bindex.qxd 3/22/05 12:11 PM Page 265

Index

φ(n). See Euler’s totient arithmetic progressions, Billingsley, Patrick, (phi) function of primes, 13–14, 201–202 σ (n), 51–54, 210 44–45 Binet, H., 143 π (n), 106, 111, 129, 134, Artin’s conjecture, 38, binomial coefficients, 138–139, 165, 188, 229 Pascal’s triangle and, 183–187, 209 Association for Comput- 165–166 π. See entry listed alpha- ing Machinery, 87–88 binomial theorem, 91 betically in the p Aurifeuille, Léon blue , 210–211 section François Antoine, Bonse’s inequality, 21 14–15 Brennen, Jack, 234 abc conjecture, 6, 17, 236 Aurifeuillian factoriza- Brent, R. P., 96 absolute , tion, 14–15 Brickell, Ernie, 168–169 24 average prime, 15 Brier numbers, 21–22 abundant numbers, 7–8, Brillhart, John, 96, 169, 40, 49, 170, 171, 235 Babbage, Charles, 19, 204 Adleman, Leonard, 168, 77–78, 140, 238 Brocard’s conjecture, 6, 212–216 Bang’s theorem, 16 22 Agrawal, Manindra, 8–9 Barlow, Peter, 171 Brown, J. L., 217 AKS algorithm for pri- Basel problem, 72–73 Brun, Viggo, 17, 22–23, mality testing, 8–9, 88 Bass, Thomas, 213 123 Alcuin, 170 Bateman, Paul, 156 Brun’s constant, 22–23 aliquot sequences (socia- Bateman’s conjecture, 16 Buss, Frank, 23 ble chains), 9–11, Baxter, Lew, 205 Buss’s function, 23 234, 235 Bays, Carter, 139 almost perfect numbers, Beal, Andrew, 16–17 Caldwell, Chris, 81, 134, 171 Beal’s conjecture, 16–17 189, 204, 262 almost-primes, 11, 217 Beauregard, R. A., 219 Cameron, Michael, 115, American Mathematical Beeger, N. G. W. H., 236 154, 203 Society, 17 Bell, E. T., 83 Carmichael, Robert , 9, Bencze, Mihàly, 103 Daniel, 24, 63, 64, 11–13, 70 Benford, Frank, 18 75–77, 105 analytic theory, Benford’s law, 17–18 Carmichael numbers, 45 COPYRIGHTEDBen Gerson, Levi, 25 MATERIAL23–24, 92, 116, 121, Anderson, A. E., 189 Bernoulli, Jacob, 72 179, 191, 192; prime Anderson, J. K., 231 Bernoulli numbers, pretenders and, 186; Andrica, Dorin, 13 19–20, 99 smooth numbers and, Andrica’s conjecture, 13 Bertrand, Joseph, 20–21 223; strong pseudo- APR , 213 Bertrand’s postulate, primes and, 193 Arenstorf, R. F., 231, 264 20–21 Carmichael’s totient Arithmetica (Diophan- Bible, references to function conjecture, tus), 42–44 primes in, 11 75 arithmetic mean, of divi- Bicknell-Johnson, Mar- Carmody, Phil, 81, 127, sors, 51 jorie, 102 231 bindex.qxd 3/22/05 12:11 PM Page 266

266 • Index

Catalan, Eugène Charles, and, 26–27; factoriza- Cunningham project, 24, 155 tion and, 87 34–35, 135, 264 Catalan’s conjecture, conjectures: defined, 2, cyclic numbers, 37–38 24–25, 100 30–31; errors in, Catalan’s Mersenne con- 63–64, 75–77. See also Dase, Zacharias, 111 jecture, 25 individual names of data encryption standard Cataldi, Pietro Antonio, conjectures (DES), 216 152, 171 consecutive composite Davis, Ken, 81 Champernowne, David, numbers, 67 De Bessy, Frenicle, 26 consecutive 70–71 Champernowne’s con- sequence, 32 decimal expansion, of π, stant, 26 consecutive numbers, 32 174–175 champion numbers, 26 consecutive primes, decimals, recurring (peri- Chen, Jing-Run, 218, sums of, 32 odic), 36–40, 188, 229 230 contests, 60, 61, 117, decryption, 193. See also Chinese remainder theo- 149, 214–215; Beal’s public key encryption rem, 26–27 conjecture, 16–17; DeCSS code, 126–127 cicadas, 27 Clay prizes, 28–29; deficient numbers, 7, 40, circle, Gauss’s problem, EFF Cooperative 49, 170, 171, 217 113 Computing Awards, delectable primes, 40 circles, prime, 27 55–56, 115, 153, 203; Deleglise, Marc, 183 circular primes, 28, 174 New RSA Factoring De Méziriac, Bachet, 93 Clay Insti- Challenge, 215–216; Demlo numbers, 40–41 tute, 28–29, 47 Riemann zeta func- Descartes, René, 172, Cocks, Clifford, 212 tion, 47 173 Cogitata Physico-Mathe- convenient numbers, descriptive primes, 41 matica (Mersenne), 70–72 Dickson, Leonard 152 Conway, John, 128, 131, Eugene, 41–42 Cohen, Fred, 213 186 Dickson’s conjecture, Cohen, Henri, 10, 176 Conway’s prime-produc- 41–42, 55, 126 Cole, Frank Nelson, 83, ing machine, 33, Difference Engine, 153 107 77–78, 140 composite numbers, 1; Copeland-Erdös con- Diffie, Whitfield, 193 consecutive, 67; stant, 29 Digital Millenium Copy- gaps between Cosgrave, John, 97 right Act (DMCA), primes and, Costello, Pat, 222 127 109–110; highly Course of Pure Mathe- digit properties, 42 composite numbers, matics, A (Hardy), Diophantus, 42–44 199–200 121 Dirichlet, Gustav Peter compositorial, 29 cousin primes, 33–34 Lejeune, 44–45 computers, 2, 178; Cramér, Harald, 148 Dirichlet’s theorem, ENIAC, 135; first Cramér’s conjecture, 44–45 algorithm using, 19; 109 Disquisitiones Arithmeti- used for Mersenne Crandall, Richard, 96 cae (Gauss), 110 primes, 153–155; Creyaufmueller, Wolf- distributed computing, World War II and, gang, 10 13, 45–47 193. See also distrib- “crowd,” 10 distribution of primes, uted computing cryptography. See public Gauss on, 111–112 computer viruses, incep- key encryption divisibility properties, of tion of, 213 Cullen, J., 34 Fibonacci numbers, concatenation of primes, Cullen primes, 34, 240 102–103 29 Cunningham, Allan divisibility tests, 48 congruences: Chinese Joseph, 34, 240 function, d(n), remainder theorem Cunningham chains, 35 49–51 bindex.qxd 3/22/05 12:11 PM Page 267

Index • 267

(factors), 48–54; Euler, Leonhard, 12, 25, ized Fermat num- congruences and, 51; 52–53, 69–79, 95, 117, bers, 97, 190; infi- and partitions, 53; prime 128, 130, 157, 171; nite descent, 90, 98; factors, 54. See also Basel problem and, perfect numbers, aliquot sequences 72; Carmichael’s 172; sums of (sociable chains) totient function con- squares, 92–94. See DNA computer, 213 jecture and, 75–77; also Fermat’s Last Donnelly, Harold, 75 Euler’s constant, 52, Theorem; Fermat’s Dubner, Harvey, 13, 24, 73, 156; Euler’s con- Little Theorem 101, 110, 164–165, venient numbers, Fermat’s Last Theorem, 195, 204–205 70–72; Euler’s qua- 2, 6, 31, 64, 90, Dudley, Underwood, dratic, 77–78, 106, 97–99; Beal’s conjec- 221 140, 196–198, 232; ture and, 16–17; Dusart, Pierre, 185 Euler’s totient (phi) Dirichlet and, 44; Dyson, Freeman, function, 45, 63, 64, Euler on, 70; Fermat- 241–242 70, 74–77, 130, 214, Catalan equation 217, 264; on factoriza- and, 100; Gauss on, Eberhart’s conjecture, tion, 84; on Fermat’s 111; Wieferich 157 Little Theorem, 70, primes, 235 economical numbers, 55 92, 233; Lucky Num- Fermat’s Little Theorem, Edwards, Harold, 209 bers of, 78–79; 8–9, 85, 90–92, Electronic Frontier Foun- Mersenne primes, 178–179; Carmichael dation, 55–56, 115, 152; obstinate num- numbers and, 23–24; 153, 203 bers and, 175; recip- Euler on, 70, 92, 233; elementary proof, of rocals of primes, Fermat quotient, 92; theo- 73–74 Giuga on, 116; Lucas- rem, 182–183 Lehmer test and, 146; Elements (Euclid), 64–66, primes, 29, prime pretender vs., 170 80–81 186; pseudoprimes, elliptic curve method: , 80, 236–237; 190–191; RSA algo- for factorization, 87; factors of, 80; strong rithm and, 214; for proving primality, law of small numbers Wieferich primes and, 56–57 and, 226 235 emirp, 57 factorial sums, 81–82 Fibonacci numbers, 18, Entropia, 115 factorization, methods 99, 101–106, 228; Eratosthenes, sieve of, of, 82–89, 95–97 divisibility properties, 58–59, 84, 147, 178 Farey, John, 209–210 102–103, 150–151; Erdös, Paul, 17, 59–63, Feit-Thompson conjec- Fibonacci composite 75, 98, 168; ture, 89 sequences, 105–106, Copeland-Erdös con- Fermat, Pierre de, 12, 142–144, 167; Lucas stant, 29; elementary 89–100; on factor- and, 104–106, proof by, 183; Erdös ization, 83, 84, 144–145 numbers, 61–62; 85–86; Fermat- Filz, Antonio, 27 Erdös-Kac theorem Catalan conjecture, Findley, Josh, 155, 203, of, 200–202; on good 17, 100; Fermat fac- 264 primes, 119; on odd torization, 95–97; Forbes, Tony, 13, 35, 46, numbers, 176 Fermat numbers, 46, 132, 195 Erdös-Kac theorem, 68, 84, 94–95, 97, Ford, Kevin, 219 200–201, 202 106, 110, 117, 128, formulae for primes, 31, errors, 63–64, 75–77 169, 175, 225, 264; 106–108, 160–161 Euclid, 44–45, 64–69, Fermat pseudo- Fortunate numbers, 170 primes, 190–191; 108–109 Euclidean algorithm, Fermat’s conjecture, Fortune’s conjecture, 68–69 31, 94–95; General- 108–109 bindex.qxd 3/22/05 12:11 PM Page 268

268 • Index

Franel, J., 210 good primes, 119 Hilbert’s 10th problem, Franke, Jens, 204 Gostin, G. B., 96 150–151 frugal numbers, 55 Gourdon, Xavier, 183, Hilbert’s 23 problems, Fung, Gilbert, 136 185 28, 107, 124–125, Graham, R. L., 105 150–151 Gage, Paul, 153, 154 Gram, J. P., 207 History of the Theory of Gallot, Yves, 22, 190, Grand Internet Obstinate Numbers (Dickson), 240, 264 Number Search, 176 41–42 gaps between primes, graph, prime number, Hodges, Laurent, 13, 109, 109–110, 139, 180–181 118–119 147, 183, 218–219, greatest common factor Hoggatt, Jr., Vernon, 102 230 (GCD), 68–69 Holey primes, 229 Gardner, Martin, 149, Green, Ben, 14 home prime, 125–126 214–215 Greenwood, Thomas, Honaker, Jr., G. L., 41 Gauss, Johann Carl 228 Hudalricus Regius, 152, Friedrich, 3–4, 45, grid computing, 208 171 65, 110–113, Grimm’s problem, 119 Hudson, Richard, 139 128–130, 183–184, Gutzwiller, M. C., 241 Hurwitz, A., 96, 154, 206; Fermat’s Last Guy, Richard K., 75, 106, 204 Theorem and, 189; 156, 226 hypothesis H, 126, 195 Germain and, 223; on primality testing, Hadamard, Jacques, 129, Iamblichus, 11, 171 177–178; prime num- 134, 182 Ibn al-Banna, 12 ber theorem and, Hajratwala, Nayan, 115, Ibn Khaldun, 11–12 182; quadratic reci- 154 Ibn Qurra, Thabit, 12 procity law, 3, Hardy, G. H., 30, 49, 61, iccanobiF prime, 228 197–198 108, 119–122, 123, illegal prime (DeCSS Gaussian , 99 130, 138, 184, 200; code), 126–127 Gaussian primes, examples of trivial inconsummate numbers, 112–113Germain, properties, 228; 128 Sophie, 99, Ramanujan and, induction, 128–130 223–224 198–200; Riemann infinite coprime Ghory, Imran, 13 hypothesis and, 207; sequence, 104 Gilbreath, N. O., on twin primes, 230 infinity, of prime num- 113–114 Hardy-Littlewood con- bers, 16, 66–68 Gilbreath’s conjecture, jectures, 33, 79, Institute of Electrical and 113–114 121–122 Electronics Engineers, Gilmore, John, 55 harmonic mean, of divi- 87–88 GIMPS (Great Internet sors, 51 Intel Pentium micro- Mersenne Primes Harshad numbers, 163 processor, 22 Search), 56, 115, Hawkins, David, Introduction to Arith- 153–155, 203 148–149 metic (Nicomachus), Girard, Albert, 93 Heath-Brown, Roger, 137 7 Giuga’s conjecture, 116; Helenius, F. W., 173 Introduction to the The- Giuga numbers, 116 Hellman, Martin, 168, ory of Numbers, An Goldbach, Christian, 193 (Hardy, Wright), 117–119 Helm, Louis K., 46–47 121 Goldbach’s conjecture, Heuer, Daniel, 189 irrationality of 2, 23, 64, 117–119, 125, heuristic reasoning, 65–66 130, 137 123–124 Ivanov, Plamen, 201 golden theorem, 197 highly composite num- Goles, Eric, 27 bers, 199–200 Jacquard, Joseph-Marie, Golomb, Solomon, Hilbert, David, 28, 140 176–177, 185–186 124–125, 241 James Bond primes, 228 bindex.qxd 3/22/05 12:11 PM Page 269

Index • 269

Jones, J. P., 150 Lehmer, Emma, 35, 38, Matijasevic, Yuri, 107, jumping champions, 131 123–124, 130, 135, 125, 150–151 227 McDaniel, Wayne, 222 Kac, Mark, 200–201, 202 Lehmer phenomenon, McIntosh, Richard, 236 Kaprekar, D., 163 209 McLaughlin, P. B., 96 Kayal, Neeraj, 8 Leibniz, G. W., 237 Meissner, W., 236 Keller, Wilfrid, 82, 96–97, Lemmermeyer, Frank, Merkle, Ralph, 168–169 101 197 Mersenne, Marin, 70, Kenney, Margaret, 194 Lenstra, Arjen, 46, 96, 151, 152 Kerchner, Chip, 190 214 Mersenne numbers, 42, Kitchen, Craig, 97 Lenstra, H.W., Jr., 96 46, 55, 144, 151–159; Kleinjung, Thorsten, 204 Lewis, Kathy, 222 Catalan’s Mersenne knots, prime and com- Linnik, Yuri conjecture, 25; dis- posite, 132–134 Vladimirovich, 137 tributed computing Knuth, Donald, 105–106 Linnik’s constant, 137 search for, 46; Eber- Koch, Elise, 206 Liouville, Joseph, hart’s conjecture, 157; Kraitchik, Maurice, 87, 137–138 factorization of, 84; 88, 96, 146 Littlewood, John Eden- factors of, 157–158; k-tuples conjecture, sor, 120–122, Fibonacci numbers prime, 35, 42, 138–139, 183, 225, and, 104; Lucas- 131–132 230 Lehmer test and, 135, Kulik, J. P., 63 Littlewood’s theorem, 146, 158–159; Kumar, Pradeep, 201 138–139 con- Kummer, Ernst Eduard, Lovelace, Ada, 19 jectures, 155–159; 99 Lucas, Édouard, 3, 15, Mersenne primes, 55, 25, 95, 96, 104–106, 90, 115, 151, 152–159, Lagrange, J. L., 196, 237 139–147, 152; Lucas- 153–155, 171, 203, Lambert, Johann H., 188 Lehmer test, 135, 146, 264; New Mersenne Landau, Edmund, 134, 158–159; Lucas num- conjecture, 88, 156; 208, 210 bers, 15, 104; Lucas Tower of Hanoi and, Landry, Fortuné, 15, 95 pseudoprimes, 142; 141; Wieferich primes Laws (Plato), 48–49 Lucas sequence, and, 236 left-truncatable prime, 142–144; Lucas’s Mertens, Franz, 211 134 game of calculation, Mertens conjecture, 211 legal issues, of DeCSS 145–146; on primality Mertens constant, 159 code, 126–127 testing, 144–147 Mertens theorem, Legendre, A. M., 77, lucky numbers, 159–160 134–135, 181–182, 147–149 Mihailescu, Preda, 25 197–198 Lucky Numbers of Euler, Mills, W. H., 160–161 Legendre on factoriza- 78–79 Mills’ theorem, 160–161 tion, 86–87 Milnor, John, 2–3 Lehmer, Derrick Henry magic squares, 39–40, Mishima, Hisanori, (D. H.), 34–35, 38, 63, 70, 140, 149–150 239–240 75, 123–124, 130, Manasse, Mark, 46, 96 Mollin, Richard, 177 135–137; Lucas- Marchal, Leon, 81 Montgomery, Hugh, Lehmer test, 135, 146, Markus, Mario, 27 241–242 158–159; on Pock- Martin, Greg, 76 Montgomery-Odlyzko lington’s theorem, Masser, David, 6 law, 242–243 175; on prime num- ’s Apol- Moorhead, J. C., 169 ber theorem, 185; on ogy, A (Hardy), 120, Morrison, M., 96, 169 pseudoprimes, 191 199, 228 Moser, Leo, 32 Lehmer, Derrick Norman Mathematics: Queen Mountain primes, 229 (D. N.), 10, 63, 112, and Servant of Sci- multidigital numbers, 135, 173, 227 ence (Bell), 83 163 bindex.qxd 3/22/05 12:11 PM Page 270

270 • Index

multiplication, fast, 1, pentagonal numbers, 56–57; Euler on, 70; 162–163 52–53 Lucas on, 144–147; multiply perfect num- Pépin, Jean, 169 probabilistic, bers, 172–174 Pépin’s test, 95, 169, 175 179–180 perfect numbers, 7, 9, prime circles, 27 Nash, Chris, 217 49, 68, 90, 170–174; prime counting function, Nelson, Harry, 149 multiply perfect, 181–186; record cal- Newcomb, Simon, 18 172–174; odd, 172; culations, 183–185 Newman, D. J., 76 and, 217; prime decade, 132 New RSA Factoring Chal- triangular, 228; uni- prime factors, 54 lenge, 215–216 tary, 234 prime knots, 132–134 Nicely, Thomas R., 22, permutable primes, 174 PrimeNet system, 115, 110, 231 Pervushin, I. M., 96, 152 154, 155 Nickel, Laura, 153, 154 physics, 241–243 prime numbers: defined, Nicomachus, 7, 171 π, decimal expansion of, 1–2; of form 4n + 3, Niven numbers, 163 174–175 67, 138–139, 141, 225; Noll, Curt, 153, 154 Pickover, Clifford, 176 of form x 2 + y 2, normal law, 200 Pime, 229 92–94; graph of, Norris, David A., 46–47 Pisano, Leonardo 180–181; “mixed Nyman, Bertil, 110 (Fibonacci), 101–106 bag,” 161–162; Plato, 48–49 patents on, 168–169; obstinate numbers, Plouffe, Simon, 20, 220 records, 203–204 175–176 p(n), estimating/calculat- prime numbers race, odd numbers: obstinate ing, 185 138–139, 225 numbers, 175–176; Pocklington, Henry prime number theorem, Riemann hypothesis Cabourn, 175 61, 63, 98, 134, 149, and, 210–211; written Pocklington’s theorem, 160, as p + 2a2, 164 146, 175 181–186; distribution of odd perfect numbers, Poincaré, Henri, 2, 30, zeros in, 207 172 200 prime periods, 27 Odlyzko, Andrew, 208, Polignac, Alphonse de, prime pretenders, 186 241–243 175–176 prime pyramid, 194–195 Oesterlé, Joseph, 6 Polignac’s conjectures, PRIMES is in P, 8–9 Oltikar, Sham, 222 175–176 primitive prime factors, On-Line Encyclopedia Pólya, George, 31, 16, 187 (Sloane), 220–221, 123–124, 189, primitive roots, 38, 228, 263 241–242 187–188 Opperman’s conjecture, polynomial congruences, primes, 26, 29, 164 196 108–109, 131, polynomials, 45, 122, 188–189 Paganini, Nicolo, 12 196 prizes. See contests Paganini’s amicable pair, Pomerance, Carl, 57, 77, probabilistic primality 9, 12 86, 88, 156, 179; APR tests, 179–180 palindromic primes, primality test, 213; probability theory, 89 164–165 Ruth-Aaron numbers, num- pandigital primes, 165 216–217; smooth bers, 179 Pascal, Blaise, 89 numbers and, 223 proper divisors, 40 Pascal’s triangle, 20, 141, Poulet, Paul, 10 Proth, François, 189–190 165–168 powerful numbers, 6, 62, Proth.exe program, 190 patents on prime num- 176–177 Prothro, Edward, 175 bers, 168–169 primality testing, 95, Proth’s theorem, Paxson, G. A., 96 177–180, 213; AKS 189–190 Pell equation, 89, 177 algorithm for, 8–9; pseudoperfect numbers, Penrose, Roger, 242 elliptic curve method, 190, 235 bindex.qxd 3/22/05 12:11 PM Page 271

Index • 271

pseudoprimes, 24, 91, Riemann hypothesis, 2, Selberg, Atle, 61, 98, 183 130, 178–179, 28, 31, 47, 63–64, 73, Selfridge, J. L., 154 190–192 117, 125, 138, Selfridge, John, 96, 204, pseudoprimes, strong, 182–185, 203, 219 192–193 206–212, 241–243; Selfridge, Paul, 156, 176 public key encryption, 3, Farey sequence and, Selvan, A. M., 242 29, 120, 168–169, 209–210; Hardy on, semiprimes, 11, 42, 59, 193–194, 212–216 120; Mertens conjec- 118, 121, 215, P versus NP problem, 29 ture, 211; σ (n), sum 217–218, 230 pyramid, prime, 194–195 of divisors function, Seventeen or Bust pro- Pythagoras, 11, 243 210; squarefree num- ject, 46–47 Pythagoras’s theorem, 5, bers, 210–211; zeta sexy primes, 33, 218 43 function, 20, 47, 182, Shafer, Michael, 115, Pythagorean triangles, 202–203, 207, 155 prime, 195 211–212, 225, Shamir, Adi, 168, 241–243 212–216 quadratic reciprocity, law Riesel, H., 96, 154 Shank’s conjecture, of, 3, 70, 100, 110, Riesel numbers, 21, 212, 218–219 125, 134, 197–198 264 Siamese primes, 219 quadratic residues, 87, right-truncatable prime, Sierpinski, Waclav, 76, 195–196 212 79, 126, 190, 195, quadratic sieve, 88, Rivera, Carlos, 190, 231, 219 214–215 264 Sierpinski numbers, 21, quantum computation, Rivest, Ronald, 168, 46, 219; Sierpinski’s 88–89 212–216 φ (n) conjecture, 219; quasi-perfect numbers, Robinson, D. E., 189 Sierpinski’s quadratic, 171 Robinson, R. M., 154 219; Sierpinski Quine, W. V., 98 Roonguthai, Warut, 35 strings, 219 Rouse Ball, W. W., 149 Sierpinski’s gasket, 141, radical of n, 6 RSA algorithm, 3, 168, 167 Ramanujan, Srinivasa, 194, 212–216; New sieves, 136; of Eratos- 120–121, 198–200, RSA Factoring Chal- thenes, 58–59, 84, 200 lenge, 215–216 147, 178; quadratic, randomness, of primes, RSA Data Security, 168, 88, 214–215; Special 60, 200–203 213 Number Field, 158 “random primes,” Rumely, R. S., 213 Silverman, J. H., 236 148–149 Ruth-Aaron numbers, Singmaster, David, 167 reciprocals of primes, 216–217 Sloane, Neil, 220–221, 73–74, 159, 238–240 228, 263 record primes, 203–204, Sacks, Oliver, 83 Sloane sequences, 174, 234 safe primes, 224 195 recursive sequence of Sato, D., 150 Slowinski, David, 153, primes, 67–68 Saxena, Nitin, 8 154 red numbers, 210–211 Schafer, Michael, 203 Smith numbers, 221–222, primes, 28, Scherk, H. F., 217 229; Smith brothers, 38–39, 157, 204–205, Scherk’s conjecture, 217 222 229; Demlo numbers, Schinzel, Andrzej, 22, smooth numbers, 24, 41; digit properties, 126, 195 222–223 42; Smith numbers Schlafly, Roger, 169 sociable chains. See and, 222 Schleybl, J., 171 aliquot sequences Rhonda numbers, 206, Schneider, Walter, 41, 264 Sophie Germain primes, 222 Schönage, A., 208 35, 42, 103, 157, 190, Riemann, Georg Friedrich Schulz, Oliver, 27 223–224 Bernhard, 206–207 Schur, Issia, 196 Sorli, Ron M., 173 bindex.qxd 3/22/05 12:11 PM Page 272

272 • Index

squarefree numbers, 6, triangular numbers, 32, Wiens, D., 150 20, 24, 62, 224–225; 111, 129, 228 Wilansky, Albert, 221 Fermat numbers, 85; trivia, 221–222, 228–229 Wiles, Andrew, 2, 6, 31, Mersenne numbers trivial zeros, 207 64, 98 and, 236; Riemann truncatable primes, 40 Willans, C. P., 107 hypothesis and, Turán, Paul, 60, 61 Williams, Hugh, 136, 210–211 Turing, Alan, 193 204 square root of 2, irra- twin primes, 22–23, 33, Wilson, John, 236–237 tional, 65–66 42, 64, 119, 122, Wilson’s theorem, 107, Stanley, Eugene, 201 229–232, 264; Poli- 178, 236–237; Wilson Stern prime, 118, 225 gnac’s conjectures primes, 237 Sternach, R. D. von, and, 175; Wilson’s Wirth, Tobias, 204 202 theorem and, 237 Wolstenholme, Joseph, Stieltjes, T., 63–64 238 Stirling, James, 80 Ulam, Stanislav, 147–149, Wolstenholme’s numbers strong law of small num- 232 and theorems, bers, 23, 31, 82, 106, Ulam spiral, 232–233, 238–240, 263 225–227 264 Woltman, George, 115, strong pseudoprimes, unique factorization, 65 154, 155, 173 192–193 unitary divisors, Woodall, H. J., 240 sum of divisors function, 233–234; perfect, 234 Woodall prime, 240 210 untouchable numbers, World Integer Factoriza- Sun, Z. H. and Sun, Z. 235 tion Center, 239–240 W., 99–100 World War II, 193 Sun Tsu Suan-ching, 26 Vallée Poussin, C.-J. de Wrathall, C. P., 96 superabundant num- la, 129, 134 Wright, E. M., 121, 161 bers, 8 Varona, Juan, 11 Wright’s theorem, 161 Suryanarayan, E. R., 219 Vieta, 90 Sylvester, J. J., 32 Vinogradov, I. M., 137 Xiao-Song Lin, 243 Szekeres, George, 61 Wada, H., 150 Yarborough primes, 229 Tao, Terence, 14 Wagstaff, Stan, 156–157 Yates, Samuel, 179, 221 Tchebycheff, P., 135 Wall, D. D., 99–100 Young, Jeffrey, 204 Te Riele, Herman, 12 Wall-Sun-Sun primes, Tijdeman, Robert, 25 99–100 zeta function. See Rie- titanic primes, 179 Wayland, Keith, 222 mann hypothesis, Toplic, Manfred, 13, 240 Wedeniswski, Sebastian, zeta function Total Destiny, 88 47 ZetaGrid distributed totient function. See weird numbers, 60, 235 computing network, Euler’s totient func- Western, A. E., 95, 96, 28, 47, 208 tion 169 zeta mysteries, 241–243. Tower of Hanoi, Wieferich, Arthur Joseph See also Riemann 140–142 Alwin, 99, 235 hypothesis “Traveling Salesman” Wieferich primes, 6, 25, Zimmerman, Paul, 183 problem, 213 95, 155, 235–236 Zwillinger, Dan, 119