 handsome, 44 powerful, 92 happy, 26 prime, 87 happy couple, 52 prime digital sum, 93 Harshard, 64 primeness, 134 hexagonal, 7 prime octets, 120 highly composite, 96 prime quartets, 120 Hilbert primes, 92 prime triples, 120 impractical, 160 primitive abundant, 158 isolated prime, 127 primitive friendly pair, 157 least residue, 161 primitive Pythagorean triple, 76 left-truncatable primes, 121 primitive semiperfect, 151 Lucas, 28 primorial, 122 lucky, 124 pseudoperfect, 154 k-friendly, 157 pseudoprime, 155 k-hyperperfect, 151–152 pyramidal, 11 k-perfect, 150 Pythagorean, 78 k-pseudoprime, 155 Pythagorean triple, 76 k-Smith, 93 quasiperfect, 156 k-transposable, 43 Ramanujan, 135 Kaprekar, 44 reduced Niven, see riven left-truncatable prime, 121 refractorale, 134 lucky, 124 relatively prime, 67 m-gonal, 10 repdigit, 193 m-superperfect, 157 repunit, 122 m-triangular, 21 reversible prime, 121 Mersenne, 140 right-truncatable prime, 121 minimal, 97 riven, 86 multidigital, 64 Ruth-Aaron, 134 multiperfect, 150 sad, 26 multiplicatively unitary perfect, 169 self, 39 multiplicatively unitary superperfect, 160 semiperfect, 154 n-almost prime, 133 semiprime, 133 narcisstic, 44 silverbach, 128 neve, 52 Smith, 92 Niven, 64 sociable, 99 nonsquarefree, 133 solitary, 157 nth harmonic, 95 Sophie Germain prime, 122 nth order figurate, 13 square, 3 oblong, 2 square dance, 47–48 octagonal, 7 squarefree, 92 octahedral, 22 squarefull, 133 oddly-even, 51 star, 20 oddly-odd, 51 Stern, 52 Ore, 156 strobogrammatic, 52 palindromic, 43 superabundant, 157 palindromic prime, 121 superbiprime, 133 pandigital, 43 superperfect, 157 pentagonal, 7 tau, 134 pentagonal-type, 98 tetrancci, 38 perfect, 136 tetrahedral, 13 permutation prime, 121–122 triangular, 2 pfibonacci, 53 tribonacci, 38 phappy, 52 trimorphic, 43–44

428 Index

twin primes, 119 plaintext, 226 Ulam, 36 Plato, 5, 11, 59, 80, 102, 136 unitary perfect, 157 Plimpton 322, 76 unitary amicable, 157 Plutarch, 4 unitary nonrepetitive, 156 Polya, George, 100 untouchable, 157 Polybius, 227 vampire, 135 Pomerance, Carl, 155, 157 weird, 155 Pope Sixtus IV,16 Woodall, 135 Pope Sylvester II, see Gerbert zigzag, 48 Porges, Arthur, 19, 26 Zuckerman, 86 Porta, Giovanni, Battista, 239 Poulet, Paul, 99, 151 numeri idonei, 111 Powers, R.E., 141 Odlyzko, Andrew, 106 prime counting function, 117 Octavian, see Caesar, Augustus primeness of a number, 134 order of number, 213 Prime Number Theorem, 117 Ore, Oystein, 158 primitive root, 215 Orestes, 14 principle of mathematical induction, 40 Ostrowski, Alexander, 299 Probus, Valerius, 228 Ostura, 166 Proetus, King, 227 Oughtred, William, 33 properties of division, 57 Ozanam, 22 pseudo-Smarandache function, 86 rule, 22 Ptolemy, 14 public-key encription system, 251–252 Paganini, Nicolo, 148 Putnam, Hillary, 181 Paoli, P. 190 Pythagorus of Samos, 1, 2, 4, 76, 147 parity, 19 box, 80 Parkin, T.R., 270 triangle, 78 partition triple, 76 of a number, 306 of a set, 161–162 quadratic nonresidue, 201 perfect, 320 quadratic reciprocity law, 207 Pascal, Blaise, 31, 34, 40 quadratic residue, 201 triangle, 31–33 Quadrivium,11 Paton, W.R., 45 Quarra, Thabit ibn, 10, 147, 156 Peirce, Charles Sanders, 223 rule, 147–148 Pell, John, 109, 274 Qin Jiushao, 72, 186 equation, 274 sequence, 52, 81 Rademacher, Hans, 323 Pentagonal Number Theorem, 316–317 Rahn, Johann Heinrich, 109 Peppin, J.T.F., 146, 214 Raleigh, Sir Walter, 1 primality test, 214 Ramanajan, Srinivasa, 49, 96–97, 269, Pepys, Samuel, 276 321, 323 Perec, George, 230 sum, 180 Periander, 227 Rame´e, Pierre de la, 138 Perrin, Emily, 132 Recorde, Robert, 150 Perrone, Oskar, 130 reduced residue system, 174 persistence, 36 Reeds, James, 227 Pervushin, Ivan Mikheevich, 141 Regiomontanus, 15, 16, 138, 189, 190 Petrus Ramus, see Pierre de la Rame´e Regius, Hudalrichus, 138 Peurbach, Georges, 15 Regula Nicomachi,7 Philip of Opus, 11 relatively prime, see coprime Philip II of Spain, 240 repetend, 85 Piazzi, Guseppe, 25 repunit, 122 Pick, Georg Alexander, 131 residue classes, 161 formula, 131 Rohnius, see Johann Heinrich Rahn Pinch, Richard, 155 Ribit, Kenneth, 271

Index 429

Richert, Hans-Egon, 121 sieve of Eratosthenes, 87–88 Ridpath, Ian, 164 Siksek, Samir, 26 Riemann, Bernhard, 117 Simonetta, Cicco, 234 hypothesis, 119 Simson, Robert, 43, 292 zeta function, 118, 126 skytale, 227 Riesel, Hans, 141 Sloane, Neil, 36 ring, 192 Slowinski, David, 142 Rivist, Ronald, L., 252 Sluse, Rene´Franc¸ois de, 34 Roberval, Gilles Persone de, 22 Smarandache function, 83 Robinson, Julia, 181 Smith, Harold, 93 Robinson, Raphael M., 141 Smith, Michael, 93 Rogers, Will, 161 sociable chain, 99 Rotkiewicz, Andrezi, 155 Spencer, Gordon, 142 RSA system, 252 Speusippus, 11 rule of the virgins, 188 Srinivasan, A.K., 155 Stanley, Richard, 321 Sachs, A., 76 theorem, 321 Sadler, Bonnie, 21 Steiner, Ralph, 121 Saint Augustine, 2 Sterling, James, 96 Sarrus, Pierre Frederic, 155 Sterling’s formula, 96 Saunderson, Nicholas, 72, 75, 95, 294 Stern, M.A., 52 algorithm, 73–75 Stevin, Simon, 16 Schlegel, V., 188 Stifel, Michael, 13, 14, 33, 80, 95, 138, 169 Schneeberger, William, 282 Strass, E.G., 93 Schnirelmann, Lev Genrikhovich, 125 Suetonius, 228 Schwenter, Daniel, 286 Suryanarayana, D., 157 Seelhoff, P., 148 Suzanne set, 93 Selberg, Atle, 118 Synesius of Cyrene, 14 Selfridge, John, 34, 100 Sylvester, Joseph James, 83, 126, 144, 178, Senior Wrangler, 96–97 223, 273, 312, 314 sequences theorem, 314 Chowla, 102 digital root, 35 Talmud, 169 digital sum, 35 Tamerlane, 147 Fibonacci, 27 Taniyama, Yutaka, 271 Fibonacci-type, 28 Tartaglia, 33, 92, 138 Galileo, 23–24 triangle, 33 Hofstader product, 53 Taylor, Richard, 271 Hofstader sum, 52 te Reile, Herman J.J., 106, 144 look and say, 23 Thales of Miletus, 2 Lucas, 28 Thanigasalam, Kandiah, 270 Kaprekar, 39 Theodoric, 10 MacMahon, 52 Theodorus of Cyrene, 59 Pell, 52, 81 Theodosius, 14 persistence, 36 Theophilus, Bishop of Alexandria, 14 psi-, 103 Theon of Alexandria, 14 Sidney, 27 Theon of Smyrna, 5, 7, 59, 63, 137, 274 Sidney product, 27 Thrasybulus, 227 superincreasing, 25 Thurber, James, 136 Serre, Jean-Pierre, 271 Titius, Johann, 25 Sestri, Giovanni, 242 Torricelli, Evangelista, 110 Shafer, Michael, 142 Tower of Hanoi, 27–28 Shalaﬂy, A., 174 triangle inequality, 56 Shamir, Adi, 252 Trigg, Charles W. Shanks, William, 122 constant, 38 Shimura, Goro, 271 operator, 38 Sierpin´ski, W., 99 Trithemius, Johannes, 237

430 Index

Trivium,11 problem, 267–268 Tuckerman, Bryant, 141 Washington, George, 243 Twain, Mark, see Clemens, Samuel Watkins, William, 146 twin primes, 119 Watson, G.N., 21 Weierstrass, Karl, 56 Ulam, Stanislaw, 36, 124 Welsh, Luther, 142 spiral, 131–132 Wertheim, G., 22 unitary divisor, 157 Western, A.E., 146, 220 Wheatstone, Charles, 242 Vachette, A., 188 cipher, 242 Vatsyanyana, 232 Whewell, William, 45, 47 valuation puzzle, 45 Archimedean, 299 Whiston, William, 95 nonArchimedean, 298–299 Wilanski, Albert, 92 Vandiver, Harry S., 125 Wiles, Andrew, 271–272 van Schooten Fritz, 95, 109 Wilkins, John, 240 Vergil’ Aeneid,19 William and Mary, 240 Vespucci, Amerigo, 15 Williams, Hugh Cowie, 275 Vie`te, Franc¸ois, 16, 76, 240 Wilson, John, 198 Vigene´re, Blaise de, 240 theorem, 199 tableau, 240 Wooden Spoon, 96 Vinogradov, Ivan Matveevich, 125, 270 Wolstenholme, Joseph, 200, 212 von Mangolt, Hans Carl Fredrich Woltman, George, 142 function, 108 Wundt, Wilhelm, 30 Voroni, Georgi, 183 Wylie, Alexander, 185 formula, 183 Xylander, see William Holtzman Wagon, Stan, 174 Walker, Gilbert, 96 Wall, Charles, 149, 157 Yang Hui, 31 Wallis, John, 13, 33, 91, 97, 100–101, 102, Yi Xing, 186, 189 240, 276, 283 Young, Alfred, 323 Walpole, Horace, 75 tableau, 323 Walpole, Robert, 75 Wang Xun, 7 Zarnke, C.R., 275 Wantzel Pierre Laurent, 138 Zeckendorf representation, 54 Waring, Edward, 95, 116, 125, 198, 199, Zhang Quijian, 188 267, 268 Zhu Shijie, 7, 31