Computers as novel mathematical reality. III. Mersenne numbers and divisor sums
Abstract
Nowhere in mathematics is the progress resulting from the advent of computers is as apparent, as in the additive number theory. In this part, we describe the role of computers in the investigation of the oldest function studied in mathematics, the divisor sum. The disciples of Pythagoras started to systematically explore its behaviour more that 2500 years ago. A description of the trajectories of this function — perfect numbers, amicable numbers, sociable numbers, and the like — constitute the contents of several problems
stated over 2500 years ago, which still seem completely inaccessible. A theorem due to Euclid and Euler reduces classification of even perfect numbers to Mersenne primes. After 1914 not a single new Mersenne prime was ever produced manually, since 1952 all of them have been discovered by computers. Using computers, now we construct hundreds or thousands times more new amicable pairs daily, than what was constructed by humans over several millenia. At the end of the paper, we discuss yet another problem posed by Catalan and Dickson
References
M. M. Artyukhov, “K problemam teorii druzhestvennykh chisel” [On problems of the theory of amicable numbers], Acta Arith, vol. 27, pp. 281–291, 1975 (in Russian).
W. Borho, “Befreundete Zahlen. Ein zweitausend Jahre altes Thema der elementaren Zahlentheorie,” in Lebendige Numbers, Moscow: Mir, 1985, pp. 11–41 (in Russian).
N. A. Vavilov, “Computers as novel mathematical reality. I. Personal Account,” Computer tools in education, no. 2, pp. 5–26, 2020 (in Russian); doi: 10.32603/2071-2340-2020-2-5-26
N. A. Vavilov, “Computers as novel mathematical reality. II. Waring Problem,” Computer tools in education, no. 3, pp. 5–55, 2020 (in Russian); doi: 10.32603/2071-2340-2020-3-5-55
N. A. Vavilov, “Computers as novel mathematical reality. IV. Goldbach’s conjecture,” Computer tools in education, [In Print], 2021 (in Russian).
N. A. Vavilov and V. G. Khalin, Zadachi po kursu Matematika i Komp’yuter. Vyp. 1. Arifmetika i teoriya chisel [Exercises for the course "Mathematics and Computers". Issue 1, Arithmetics and Number Theory], St. Petersburg, Russia: OTsEiM, 2005 (in Russian).
N. A. Vavilov and V. G. Khalin, Dopolnitel’nye zadachi po kursu Matematika i Komp’yuter [Supplementary exercises for the course "Mathematics and Computers"], St. Petersburg, Russia: OTsEiM, 2007 (in Russian).
N. A. Vavilov, V. G. Khalin, and A. V. Yurkov, Mathematica dlya nematematika [Mathematica for nonmathematician], [In Print], St. Petersburg, Russia, 2020 (in Russian).
E. V. Vengerova, Memuareski, Moscow: Tekst, 2016 (in Russian).
S. Yates, Repunits and Repetends, Moscow: Mir, 1992 (in Russian).
I. S. Gradstein, “On odd Perfect Numbers,” Mat. Sb., vol. 32, no. 3, pp. 476–510, 1925 (in Russian).
G. M. Edwards, Fermat’s last theorem. A genetic introduction to algebraic number theory, Moscow: Mir, 1980 (in Russian).
H. L. Abbott, C. E. Aull, E. Brown, and D. Suryanarayana, “Quasiperfect numbers,” Acta Arith., vol. 22, pp. 439–447, 1973; doi: 10.4064/aa-22-4-439-447
J. Alanen, O. Ore, and J. Stemple, “Systematic computations on amicable numbers,” Math. Comput., vol. 21, pp. 242–245, 1967; doi: 10.1090/S0025-5718-1967-0222006-7
R. C. Archibald, “Mersenne’s numbers,” Scripta Math., vol. 3, pp. 112–119, 1935.
M. M. Artuhov, “On the problem of h-fold perfect numbers,” Acta. Arith., vol. 23, pp. 249–255, 1972.
E. Bach and J. Shallit, Algorithmic number theory: Efficient algorithms, vol. 1, Cambridge, MA: MIT Press, 1996.
T. Bang, “Store primtal,” Nordisk Mat. Tidskr., vol. 2, pp. 157–168, 1954.
C. B. Barker, “Proof that the Mersenne number M167 is composite,” Bull. Amer. Math. Soc., vol. 51, no. 6, p. 389, 1945; doi: 10.1090/S0002-9904-1945-08362-1
P. T. Bateman, J. L. Selfridge, and S. S. Jr. Wagstaff, “The new Mersenne conjecture,” Amer. Math. Monthly, vol. 96, no. 2, pp. 125–128, 1989; doi: 10.1080/00029890.1989.11972155
S. Battiato and W. Borho, “Are there odd amicable numbers not divisible by three?” Math. Comput., vol. 50, pp. 633–637, 1988; doi: 10.2307/2008630
S. Battiato and W. Borho, “Breeding amicable numbers in abundance II,” Math. Comput., vol. 70, pp. 1329–1333, 2001; doi: 10.1090/S0025-5718-00-01279-5
W. E. Beck and R. M. Najar, “Reduced and augmented amicable pairs to 108,” Fibonacci Quart, vol. 31, pp. 295–298, 1993.
A. H. Beiler, Recreations in the theory of numbers — the queen of mathematics entertains, 2nd ed., New York:
Dover Publications, 1966.
M. Benito and J. L. Varona, “Advances in aliquot sequences,” Math. Comput., vol. 68, no. 225, pp. 389–393, 1999; doi: 10.1090/S0025-5718-99-00991-6
M. Benito, W. Creyaufmuller, J. L. Varona, and P. Zimmermann, ¨ “Aliquot sequence 3630 ends after reaching 100 digits,” Experiment. Math., vol. 11, no. 2, pp. 201–206, 2002.
H. A. Bernhard, “On the least possible odd perfect number,” Amer. Math. Monthly, vol. 56, pp. 628–629, 1949; doi: 10.2307/2304736
C. E. Bickmore, “On the numerical factors of a n −1,” Messenger of Math., vol. 25, no. 2, pp. 1–44, 1895–1896.
C. E. Bickmore, “On the numerical factors of a n − 1. (Second notice),” Messenger of Math., vol. 26, pp. 1–38, 1896–1897.
K. Blankenagel and W. Borho, “New amicable four-cycles II,” Int. J. Math. Sci. Comput., vol. 5, no. 1, pp. 49–51,
K. Blankenagel, W. Borho, and A. vom Stein, “New amicable four-cycles,” Math. Comput., vol. 72, no. 244, pp. 2071–2076, 2003; doi: 10.1090/s0025-5718-03-01489-3
W. Borho, “Uber die Fixpunkte der ¨ k-fach iterierten Teilersummenfunktion,” Mitt. Math. Gesellsch. Hamburg,
vol. 9, no. 5, pp. 34–48, 1969 (in German).
W. Borho, “Bemerkung zu einer Arbeit von H.-J. Kanold,” J. Reine Angew. Math., vol. 243, pp. 219–220, 1970 (in German).
W. Borho, “On Thabit ibn Kurrah’s formula for amicable numbers,” Math. Comput., vol. 26, pp. 571–578, 1972
(in German).
W. Borho, “Befreundete Zahlen mit gegebener Primteileranzahl,” Math. Ann., vol. 209, pp. 183–193, 1974 (in German).
W. Borho, “Eine Schranke fur befreundete Zahlen mit gegebener Teileranzahl, ¨ ” Math. Nachr., vol. 63, pp. 297–301, 1974 (in German).
W. Borho, “Some large primes and amicable numbers,” Math. Comput., vol. 36, no. 153, pp. 303–304, 1981; doi: 10.1090/S0025-5718-1981-0595068-2
W. Borho and K. Blankenagel, “Befreundete Zyklen,” Mitt. Math. Gesellsch. Hamburg, vol. 35, pp. 67–75, 2015
(in German).
W. Borho and H. Hoffmann, “Breeding amicable numbers in abundance,” Math. Comput., vol. 46, pp. 281–293,
; doi: 10.1090/S0025-5718-1986-0815849-1
W. Bosma and B. Kane, “The aliquot constant,” Quart. J. Math., vol. 63, no. 2, pp. 309–323, 2012; doi: 10.1093/qmath/haq050
P. Bratley, F. Lunnon, and J. McKay, “Amicable numbers and their distribution,” Math. Comput., vol. 24, pp. 431–432, 1970; doi: 10.1090/S0025-5718-1970-0271005-8
P. Bratley and J. McKay, “More amicable numbers,” Math. Comput., vol. 22, pp. 677–678, 1968; doi: 10.1090/S0025-5718-1968-0225706-9
A. Brauer, “On the non-existence of odd perfect numbers of form p αq2 1 q 2 2···q 2 t−1 q 4 t,” Bull. Am. Math. Soc., vol. 49, pp. 712–718, 1943; doi: 10.1090/S0002-9904-1943-08007-X
A. Brauer, “Note on the non-existence of odd perfect numbers of form p αq 2 1 q 2 2 ···q 2 t−1 q 4 t,” Bull. Am. Math. Soc., vol. 49, p. 937, 1943; doi: 10.1090/S0002-9904-1943-08067-6
R. P. Brent and G. L. Cohen, “A new lower bound for odd perfect numbers,” Math. Comput., vol. 53, pp. 431–437, S7–S24, 1989; doi: 10.1090/S0025-5718-1989-0968150-2
R. P. Brent, G. L. Cohen, and H. J. J. te Riele, “Improved techniques for lower bounds for odd perfect numbers,” Math. Comput., vol. 57, no. 196, pp. 857–868, 1991; doi: 10.1090/S0025-5718-1991-1094940-3
S. Brentjes and J. P. Hogendijk, “Notes on Thabit ibn Qurra and his rule for amicable numbers, ¯ ” Historia Math., vol. 16, pp. 373–378, 1989; doi:10.1016/0315-0860(89)90084-0
J. Brillhart, “Some miscellaneous factorizations,” Math. Comput., vol. 17, pp. 447–450, 1963.
J. Brillhart, “On the factors of certain Mersenne numbers. II,” Math. Comput., vol. 18, pp. 87–92, 1964; doi:
1090/S0025-5718-1964-0159776-X
J. Brillhart and G. D. Johnson, “On the factors of certain Mersenne numbers,” Math. Comput., vol. 14, pp. 365–369, 1960; doi: 10.1090/S0025-5718-1960-0123507-6
J. Brillhart, D. H. Lehmer, and J. L. Selfridge, “New primality criteria and factorizations of 2m ±1,” Math. Comput., vol. 29, pp. 620–647, 1975.
J. Brillhart, D. H. Lehmer, J. L. Selfridge, B. Tuckerman, and S. S. Wagstaff Jr., “Factorizations of bn ± 1, b = 2, 3, 5, 6, 7, 10, 11, 12 up to high powers,” 2nd ed., Contemp. Math., vol. 22, 1988.
J. Brillhart and J. L. Selfridge, “Some factorizations of 2n ±1 and relate d results,” Math. Comput., vol. 21, pp. 87–96,1967; corrigendum, ibid., vol. 21, p. 751, 1967.
J. W. Bruce, “A really trivial proof of the Lucas—Lehmer test,” Amer. Math. Monthly, vol. 100, no. 4, pp. 370–371, 1993.
R. D. Carmichael, “Note on multiply perfect numbers,” Amer. Math. Soc. Bull., vol. 17, no. 2, p. 518, 1911.
E. Catalan, “Propositions et questions diverses,” Bull. Soc. Math. France, vol. 16, pp. 128–129, 1888.
P. A. Cataldi, Trattato d`e numeri perfetti, Bologna, IT: Giovanni Rossi, 1603 (in Italian).
P. Cattaneo, “Sui numeri quasiperfetti,” Boll. Un. Mat. Ital., vol. 6, pp. 59–62, 1951 (in Italian).
C. Shi-Chao and L. Hao, “Bounds for odd k-perfect numbers,” Bull. Aust. Math. Soc., vol. 84, no. 3, pp. 475–480,
; doi: 10.1017/S0004972711002462
C. Shi-Chao and L. Hao, “Odd multiperfect numbers,” Bull. Aust. Math. Soc., vol. 88, no. 1, pp. 56–63, 2013; doi:
1017/S0004972712000858
C. Yong-Gao and T. Cui-E, “Improved upper bounds for odd multiperfect numbers,” Bull. Aust. Math. Soc., vol. 89, no. 3, pp. 353–359, 2014; doi: 10.1017/S0004972713000488
Y. Chishiki, T. Goto, and Y. Ohno, “On the largest prime divisor of an odd harmonic number,” Math. Comput., vol. 76, no. 259, pp. 1577–1587, 2007; doi: 10.1090/S0025-5718-07-01933-3
K. Chum, R. K. Guy, M. J. Jacobson Jr., and A. S. Mosunov, “Numerical and statistical analysis of aliquot sequences,” Exp. Math., vol. 29, no. 4, pp. 414–425, 2020; doi: 10.1080/10586458.2018.1477077
G. L. Cohen, “On odd perfect numbers. II. Multiperfect numbers and quasiperfect numbers,” J. Austral. Math.
Soc. Ser. A, vol. 29, pp. 369–384, 1980; doi: 10.1017/S1446788700021376
G. L. Cohen, “Even perfect numbers,” Math. Gaz., vol. 65, pp. 28–30, 1981; doi: 10.2307/3617930
G. L. Cohen, “The non-existence of quasiperfect numbers of certain forms,” Fibonacci Quart, vol. 20, pp. 81–84, 1982.
G. L. Cohen, “On primitive abundant numbers,” J. Aust. Math. Soc. Ser. A, vol. 34, pp. 123–137, 1983; doi:
1017/S1446788700019819
G. L. Cohen, “On the largest component of an odd perfect number,” J. Aus. Math. Soc. A., vol. 42, pp. 280–286,
G. L. Cohen, S. F. Gretton, and P. Hagis Jr., “Multiamicable numbers,” Math. Comput., vol. 64, no. 212, pp. 1743–1753, 1995; doi: 10.1017/S1446788700028251
G. L. Cohen, and P. Hagis Jr., “Results concerning odd multiperfect numbers,” Bull. Malaysian Math. Soc., vol. 8, no. 1, pp. 23–26, 1985.
G. L. Cohen and M. D. Hendy, “On odd multiperfect numbers,” Math. Chron., vol. 10, pp. 57–61, 1981.
G. L. Cohen and H. J. J. te Riele, “Iterating the sum-of-divisors function,” Experiment. Math., vol. 5, no. 2,
pp. 91–100, 1996; errata: vol. 6, no. 2, p. 177, 1997.
G. L. Cohen and H. J. J. te Riele, “On φ-amicable pairs,” Math. Comput., vol. 67, no. 221, pp. 399–411, 1998.
G. L. Cohen and R. M. Sorli, “On the number of distinct prime factors of an odd perfect number. Combinatorial algorithms,” J. Discrete Algorithms, vol. 1, no. 1, pp. 21–35, 2003; doi: 10.1016/S1570-8667(03)00004-2
G. L. Cohen and R. M. Sorli, “Odd harmonic numbers exceed 1024,” Math. Comput., vol. 79, no. 272, pp. 2451–2460, 2010; doi: 10.1090/S0025-5718-10-02337-9
G. L. Cohen and R. M. Sorli, “On odd perfect numbers and even 3-perfect numbers,” Integers, vol. 12, no. 6,
pp. 1213–1230, 2012.
G. L. Cohen and R. J. Williams, “Extensions of some results concerning odd perfect numbers,” Fibonacci Quarterly, vol. 23, pp. 70–76, 1985.
H. Cohen, “On amicable and sociable numbers,” Math. Comput., vol. 24, pp. 423–429, 1970; doi: 10.1090/S0025-5718-1970-0271004-6
H. Cohen, “A course in computational algebraic number theory,” in Graduate Texts in Mathematics, vol. 138,
Berlin: Springer-Verlag, 1993.
P. Cohen, K. Cordwell, A. Epstein, K. Chung-Hang, A. Lott, and S. J. Miller, “On near-perfect numbers,” Acta Arith., vol. 194, no. 4, pp. 341–366, 2020; doi: 10.4064/aa180821-11-10
F. N. Cole, “On the factoring of large numbers,” Bull. Amer. Math. Soc., vol. 10, pp. 134–137, 1903.
W. N. Colquitt and L. Welsh Jr., “A new Mersenne prime,” Math. Comput., vol. 56, no. 194, pp. 867–870, 1991; doi: 10.1090/S0025-5718-1991-1068823-9
J. H. Conway and R. K. Guy, The book of numbers, New York: Copernicus, 1996.
R. J. Cook, “Bounds for odd perfect numbers,” in Number theory, Providence, RI: Amer. Math. Soc., pp. 67–71,
P. Costello, “Amicable pairs of Euler’s first form,” J. Rec. Math., vol. 10, pp. 183–189, 1977–1978.
P. Costello, “Amicable pairs of the form (i, 1),” Math. Comput., vol. 56, pp. 859–865, 1991; doi: 10.1090/S0025-5718-
-1068822-7
P. Costello and R. A. C. Edmonds, “Gaussian amicable pairs,” Missouri J. Math. Sci., vol. 30, no. 2, pp. 107–116,
; doi: 10.35834/mjms/1544151688
R. Crandall and M. A. Penk, “A search for large twin prime pairs,” Math. Comput., vol. 33, pp. 383-388, 1979.
R. Crandall and C. Pommerance, Prime numbers. A computational perspective, 2nd ed., New York: Springer, 2005.
J. T. Cross, “A note on almost perfect numbers,” Math. Mag., vol. 47, pp. 230–231, 1974; doi: 10.2307/2689220
A. Cunningham, “On Mersenne’s numbers,” Proc. London Math. Soc., vol. s2-9, no. 1, 1911.
A. Cunningham, “On Mersenne’s numbers,” Proc. 5. Intern. Math. Congr. no. 1, pp. 384–386; Brit. Ass. Rep.
Dundee, vol. 82, pp. 406–407, 1913.
A. Cunningham, “On Lucas’s process applied to composite Mersenne’s numbers,” Lond. M. S. Proc., vol. 32, no. 2,
p. 17, 1919.
L. Dai, H. Pan, and C. Tang, “Note on odd multiperfect numbers,” Bull. Aust. Math. Soc., vol. 87, no. 3, pp. 448–451,
© КОМПЬЮТЕРНЫЕ ИНСТРУМЕНТЫ В ОБРАЗОВАНИИ. №4, 2020 г.
Компьютер как новая реальность математики
; doi: 10.1017/S0004972712000731
G. G. Dandapat, J. L.Hunsucker, and C. Pomerance, “Some new results on odd perfect numbers,” Pacific J. Math.,
vol. 57, pp. 359–364, 1975.
J. S. Devitt, R. K. Guy, and J. L. Selfridge, “Third report on aliquot sequences,” Proc. of the Sixth Manitoba
Conference on Numerical Mathematics (Univ. Manitoba, Winnipeg, Man., 1976), pp. 177–204, Congress. Numer.,
Winnipeg, Man., 1977.
K. Devlin, Mathematics. The new golden age, 2nd ed., New York: Columbia Univ. Press, 1999.
L. E. Dickson, “Finiteness of the odd perfect and primitive abundant numbers with n distinct prime factors,”
Amer. J. Math., vol. 35, pp. 413–422, 1913; doi: 10.2307/2370405
L. E. Dickson, “Theorems and tables on the sum of the divisors of a number,” Quart. J. Math., vol. 44, pp. 264–296,
L. E. Dickson, “Amicable number triples,” Amer. Math. Monthly, vol. 20, pp. 84–91, 1913; doi: 10.1080/00029890.
11997926
L. E. Dickson, History of the theory of numbers, vol. 1, Chelsea, 1952.
J. Dieudonn´e, “Pour l’honneur de l’esprit humain. Les math´ematiques aujourd’hui. Histoire et Philosophie des
Sciences. Librairie Hachette,” Paris, 1987 (in French).
S. Drake, “The rule behind ’Mersenne’s numbers’, Physis. Riv. Internaz. Storia Sci., vol. 13, pp. 421–424, 1971.
H. Dubner, “New amicable pair record,” Oct. 14, 1997. [Online]. Available: https://listserv.nodak.edu/scripts/wa.
exe?A2=ind9710&L=NMBRTHRY&F=&S=&P=695.
J. R. Ehrman, “The number of prime divisors of certain Mersenne numbers,” Math. Comput., vol. 21, pp. 700–704,
; doi: 10.1090/S0025-5718-1967-0223320-1
Z. Engberg and P. Pollack, “The reciprocal sum of divisors of Mersenne numbers,” Acta Arith., vol. 197, no. 4,
pp. 421–440, 2021.
P. Erd˝os, “On amicable numbers,” Publ. Math. Debrecen, vol. 4, pp. 108–111, 1955–1956.
P. Erd˝os, “On perfect and multiply perfect numbers,” Ann. Mat. Pura Appl., vol. 42, no. 4, pp. 253–258, 1956.
P. Erd˝os, “On the sum P
d|2
n−1 d
−1
,” Israel J. Math., vol. 9, pp. 43–48, 1971.
P. Erd˝os, “On asymptotic properties of aliquot sequences,” Math. Comput., vol. 30, pp. 641–645, 1976; doi:
1090/S0025-5718-1976-0404115-8
P. Erd˝os, P. Kiss, and C. Pomerance, “On prime divisors of Mersenne numbers,” Acta Arith., vol. 57, pp. 267–281,
E. B. Escott, “Amicable numbers,” Scripta Math., vol. 12, pp. 61–72, 1946.
L. Euler, “De numeris amicabilibus,” Opuscula varii argumenti, pp. 23–107, 1750.
I. S. Eum, “A congruence relation of the Catalan—Mersenne numbers,” Indian J. Pure Appl. Math., vol. 49, no. 3,
pp. 521–526, 2018; doi: 10.1007/s13226-018-0281-8
E. Fauquembergue, “Nombres de Mersenne, Sphinx-Œdipe,” vol. 9, pp. 103–105, 1914; vol. 15, pp. 17–18, 1920
(in French).
E. Fauquembergue, “Plus grand nombre premier connu (question 176, de G. de Rocquigny),” Interm´ed. des math.,
vol. 24, no. 33, 1917 (in French).
A. Ferrier, Librairie Vuibert, Table donnant, jusqu’a 100.000, les nombres premiers et les nombres compos ` ´es
n’ayant pas de diviseur inf´erieur a 17, avec, pour chacun d ` ’eux, son plus petit diviseur, Paris: Librairie Vuibert,
(in French).
A. Flammenkamp, “New sociable number,” Math. Comput, vol. 56, pp. 871–873, 1991; doi: 10.1090/S0025-5718-
-1052094-3
S. Fletcher, P. P. Nielsen, and P. Ochem, “Sieve methods for odd perfect numbers,” Math. Comput., vol. 81, no. 279,
pp. 1753–1776, 2012; doi: 10.1090/S0025-5718-2011-02576-7
B. Franqui and M. Garc´ıa, “Some new multiply perfect numbers,” Amer. Math. Monthly, vol. 60, pp. 459–462,
; doi: 10.1080/00029890.1953.11988324
B. Franqui and M. Garc´ıa, “57 new multiply perfect numbers,” Scripta Math., vol. 20, pp. 169–171, 1954.
M. Garc´ıa, “New amicable pairs,” Scripta Math., vol 23, pp. 167–171, 1957.
M. Garc´ıa, “New amicable pairs of Euler’s first form with greatest common factor a prime times a power of 2.,”
Nieuw Arch. Wisk. 4 , vol. 17, no. 1, pp. 25–27, 1999.
M. Garc´ıa, “A million new amicable pairs,” J. Integer Seq., vol. 4, no. 2, article 01.2.6, pp. 1–3, 2001.
V. Garc´ıa, “The first known type (7, 1) amicable pair,” Math. Comput., vol. 72, no. 242, pp. 939–940, 2003; doi:
1090/S0025-5718-02-01450-3
M. Garc´ıa, J. M. Pedersen, and H. J. J. te Riele, “Amicable pairs, a survey, High primes and misdemeanours:
lectures in honour of the 60th birthday of Hugh Cowie Williams,” Fields Inst. Commun., vol. 41, pp. 179–196,
M. Gardner, “Perfect, Amicable, Sociable,” in Mathematical Magic Show: More Puzzles, Games, Diversions,
Illusions and Other Mathematical Sleight-of-Mind from Scientific American, New York: Vintage, 1978, pp. 160–171.
D. B. Gillies, “Three new Mersenne primes and a statistical theory,” Mathematics of Comput., vol. 18, no. 85,
pp. 93–97, 1964; doi: 10.1090/S0025-5718-1964-0159774-6
АЛГОРИТМИЧЕСКАЯ МАТЕМАТИКА 51
Вавилов Н. А.
S. Gimbel and J. H. Jaroma, “Sylvester: ushering in the modern era of research on odd perfect numbers,” Integers:
Electronic Journal of Combinatorial Number Theory, vol. 3, pp. 1–26, 2003.
A. A. Gioia and A. M. Vaidya, “Amicable numbers with opposite parity,” Amer. Math. Monthly, vol. 74, pp. 969–973,
; doi: 10.2307/2315280
V. A. Golubev, “Nombres de Mersenne et caract`eres du nombre 2,” Mathesis, vol. 67, pp. 257–262, 1958.
I. J. Good, “Conjectures concerning the Mersenne numbers,” Math. Comput., vol. 9, pp 120–121, 1955; doi:
1090/S0025-5718-1955-0071444-6
T. Goto and Y. Ohno, “Odd perfect numbers have a prime factor exceeding 108
,” Math. Comput., vol. 77, no. 263,
pp. 1859–1868, 2008; doi: 10.1090/S0025-5718-08-02050-4
A. Granville, Number theory revealed: a masterclass, Providence, RI: Amer. Math. Soc., vol. 126, 2019.
O. Grun, ¨ “Uber ungerade vollkommene Zahlen, ¨ ” Math. Zeit., vol. 55, pp. 353–354, 1952 (in German); doi:
1007/BF01181133
A. W. P. Guy and R. K. Guy, “A record aliquot sequence,” in Mathematics of Computation 1943–1993: a half-century
of computational mathematics (Vancouver, BC, 1993), vol. 48, Providence, RI: Amer. Math. Soc., 1994, pp. 557–559.
R. K. Guy, “The strong law of small numbers,” Amer. Math. Monthly, vol. 95, no. 8, pp. 697–712, 1988; doi:
1080/00029890.1988.11972074
R. K. Guy, “Graphs and the strong law of small numbers,” Proc. 6 Intern. Conf Theory Appl. Graphs, Kalamazoo,
MI, 1988.
R. K. Guy, “The second strong law of small numbers,” Math. Magazine, vol. 63, no. 1, pp. 3–20, 1990; doi:
1080/0025570X.1990.11977475
R. K. Guy, “Unsolved problems in number theory,” in Problem Books in Mathematics, 3rd ed., New York, SpringerVerlag, 2004.
R. K. Guy, D. H. Lehmer, J. L. Selfridge, and M. C. Wunderlich, “Second report on aliquot sequences,” in Proc.
of the Third Manitoba Conference on Numerical Mathematics, Winnipeg, Man: Utilitas Math., 1974, pp. 357–368.
R. K. Guy and J. L. Selfridge, “Interim report on aliquot series,” in Proc. of the Manitoba Conference on Numerical,
Univ. Manitoba, Winnipeg, Man., 1971, pp. 557–580.
R. K. Guy and J. L. Selfridge, “What drives an aliquot sequence?” Math. Comput., vol. 29, pp. 101–107, 1975; doi:
1090/S0025-5718-1975-0384669-X
P. Jr. Hagis, “On relatively prime odd amicable numbers,” Math. Comput., vol. 23, pp. 539–543, 1969; doi:
1090/S0025-5718-1969-0246816-7
P. Jr. Hagis, “Lower bounds for relatively prime amicable numbers of opposite parity,” Math. Comput., vol. 24,
pp. 963–968, 1970; doi: 10.1090/S0025-5718-1973-0325507-9
P. Jr. Hagis, “Unitary amicable numbers,” Math.Comput., vol.25, pp. 915–918, 1971; doi: 10.1090/S0025-5718-1971-
-2
P. Jr. Hagis, “Relatively prime amicable numbers with twenty-one prime divisors,” Math. Mag., vol.45, pp. 21–26,
; doi: 10.1090/S0025-5718-1970-0276167-4
P. Jr. Hagis, “A lower bound for the set of odd perfect numbers,” Math. Comput., vol.35, pp. 1027–1031, 1973; doi:
1090/S0025-5718-1973-0325507-9
P. Jr. Hagis, “On the number of prime factors of a pair of relatively prime amicable numbers,” Math. Mag.,
vol. 48, pp. 263–266, 1975.
P. Jr. Hagis, “On the largest prime divisor of an odd perfect number,” Math. Comput., vol. 29, pp. 922–924, 1975.
P. Jr. Hagis, “Outline of a proof that every odd perfect number has at least eight prime factors,” Math. Comput.,
vol. 35, pp. 1027–1031, 1980.
P. Jr. Hagis, “On the second largest prime divisor of an odd perfect number,” in Analytic Number Theory, Lecture
Notes in Math, vol. 899, Berlin: Springer-Verlag, 1981, pp. 254–263; doi: 10.1007/BFb0096466
P. Jr. Hagis, “Sketch of a proof that an odd perfect number relatively prime to 3 has at least eleven prime factors,”
Math. Comput., vol. 40, pp. 399–404, 1983.
P. Jr. Hagis, “Lower bounds for unitary multiperfect numbers,” Fibonacci Quart., vol.22, no. 2, pp. 140–143, 1984;
doi: 10.1090/S0025-5718-1970-0276167-4
P. Jr. Hagis, “The third largest prime factor of an odd multiperfect number exceeds 100,” Вull. Malaysian Math.
Soc., vol. 9, no. 2, pp. 43–49, 1986.
P. Jr. Hagis, “A systematic search for unitary hyperperfect numbers,” Fibonacci Quart., vol. 25, no. 1, pp. 6–10,
P. Jr. Hagis, “Odd nonunitary perfect numbers,” Fibonacci Quart., vol. 28, no. 1, pp. 11–15, 1990.
P. Jr. Hagis, “A new proof that every odd triperfect number has at least twelve prime factors. A tribute to Emil
Grosswald: number theory and related analysis,” Contemp. Math., vol. 143, pp. 445–450, 1993.
P. Jr. Hagis and G. L. Cohen, “Some results concerning quasiperfect numbers,” J. Austral. Math. Soc., ser. A,
vol. 33, pp. 275–286, 1982; doi: 10.1017/S1446788700018401
P. Jr. Hagis and G. L. Cohen, “Every odd perfect number has a prime factor which exceeds 106
,” Math. Comput.,
vol. 67, no. 223, pp. 1323–1330, 1998.
P. Jr. Hagis and G. Lord, “Quasi-amicable numbers,” Math. Comput., vol. 31, no. 138, pp. 608–611, 1977; doi:
© КОМПЬЮТЕРНЫЕ ИНСТРУМЕНТЫ В ОБРАЗОВАНИИ. №4, 2020 г.
Компьютер как новая реальность математики
1090/S0025-5718-1977-0434939-3
P. Jr. Hagis and W. L. McDaniel, “A new result concerning the structure of odd perfect numbers,” Proc. Amer.
Math. Soc., vol. 32, pp. 13–15, 1972.
P. Jr. Hagis and W. L. McDaniel, “On the largest prime divisor of an odd perfect number,” Math. Comput., vol. 27,
pp. 955–957, 1973.
P. Jr. Hagis and W. L. McDaniel, “On the largest prime divisor of an odd perfect number II,” Math. Comput.,
vol. 29, pp. 922–924, 1975; doi: 10.1090/S0025-5718-1975-0371804-2
H. Harborth, “Eine Bemerkung zu den vollkommenen Zahlen,” vol. 31, pp. 115–117, 1976.
K. G. Hare, “More on the total number of prime factors of an odd perfect number,” Math. Comput., vol. 74,
pp. 1003–1008, 2005; doi: 10.1090/S0025-5718-04-01683-7
G. Haworth, Mersenne numbers, Reading, UK: University of Reading, 1990.
D. R. Brown, “Odd perfect numbers,” Math. Proc. Cambridge Philos. Soc., vol. 115, pp. 191–196, 1994; doi:
1017/S0305004100072030
M. R. Heyworth, “A conjecture on Mersenne’s conjecture,” New Zealand Math. Mag., vol. 19, pp. 147–151, 1982.
M. R. Heyworth, “Continued fractions in a search for odd perfect numbers,” N. Z. Math. Mag., vol. 19, pp. 63–69,
J. P. Hogendijk, “Thabit ibn Qurra and the pair of amicable numbers 17296; 18416, ¯ ” Historia Mathematica, vol. 12,
pp. 269–273, 1985; doi: 10.1016/0315-0860(85)90025-4
J. A. Holdener, “A theorem of Touchard and the form of odd perfect numbers,” American Math. Monthly, vol. 109,
pp. 661–663, 2002.
B. Hornfeck and E. Wirsing, “Uber die H ¨ ¨aufigkeit vollkommener,” Zahlen. Мath. Ann., vol. 133, pp. 431–438, 1957
(in German).
A. Hurwitz, “New Mersenne primes,” Math. Comput., vol. 16, pp. 249–251, 1962.
D. E. Iannucci, “The second largest prime divisor of an odd perfect number exceeds ten thousand,” Math. Comput,
vol. 68, pp. 1749–1760, 1999; doi: 10.1090/S0025-5718-99-01126-6
D. E. Iannucci, “The third largest prime divisor of an odd perfect number exceeds one hundred,” Math. Comput.,
vol. 69, pp. 867–879, 2000; doi: 10.1090/S0025-5718-99-01127-8
D. E. Iannucci and R. M. Sorli, “On the total number of prime factors of an odd perfect number,” Math. Comput.,
vol. 72, no. 244, pp. 2077–2084, 2003.
V. Imchenetski and V. Bouniakowsky, “Sur un nouveau nombre premier, annonc´e par le p`ere PERVOUCHINE,”
Extrait d’un rapport a l ` ’Acad´emie Bull. Acad. Imp´eriale Sci., vol. 31, pp. 532–533, 1887 (in French). [Online].
Available: https://archive.org/stream/mobot31753003685184#page/n3/mode/1up/search/Den
P. M. Jenkins, “Odd perfect numbers have a prime factor exceeding 107
,” Math. Comput., vol. 72, pp. 1549–1554,
R. P. Jerrard and N. Temperley, “Almost perfect numbers,” Math. Mag., vol. 46, pp. 84–87, 1973; doi:
1080/0025570X.1973.11976282
P. Jobling, “Large Amicable Pairs,” Jun. 6, 2004. [Online]. Available: https://listserv.nodak.edu/scripts/wa.exe?A2=
ind0306&L=nmbrthry&P=R97&D=0
P. A Jobling, “New Largest Known Amicable Pair.” Mar. 10, 2005. [Online]. Available: https://listserv.nodak.edu/
cgi-bin/wa.exe?A2=ind0503&L=nmbrthry&F=&S=&P=552.
H.-J. Kanold, “Untersuchungen uber ungerade vollkommene Zahlen, ¨ ” J. Reine Angew. Math, vol. 183, pp. 98–109,
(in German).
H.-J. Kanold, “Folgerungen aus dem vorkommen einer Gauss’schen Primzahl in der Primfactorenzerlegung
einer ungeraden vollkommenen Zahl,” J. reine angew. Math., vol. 186, pp. 25–29, 1949 (in German).
H.-J. Kanold, “Uber befreundete Zahlen. I, ¨ ” Math. Nachr. vol. 9, pp. 243–248, 1953 (in German); doi:
1002/mana.19530090408
H.-J. Kanold, “Uber befreundete Zahlen, II, ¨ ” Math. Nachr., vol. 10, pp. 99–111, 1953 (in German); doi:
1002/mana.19530100108
H.-J. Kanold, “Uber mehrfach vollkommene Zahlen, ¨ ” J. reine angew. Math., vol. 197, pp. 82–96, 1957 (in German).
H.-J. Kanold, “Uber "quasi-vollkommene Zahlen ¨ ”, Abh. Braunschweig Wiss. Ges., vol. 40, pp. 17–20, 1988 (in
German).
I. Kaplansky, “Lucas’ tests for Mersenne numbers,” Amer. Math. Monthly, vol. 52, pp. 188–190, 1945.
E. Karst, “New factors of Mersenne numbers,” Math. Comput., vol. 15, p. 51, 1961; doi: 10.1090/S0025-5718-1961-
-0
E. Karst, “Faktorenzerlegung Mersennescher Zahlen mittels programmgesteuerter Rechenger¨ate,” Numer. Math.,
vol. 3, pp. 79–86, 1961.
E. Karst, “Search limits on divisors of Mersenne Numbers,” Nordisk Tidskr. Informationsbehandling, vol. 2,
pp. 224–227, 1962.
E. Karst, “A remarkable quartic yielding certain divisors of Mersenne numbers,” Nordisk Tidskr. Informationsbehandling, vol. 3, pp. 122–123, 1963.
M. Kishore, “On odd perfect, quasiperfect, and odd almost perfect numbers,” Math. Comput., vol. 36, pp. 583–586,
АЛГОРИТМИЧЕСКАЯ МАТЕМАТИКА 53
Вавилов Н. А.
; doi: 10.2307/2007662
M. Kishore, “Odd perfect numbers not divisible by 3,” Math. Comput., vol. 40, pp. 405–411, 1983.
M. Kobayashi, P. Pollack, and C. Pomerance, “On the distribution of sociable numbers,” J. Number Theory, vol. 129,
no. 8, pp. 1990–2009, 2009.
P. Acquaah and S. Konyagin, “On prime factors of odd perfect numbers,” Intern. J. Number Theory, vol. 8, no. 6,
pp. 1537–1540, 2012; doi: 10.1142/S1793042112500935
M. Kozek, F. Luca, P. Pollack, and C. Pomerance, “Harmonious pairs,” Int. J. Number Theory, vol. 11, no. 5,
pp. 1633–1651, 2015.
M. Kraitchik, “Factorisation de 2
n ±1,” Sphinx, vol. 8, pp. 148–150, 1938.
M. Kraitchik, “On the factorization 2
n ±1,” Scripta Mathematica, vol. 18, pp. 39–52, 1952.
S. Kravitz, “Divisors of Mersenne numbers 10, 000 < p < 15, 000,” Math. Comput., vol. 15, pp. 292–293, 1961.
U. Kuhnel, ¨ “Verscharfung der notwendigen Bedingungen fur die Existenz von ungeraden vollkommenen Zahlen, ¨ ”
Math. Z., vol. 52, pp. 202–211, 1949; doi: 10.1007/BF02230691
E. J. Lee, “Amicable numbers and the bilinear diophantine equation,” Math. Comput., vol. 22, pp. 181–187, 1968.
E. J. Lee, “On divisibility by nine of the sums of even amicable pairs,” Math. Comput., vol. 23, pp. 545–548, 1969;
doi: 10.1090/S0025-5718-1969-0248074-6
E. J. Lee and J. S. Madachy, “The history and discovery of amicable - Part. 1,” J. Recreational Math., vol. 5, no. 2,
pp. 77–93, 1972; Errata: vol. 6, no. 2, p. 164, 1973; errata: vol. 6, no. 3, p. 229, 1973.
E. J. Lee and J. S. Madachy, “The history and discovery of amicable - Part. 2,” J. Recreational Math., vol. 5, no. 3,
pp. 153–173, 1972.
E. J. Lee and J. S. Madachy, “The history and discovery of amicable numbers - Part. 3,” J. Recreational Math,
vol. 5, no. 4, pp. 231–249, 1972.
D. H. Lehmer, “Note on the Mersenne number 2
−1,” Bull. Amer. Math. Soc., vol. 32, p. 522, 1926.
D. H. Lehmer, “Note on the largest Mersenne number,” Bull. Amer. Math. Soc., vol. 33, p. 271, 1927.
D. H. Lehmer, “An extended theory of Lucas’ functions,” Ann. Math., vol. 31, pp. 419–448, 1930.
D. H. Lehmer, “Note on Mersenne numbers,” Bull. Amer. Math. Soc., vol. 38, pp. 383–384, 1932; doi: 10.1090/S0002-
-1932-05396-4
D. H. Lehmer, “Some new factorizations of 2
n ±1,” Bull. Amer. Math. Soc, vol. 39, pp. 105–108, 1933.
D. H. Lehmer, “Hunting big game in the theory of numbers,” Scripta Math., pp. 229–235, 1933.
D. H. Lehmer, “On Lucas’ test for the primality of Mersenne numbers,” J. London Math. Soc., vol. 10, pp. 162–165, 1935; doi: 10.1112/jlms/s1-10.2.162
D. H. Lehmer, “On the factors of 2n ±1,” Bull. Amer. Math. Soc., vol. 53, pp. 164–167, 1947.
D. H. Lehmer, “Recent discoveries of large primes,” Mathematical Tables and Other Aids to Computation, vol. 6,
p. 61, 1952.
D. H. Lehmer, “Two new Mersenne primes,” Mathematical Tables and Other Aids to Computation, vol. 7, pp. 72, 1953.
D. H. Lehmer, “Computer technology applied to the theory of numbers,” in Studies in Number Theory,
W. J. LeVeque ed., Math. Assoc. Amer., vol. 6, pp. 117–151, 1969.
D. H. Lehmer, “Multiply perfect numbers,” Ann. of Math., vol. 2, no. 1–4, pp. 103–104, 1900/01.
F. Luca and M. T. Phaovibul, “Amicable pairs with few distinct prime factors,” Int. J. Number Theory, vol. 12,
no. 7, pp. 1725–1732, 2016; doi: 10.1142/S1793042116501050
F. Luca and C. Pomerance, “The range of the sum-of-proper-divisors function,” Acta Arithm., vol. 168, pp. 187–199, 2015.
F. Luca and H. te Riele, “φ and σ: from Euler to Erd˝os,” Nieuw Arch. Wiskd., vol. 12, no. 1, pp. 31–36, 2011.
F. Luca and H. te Riele, “Aliquot cycles of repdigits,” Integers, vol. 12, no. 1, pp. 129–140, 2012.
E. Lucas, “Sur la recherche des grands nombres premiers,” Association Fran¸caise pour l’Avancement des Sciences, vol. 5, pp. 61–68, 1876.
A. R. G. Macdivitt, “The most recently discovered prime number,” Math. Gaz., vol. 63, no. 426, pp. 268–270, 1979.
Th. E. Mason, “On amicable numbers and their generalizations,” Amer. Math. Monthly, vol. 28, pp. 195–200,
; doi: 10.1080/00029890.1921.11986031
P. J. McCarthy, “Odd perfect numbers,” Scripta Mathematica, vol. 23, pp. 43–47, 1957.
W. L. McDaniel, “On odd multiply perfect numbers,” Boll. Un. Mat. Ital., vol. 2, pp. 185–190, 1970.
W. L. McDaniel and P. Jr Hagis, “Some results concerning the non-existence of odd perfect numbers of the form pαM2β,” Fibonacci Quart., vol. 13, pp. 25–28, 1975.
M. Mercenne, “Cogitata physico mathematica,” 1644. [Online]. Available: https://gallica.bnf.fr/ark:/12148/
bpt6k81531h.r=mersenne.langFR
M. Mercenne, “Novarum observationum physico math´ematicarum,” 1647. [Online]. Available: https://gallica.bnf.fr/ark:/12148/bpt6k815336.r=mersenne.langFR
D. Minoli, “Issues in nonlinear hyperperfect numbers,” Math. Comput., vol. 34, no. 150, pp. 639–645, 1980; doi: 10.1090/S0025-5718-1980-0559206-9
D. Moews and P. C. Moews, “A search for aliquot cycles below 1010,” Math. Comput., vol. 57, pp. 849–855, 1991.
D. Moews and P. C. Moews, “A search for aliquot cycles and amicable pairs,” Math. Comput., vol. 61, pp. 935–938, 1993; doi: 10.1090/S0025-5718-1993-1185249-X235. D. Moews and P. C. Moews, “A list of amicable pairs below 2.01×1011,” Rev. Jan. 8, 1993. [Online]. Available: https://xraysgi.ims.uconn.edu:8080/amicable.txt.
D. Moews and P. C. Moews, “A list of the first 5001 amicable pairs,” Rev. Jan. 7, 1996. [Online]. Available:
https://xraysgi.ims.uconn.edu:8080/amicable2.txt.
C. G. Moreira and N. C. Saldanha, “Primos de Mersenne (e outros primos muito grandes),” in 22o Col´oquio
Brasileiro de Matematica ´ , Rio de Janeiro, Brasil: de Matem´atica Pura e Aplicada (IMPA), 1999.
L. Murata and C. Pomerance, “On the largest prime factor of a Mersenne number,” in Number theory, CRM Proc. Lecture Notes, vol. 36, pp. 209–218, 2004; doi: 10.1090/crmp/036/16
W. Narkiewicz, Classical problems in number theory, Warszawa: Polish Scientific Publishers, 1986.
H. M. Nguyen and C. Pomerance, “The reciprocal sum of the amicable numbers,” Math. Comput., vol. 88, no. 317, pp. 1503–1526, 2019.
P. P. Nielsen, “An upper bound for odd perfect numbers,” Integers, vol. 3, no. A14, pp. 1–9, 2003.
P. P. Nielsen, “Odd perfect numbers have at least nine distinct prime factors,” Math. Comput., vol. 76, pp. 2109–2126, 2007; doi: 10.1090/S0025-5718-07-01990-4
P. P. Nielsen, “Odd perfect numbers, Diophantine equations, and upper bounds,” Math. Comput., vol. 84, no. 295, pp. 2549–2567, 2015; doi: 10.1090/S0025-5718-2015-02941-X
L. C. Noll and L. Nickel, “The 25th and 26th Mersenne primes,” Math. Comput, vol. 35, no. 152, pp. 1387–1390,
P. Ochem and M. Rao, “Odd perfect numbers are greater than 101500,” Math. Comput., vol. 81, pp. 1869–1877, 2012; doi: 10.2307/23268069
P. Ochem and M. Rao, “On the number of prime factors of an odd perfect number,” Math. Comput., vol. 83,
no. 289, pp. 2435–2439, 2014; doi: 10.1090/S0025-5718-2013-02776-7
R. Ondrejka, “More on large primes,” J. Recreational Math., vol. 11, no. 2, pp. 112–113, 1978/79.
R. Ondrejka, “More very large twin primes,” J. Recreational Math., vol. 15, no. 1, 1982/83.
B. N. I. Paganini, Atti della R. Accad. Sc. Torino, vol. 2, 1866–1867, p. 362.
Parks J., “Amicable pairs and aliquot cycles on average,” Int. J. Number Theory, vol. 11, no. 6, pp. 1751–1790,
; doi: 10.1142/S1793042115500761
J. M. Pedersen, “Known amicable pairs,”. [Online]. Available: http://amicable.homepage.dk/knwnc2.htm.
J. M. Pedersen, “Various amicable pair lists and statistics,”. [Online]. Available: http://www.vejlehs.dk/staff/jmp/aliquot/apstat.htm.
J. Perrott, “Sur une proposition empirique ´enonc´ee au Bulletin,” Bull. Soc. Math. France, vol. 17, pp. 155–156,
(in French).
P. Pollack, “A remark on sociable numbers of odd order,” J. Number Theory, vol. 130, no. 8, pp. 1732–1736, 2010; doi: 10.1016/j.jnt.2010.03.011
P. Pollack, “On Dickson’s theorem concerning odd perfect numbers,” Amer. Math. Monthly, vol. 118, no. 2,
pp. 161–164, 2011; doi: 10.4169/amer.math.monthly.118.02.161
P. Pollack, “Powerful amicable numbers,” Colloq. Math., vol. 122, no. 1, pp. 103–123, 2011.
P. Pollack, “Quasi-amicable numbers are rare,” J. Integer Seq., vol. 14, p. 13, 2011.
P. Pollack, “The greatest common divisor of a number and its sum of divisors,” Michigan Math. J., vol. 60,
pp. 199–214, 2011; doi: 10.1307/mmj/1301586311
P. Pollack, “Finiteness theorems for perfect numbers and their kin,” Amer. Math. Monthly, vol. 119, pp. 670–681, 2012; errata in vol. 120, pp. 482–483, 2013.
P. Pollack, “On relatively prime amicable pairs,” Mosc. J. Comb. Number Theory, vol. 5, no. 1–2, pp. 36–51, 2015.
P. Pollack and C. Pomerance, “Prime-perfect numbers,” Integers, vol. 12, no. 6, pp. 1417–1437, 2012.
P. Pollack and C. Pomerance, “Paul ˝os and the rise of statistical thinking in elementary number theory,” Erd˝os centennial, Bolyai Soc. Math. Stud., 25, Janos Bolyai Math. Soc., Budapest ´ , pp. 515–533, 2013; doi: 10.1007/978-3-642-39286-3_19
P. Pollack and C. Pomerance, “Some problems of Erd˝os on the sum-of-divisors function,” Trans. Am. Math. Soc. Ser. B, vol. 3, pp. 1–26, 2016; doi: 10.1090/btran/10
P. Pollack, C. Pomerance, and L. Thompson “Divisor-sum fibers,” Mathematika, vol. 64, no. 2, pp. 330–342, 2018.
P. Pollack and V. Shevelev, “On perfect and near-perfect numbers,” J. Number Theory, vol. 132, no. 12, pp. 3037–3046, 2012; doi: 10.1016/j.jnt.2012.06.008
C. Pomerance, “Odd perfect numbers are divisible by at least seven distinct primes,” Acta Arith., vol. 25, pp. 265–300, 1974.
C. Pomerance, “Multiply perfect numbers, Mersenne primes, and effective computability,” Math. Ann., vol. 226, pp. 195–206, 1977; doi: 10.1007/BF01362422
C. Pomerance, “On the distribution of amicable numbers,” J. reine angew. Math., vol. 293/294, pp. 217–222, 1977.
C. Pomerance, “On the distribution of amicable numbers, II ” J. Reine Angew. Math., vol. 325, pp. 182–188, 1981.
C. Pomerance, “On primitive divisors of Mersenne numbers,” Acta Arith., vol. 46, no. 4, pp. 355–367, 1986.
C. Pomerance, “On amicable numbers,” in Analytic number theory, pp. 321–327, Springer, Cham, 2015; doi:
1007/978-3-319-22240-0_19
C. Pomerance, The first function and its iterates. Connections in discrete mathematics, Cambridge: Cambridge Univ. Press, 2018, pp. 125–138.
C. Pomerance, “The aliquot constant, after Bosma and Kane,” Q. J. Math., vol. 69, no. 3, pp. 915–930, 2018; doi: 10.1093/qmath/hay005
L. Pomey, “Sur les nombres de Fermat et de Mersenne,” Ann. Fac. Sci. Toulouse Sci. Math. Sci. Phys., vol. 16, no. 3,pp. 135–138, 1924.
P. Poulet Sur les nombres multiparfaits, Grenoble: Association Fran¸caise, 1925, p. 88.
P. Poulet La chasse aux nombres. I: Parfaits, amiables et extensions, Bruxelles: Stevens, 1929.
P. Poulet “Forty-three new couples of amicable numbers,” Scripta Math., vol. 14, p. 77, 1948.
R. E. Powers, “The tenth perfect number,” Amer. Math. Monthly, vol. 18, no. 11, pp. 195–197, 1911; doi:
2307/2972574
R. E. Powers, “The tenth perfect number,” Bull. Amer. Math. Soc., vol. 18, no. 2, p. 1, 1912.
R. E. Powers, “On Mersenne’s Numbers,” Proc. London Math. Soc., vol. 13, p. 39, 1914.
R. E. Powers, “A Mersenne prime,” Bull. Amer. Math. Soc., vol. 20, no. 2, p. 531, 1914; doi: 10.1090/S0002-9904-1914-02547-9
R. E. Powers, “Certain composite Mersenne’s numbers,” Proc. London Math. Soc., vol. 15, no. 2, p. 22, 1917.
R. E. Powers, “Note on a Mersenne number,” Bull. Amer. Math. Soc., vol. 40, p. 883, 1934; doi: 10.1090/S0002-
-1934-05994-9
C. Reid, From zero to infinity. What makes numbers , 4 ed., Washington, DC: MAA Spectrum, Mathematical
Association of America, 1992.
H. Reidlinger, “Uber ungerade mehrfach vollkommene Zahlen, ¨ ” Osterreichische Akad. Wiss. Math. Natur. ¨ ,
vol. 192, pp. 237–266, 1983 (in German).
P. Ribenboim, The book of prime number records, 2 ed., New York: Springer-Verlag, 1989.
P. Ribenboim, The new book of prime number records, New York: Springer-Verlag, 1996.
P. Ribenboim, My numbers, my friends. Popular lectures on number theory, New York: Springer-Verlag, 2000.
P. Ribenboim, The little book of bigger primes, 2 ed., New York: Springer-Verlag, 2004.
P. Ribenboim, Die Welt der Primzahlen. Geheimnisse und Rekorde, 2 ed., Springer, Heidelberg, 2011 (in German).
G. J. Rieger, “Bemerkung zu einem Ergebnis von Erd˝os uber befreundete Zahlen, ¨ ” J. reine angew. Math., vol. 261, pp. 157–163, 1973 (in German).
H. J. J. te Riele, “Four large amicable pairs,” Math. Comput., vol. 28, pp. 309–312, 1974; doi: 10.1090/S0025-5718-1974-0330033-8
H. J. J. te Riele, “Hyperperfect numbers with three different prime factors,” Math. Comput., vol. 36, no. 153,
pp. 297–298, 1981.
H. J. J. te Riele, “Perfect numbers and aliquot sequences, Computational methods in number theory, Part I,”
Math. Centre Tracts, vol. 154, pp. 141–157, 1982.
H. J. J. te Riele, “New very large amicable pairs,” Number Theory Noordwijkerhout, Springer-Verlag, 1984,
pp. 210–215; doi: 10.1007/BFb0099454
H. J. J. te Riele, “On generating new amicable pairs from given amicable pairs,” Math. Comput., vol. 42, pp. 219–223, 1984.
H. J. J. te Riele, “Rules for constructing hyperperfect numbers,” Fibonacci Quart., vol. 22, no. 1, pp. 50–60, 1984.
H. J. J. te Riele, “Computation of all the amicable pairs below 1010,” Math. Comput., vol. 47, no. S9–S40, pp. 361–368, 1986.
H. J. J. te Riele, “A new method for finding amicable pairs. Mathematics of Computation 1943–1993: a half-century of computational mathematics,” Proc. Sympos. Appl. Math., vol. 48, Providence, RI: Amer. Math. Soc. pp. 577–581, 1994.
H. J. J. te Riele, “Grootschalig rekenen in de getaltheorie,” Nieuw Arch. Wiskd., vol. 14, no. 4, pp. 236–243, 2013.
H. J. J. te Riele, W. Borho, S. Battiato, H. Hoffmann, and E. J. Lee, “Table of amicable pairs between 1010 and
,” Technical Report NM-N8603, Centrum voor Wiskunde en Informatica,, Amsterdam, 1986.
H. Riesel, “Nagra stora primtal, ˚ ” Elementa, vol. 39, pp. 258–260, 1956 (in Swedish).
H. Riesel, “A new Mersenne prime,” Math. Tables Aids Comput., vol. 12, p. 60, 1958.
H. Riesel, “Mersenne numbers,” Math. Tables Aids Comput., vol. 12, pp. 207–213, 1958; doi: 10.2307/2002023
H. Riesel, “All factors q < 108 in all Mersenne numbers 2 p −1, p prime < 104,” Math. Comput., vol. 16, pp. 478–482, 1962.
T. Roberts, “On the form of an odd perfect number,” Australian Math., vol. 35, no. 4, p. 244, 2008.
R. M. Robinson, “Mersenne and Fermat numbers,” Proc. Amer. Math. Soc. vol. 5, pp. 842–846, 1954; doi:
2307/2031878
R. M. Robinson, “Some factorizations of numbers of the form 2n ± 1,” Math. Tables Aids Comput., vol. 11, pp. 265–268, 1957.
M. I. Rosen, “A Proof of the Lucas—Lehmer Test,” Amer. Math. Monthly, vol. 95, pp. 855–856, 1988; doi:
1080/00029890.1988.11972101
W. W. Rouse Ball, “Mersenne’s numbers,” Mess., vol. 21, no. 2, pp. 34–40, 1891.
J. S´andor and B. Crstici, Handbook of number theory, vol. 2, Dordrecht: Kluwer Academic Publishers, 2004; doi: 10.1007/1-4020-2547-5
J. S´andor, D. S. Mitrinovi´c, and B. Crstici, “Handbook of number theory,” vol. 1, Dordrecht: Springer, 2006.
M. du Sautoy, The music of the primes. Searching to solve the greatest mystery in mathematics, New York: Harper Collins Publishers, 2003.
D. Scheffler and R. Ondrejka, “The numerical evaluation of the eighteenth perfect number,” Math. Comput.,
vol. 14, pp. 199–200, 1960.
A. Schinzel and W. Sierpinski, ´ “Sur certaines hypoth`eses concernant les nombres premiers,” Acta Arith, vol. 4, pp. 185–208, 1958; erratum vol. 5, p. 259, 1958.
J. L. Selfridge and A. Hurwitz, “Fermat numbers and Mersenne numbers,” Math. Comput., vol. 18, pp. 146–148, 1964; doi: 10.1090/S0025-5718-1964-0159775-8
D. Shanks and S. Kravitz, “On the distribution of Mersenne divisors,” Math. Comput., vol. 21, pp. 97–101, 1967; doi: 10.2307/2003474
W. Sierpinski, ´ “Les nombres de Mersenne et de Fermat,” Matematiche (Catania), vol. 10, pp. 80–91, 1955.
W. Sierpinski, ´ “O liczbach złod. zonych postaci ˙ (2p +1)/3, gdzie p jest liczb¸a pierwsz¸a,” Prace Mat., vol. 7, pp. 169–172, 1962 (in Polish).
W. Sierpinski, ´ Elementary theory of numbers, 2 ed., in North-Holland Mathematical Library, vol. 31., Warszawa: PWN—Polish Scientific Publishers, 1988.
V. Sitaramaiah and M. V. Subbarao, “On unitary multiperfect numbers,” Nieuw Arch. Wisk., vol. 16, no. 1–2,
pp. 57–61, 1998.
L. Skula, “Prime power divisors of Mersenne numbers and Wieferich primes of higher order,” Integers, vol. 19, no. A19, p. 4, 2019.
D. Slowinski, “Searching for the 27th Mersenne prime,” J. Recreational Math., vol. 11, no. 4, pp. 258–267, 1978/79.
R. Steuerwald, “Verscharfung einen notwendigen Bedeutung fur die Existenz einen ungeraden volkommenen ¨Zahl”, Bayer. Akad. Wiss. Math. Natur., no. 2, pp. 69–72, 1937 (in German).
R. Steuerwald, “Ein Satz uber nat ¨ urliche Zahlen mit ¨ σ(N) = 3N,” Arch. Math., vol. 5, pp. 449–451, 1954; doi:
1007/BF01898389
I. Stewart, Professor Stewart’s incredible numbers, New York: Basic Books, 2015.
I. Stewart, Do dice play God? The mathematics of uncertainty, New York: Basic Books, 2019.
M. V. Subbarao, “Odd perfect numbers: some new issues,” Period. Math. Hungar, vol. 38, no. 1–2, pp. 103–109, 1999.
M. V. Subbarao, T. J. Cook, R. S. Newberry, and J. M. Weber, “On unitary perfect numbers,” Delta (Waukesha), vol. 3, no. 1, pp. 22–26, 1972/73.
M. V. Subbarao and L. J. Warren, “Unitary perfect numbers,” Canad. Math. Bull., vol. 9, pp. 147–153, 1966; doi:
4153/CMB-1966-018-4
D. Suryanarayana, “Super perfect numbers,” Elemente Math., vol. 24, pp. 16–17, 1969.
D. Suryanarayana, “Quasiperfect numbers,” Bull. Calcutta Math. Soc., vol. 69, pp. 421–426, 1977.
Y. Suzuki, “On amicable tuples,” Illinois J. Math., vol. 62, no. 1–4, pp. 225–252, 2018; doi: 10.1215/ijm/1552442661
J. J. Sylvester, “Sur l’impossibilit´e de l’existence d’un nombre parfait impair qui ne contient pas au moins 5
diviseurs premiers distincts,” Comptes Rendus Acad. Sci. Paris, vol. 106, pp. 522–526, 1888.
J. Touchard, “On prime numbers and perfect numbers,” Scripta Math., vol. 19, pp. 35–39, 1953.
L. Troupe, “On the number of prime factors of values of the sum-of-proper-divisors function,” J. Number Theory, vol. 150C, pp. 120–135, 2015; doi: 10.1016/j.jnt.2014.11.014
B. Tuckerman, “The 24th Mersenne prime,” in Proc. Nat. Acad. Sci. U.S.A., vol. 68, pp. 2319–2320, 1971.
B. Tuckerman, “A search procedure and lower bound for odd perfect numbers,” Math. Comput., vol. 27, pp. 943–949, 1973; Corrigenda: vol. 28, pp. 887, 1974.
B. Tuckerman, “Corrigendum: "Three new Mersenne primes and a statistical theory,” Math. Comput., vol. 31,
no. 140, p. 1051, 1977; doi: 10.1090/S0025-5718-1977-0441839-1
H. S. Uhler, “First proof that the Mersenne number M157 is composite,” Proc. Nat. Acad. Sci. U.S.A., vol. 30,
pp. 314–316, 1944.
H. S. Uhler, “Note on the Mersenne numbers M157 and M167,” Bull. Amer. Math. Soc., vol. 52, p. 178, 1946.
H. S. Uhler, “A new result concerning a Mersenne number,” Mathematical Tables and Other Aids to Computation, vol. 2, pp. 94, 1946.
H. S. Uhler, “On Mersenne’s number M199 and Lucas’s sequences,” Bull. Amer. Math. Soc., vol. 53, pp. 163–164, 1947.
H. S. Uhler, “On Mersenne’s number M227 and cognate data,” Bull. Amer. Math. Soc., vol. 54, pp. 379, 1948.
H. S. Uhler, “On all of Mersenne’s numbers particularly M193,” Proc. Nat. Acad. Sci. U.S.A., vol. 34, pp. 102–103, 1948.
H. S. Uhler, “A brief history of the investigations on Mersenne’s numbers and the latest immense primes,” Scripta Mathematica, vol. 18, pp. 122–131, 1952.
H. S. Uhler, “On the 16th and 17th perfect numbers,” Scripta Mathematica, vol. 19, pp. 128–131, 1953.
H. S. Uhler, “Full values of the first seventeen perfect numbers,” Scripta Math., vol. 20, p. 240, 1954.
C. H. Viˆet, “What’s Special about the Perfect Number 6?” Amer. Math. Monthly, vol. 128, no. 1, p. 87, 2020; doi: 10.1080/00029890.2021.1839304
S. S. Jr. Wagstaff, “Divisors of Mersenne numbers,” Math. Comput., vol. 40, pp. 385–397, 1983.
S. S. Jr. Wagstaff, The Cunningham project. High primes and misdemeanours: lectures in honour of the 60th
birthday of Hugh Cowie Williams, Providence, RI: Amer. Math. Soc., pp. 367–378, 2004.
S. S. Jr. Wagstaff, “The Cunningham project”, in High primes and misdemeanours: lectures in honour of the 60th birthday of Hugh Cowie Williams, Providence, RI: Amer. Math. Soc., pp. 367–378, 2004.
S. S. Jr. Wagstaff, “The joy of factoring,” in Student Mathematical Library, vol. 68, Providence, RI: Amer. Math.
Soc., 2013.
A. Walker, “New Large Amicable Pairs,” May 12, 2004. [Online]. Available: https://listserv.nodak.edu/scripts/wa.exe?A2=ind0405&L=nmbrthry&F=&S=&P=1043
C. R. Wall, “Bi-unitary perfect numbers,” Proc. Amer. Math. Soc., vol. 33, pp. 39–42, 1972; doi: 10.1090/S0002-
-1972-0289403-9
L. J. Warren and H. G. Bray, “On the square-freeness of Fermat and Mersenne numbers,” Pacific J. Math., vol. 22, pp. 563–564, 1967; doi: 10.2140/pjm.1967.22.563
J. Watkins, Number theory. A historical approach, Princeton, NJ: Princeton University Press, 2014.
G. C. Webber, “Non-existence of odd perfect numbers of the form 3 2βp αs 2β1 1 s 2β2 2 s 2β3 3,” Duke Math. J., vol. 18, pp. 741–749, 1951; doi: 10.1215/S0012-7094-51-01867-4
A. E. Western, “On Lucas’ and Pepin’s tests for the primeness of Mersenne’s numbers,” J. London Math. Soc., vol. 7, pp. 130–137, 1932; Corrigenda: J. London Math. Soc., vol. 7, no. 4, p. 272, 1932.
H. C. Williams, Edouard Lucas and Primality Testing ´ , US: Wiley, 1998.
H. C. Williams and J. O. Shallit, “Factoring integers before computers,” in Mathematics of Computation, 1943–
: A Half-Century of Computational Mathematics. Proc. Symp. Appl. Math., vol. 48, Providence, RI: Amer. Math.
Soc., 1994, pp. 481–531.
E. Wirsing, “Bemerkung zu der Arbeit uber vollkommene Zahlen, ¨ ” Math. Ann., vol. 137, pp. 316–318, 1959
(in German); doi: 10.1007/BF01360967
H. J. Woodall, “Note on a Mersenne number,” Bull. Amer. Math. Soc., vol. 17, p. 540, 1911.
H. J. Woodall, “Mersenne’s numbers,” Manchester Mem. and Proc., vol. 56, no. 1, pp. 1–5, 1912.
G. Woltman, “GIMPS: Mathematics and Research Strategy”, in Mersenne.org. [Online]. Available: http://mersenne.
org/math.htm
G. Woltman, “On the discovery of the 38th known Mersenne prime,” Fibonacci Quart., vol. 37, pp. 367–370, 1999.
G. Woltman and S. Kurowski, “On the discovery of the 45th and 46th known Mersenne primes,” Fibonacci Quart.,
vol. 46/47, no. 3, pp. 194–197, 2008/2009.
T. Yamada, “On the divisibility of odd perfect numbers, quasiperfect numbers and amicable numbers by a high
power of a prime,” Integers, vol. 20, no. A91, pp. 1–17, 2020.
A. Yamagami, “On the numbers of prime factors of square free amicable pairs,” Acta Arith., vol. 177, no. 2,
pp. 153–167, 2017; doi: 10.4064/aa8327-7-2016
S. Yates, “Titanic primes,” J. Recreational Math., vol. 16, no. 4, pp. 250–262, 1983/84.
S. Yates, “Sinkers of the titanics,” J. Recreational Math., vol. 17, no. 4, pp. 268–274, 1984/85.
S. Yates, “Tracking titanics, in The Lighter Side of Mathematics,” Proc. Strens Mem. Conf., Calgary 1986, Math.
Assoc, of America, Washington DC, Spectrum series, 1993, pp. 349–356.
Y. Pingzhi, “An upper bound for the number of odd multiperfect numbers,” Bull. Aust. Math. Soc., vol. 89, no. 1, pp. 1–4, 2014; doi: 10.1017/S000497271200113X
Y. Pingzhi, “Zhang Zhongfeng Addition to ‘An upper bound for the number of odd multiperfect numbers,” Bull. Aust. Math. Soc., vol. 89, no. 1, pp. 5–7, 2014; doi: 10.1017/S0004972713000452
J. Zelinsky, “Upper bounds on the second largest prime factor of an odd perfect number,” International Journal of Number Theory., vol. 15, no. 6, pp. 1183–1189, 2019; doi: 10.1142/S1793042119500659
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