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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 numbers, 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 pseudoprimes, 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 amicable numbers, 9, Bencze, Mihàly, 103 Daniel, 24, 63, 64, 11–13, 70 Benford, Frank, 18 75–77, 105 analytic number 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 primality test, 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 integer 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 Mathematics 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 divisor function, d(n), remainder theorem Cunningham chains, 35 49–51 bindex.qxd 3/22/05 12:11 PM Page 267 Index • 267 divisors (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; prime number theo- 73–74 Giuga on, 116; Lucas- rem, 182–183 Lehmer test and, 146; Elements (Euclid), 64–66, factorial primes, 29, prime pretender vs., 170 80–81 186; pseudoprimes, elliptic curve method: factorials, 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.
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  • ON the PRIMALITY of N! ± 1 and 2 × 3 × 5 ×···× P

    ON the PRIMALITY of N! ± 1 and 2 × 3 × 5 ×···× P

    MATHEMATICS OF COMPUTATION Volume 71, Number 237, Pages 441{448 S 0025-5718(01)01315-1 Article electronically published on May 11, 2001 ON THE PRIMALITY OF n! 1 AND 2 × 3 × 5 ×···×p 1 CHRIS K. CALDWELL AND YVES GALLOT Abstract. For each prime p,letp# be the product of the primes less than or equal to p. We have greatly extended the range for which the primality of n! 1andp# 1 are known and have found two new primes of the first form (6380! + 1; 6917! − 1) and one of the second (42209# + 1). We supply heuristic estimates on the expected number of such primes and compare these estimates to the number actually found. 1. Introduction For each prime p,letp# be the product of the primes less than or equal to p. About 350 BC Euclid proved that there are infinitely many primes by first assuming they are only finitely many, say 2; 3;:::;p, and then considering the factorization of p#+1: Since then amateurs have expected many (if not all) of the values of p# 1 and n! 1 to be prime. Careful checks over the last half-century have turned up relatively few such primes [5, 7, 13, 14, 19, 25, 32, 33]. Using a program written by the second author, we greatly extended the previous search limits [8] from n ≤ 4580 for n! 1ton ≤ 10000, and from p ≤ 35000 for p# 1top ≤ 120000. This search took over a year of CPU time and has yielded three new primes: 6380!+1, 6917!−1 and 42209# + 1.
  • Sum of Divisors of the Primorial and Sum of Squarefree Parts

    Sum of Divisors of the Primorial and Sum of Squarefree Parts

    International Mathematical Forum, Vol. 12, 2017, no. 7, 331 - 338 HIKARI Ltd, www.m-hikari.com https://doi.org/10.12988/imf.2017.7113 Two Topics in Number Theory: Sum of Divisors of the Primorial and Sum of Squarefree Parts Rafael Jakimczuk Divisi´onMatem´atica,Universidad Nacional de Luj´an Buenos Aires, Argentina Copyright c 2017 Rafael Jakimczuk. This article is distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduc- tion in any medium, provided the original work is properly cited. Abstract Let pn be the n-th prime and σ(n) the sum of the positive divisors of n. Let us consider the primorial Pn = p1:p2 : : : pn, the sum of its Qn positive divisors is σ(Pn) = i=1(1 + pi). In the first section we prove the following asymptotic formula n 6 σ(P ) = Y(1 + p ) = eγ P log p + O (P ) : n i π2 n n n i=1 Let a(k) be the squarefree part of k, in the second section we prove the formula π2 X a(k) = x2 + o(x2): 30 1≤k≤x We also study integers with restricted squarefree parts and generalize these results to s-th free parts. Mathematics Subject Classification: 11A99, 11B99 Keywords: Primorial, divisors, squarefree parts, s-th free parts, average of arithmetical functions 332 Rafael Jakimczuk 1 Sum of Divisors of the Primorial In this section p denotes a positive prime and pn denotes the n-th prime. The following Mertens's formulae are well-known (see [5, Chapter VI]) 1 1 ! X = log log x + M + O ; (1) p≤x p log x where M is called Mertens's constant.