Advertisement

Paul Erdős and the Rise of Statistical Thinking in Elementary Number Theory

  • Paul Pollack
  • Carl Pomerance
Part of the Bolyai Society Mathematical Studies book series (BSMS, volume 25)

Abstract

It might be argued that elementary number theory began with Pythagoras who noted two-and-a-half millennia ago that 220 and 284 form an amicable pair. That is, if s(n) denotes the sum of the proper divisors of n (“proper divisor” means dn and 1 ≤ d < n), then
$$s(220) = 284\quad and\quad s(284) = 220.$$
When faced with remarkable examples such as this it is natural to wonder how special they are. Through the centuries mathematicians tried to find other examples of amicable pairs, and they did indeed succeed. But is there a formula? Are there infinitely many? In the first millennium of the common era, Thâbit ibn Qurra came close with a formula for a subfamily of amicable pairs, but it is far from clear that his formula gives infinitely many examples and probably it does not.

Keywords

Acta Arith Asymptotic Density Density Zero Statistical Thinking Abundant Number 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. au][1]
    W. R. Alford, A. Granville, and C. Pomerance, There are infinitely many Carmichael numbers, Ann. of Math. (2) 139 (1994), 703–722.CrossRefMathSciNetGoogle Scholar
  2. [2]
    F. Behrend, Über numeri abundantes. I, Sitzgsber. Akad. Berlin (1932), 322–328.Google Scholar
  3. [3]
    -, Über numeri abundantes. II, Sitzgsber. Akad. Berlin (1933), 289–293.Google Scholar
  4. [4]
    E. Catalan, Propositions et questions diverses, Bull. Soc. Math. France 16 (1888), 128–129.MathSciNetGoogle Scholar
  5. [5]
    H. Davenport, Über numeri abundantes, Sitzgsber. Akad. Berlin (1933), 830–837.Google Scholar
  6. [6]
    M. Deléglise, Bounds for the density of abundant integers, Experiment. Math. 7 (1998), 137–143.CrossRefzbMATHMathSciNetGoogle Scholar
  7. [7]
    L. E. Dickson, Theorems and tables on the sum of the divisors of a number, Q.J. Math. 44 (1913), 264–296.zbMATHGoogle Scholar
  8. [8]
    P. Erdős, On primitive abundant numbers, J. London Math. Soc. 10 (1935), 49–58.Google Scholar
  9. [9]
    -, On the density of some sequences of numbers, J. London Math. Soc. 10 (1935), 120–125.CrossRefGoogle Scholar
  10. [10]
    -, On the normal number of prime factors of p — 1 and some related problems concerning Euler’s φ-function, Quart. J. Math. Oxford Ser. 6 (1935), 205–213.CrossRefGoogle Scholar
  11. [11]
    -, On the density of some sequences of numbers, II, J. London Math. Soc. 12 (1937), 7–11.Google Scholar
  12. [12]
    -, On the density of some sequences of numbers, III, J. London Math. Soc. 13 (1938), 119–127.CrossRefGoogle Scholar
  13. [13]
    -, On the converse of Fermat’s theorem, Amer. Math. Monthly 56 (1949), 623–624.CrossRefMathSciNetGoogle Scholar
  14. [14]
    -, On almost primes, Amer. Math. Monthly 57 (1950), 404–407.CrossRefMathSciNetGoogle Scholar
  15. [15]
    -, On amicable numbers, Publ. Math. Debrecen 4 (1955), 108–111.MathSciNetGoogle Scholar
  16. [16]
    -, On perfect and multiply perfect numbers, Ann. Mat. Pura Appl. (4) 42 (1956), 253–258.CrossRefGoogle Scholar
  17. [17]
    -, On pseudoprimes and Carmichael numbers, Publ. Math. Debrecen 4 (1956), 201–206.MathSciNetGoogle Scholar
  18. [18]
    -, Remarks on number theory, II. Some problems on the σ function, Acta Arith. 5 (1959), 171–177.MathSciNetGoogle Scholar
  19. [19]
    On the sumΣd|2 n-1d-1, Actes du Congrés International des Mathé-maticiens (Nice, 1970), Tome 3, and Israel J. Math. 9 (1971), 43–48.Google Scholar
  20. [20]
    -, Über die Zahlen der form σ (n) — n undn-φ(n), Elem. Math. 28 (1973), 83–86.MathSciNetGoogle Scholar
  21. [21]
    -, On asymptotic properties of aliquot sequences, Math. Comp. 30 (1976), no. 135, 641–645.Google Scholar
  22. [22]
    P. Erdős, A. Granville, C. Pomerance, and C. Spiro, On the normal behavior of the iterates of some arithmetic functions, Analytic number theory (Allerton Park, IL, 1989), Progr. Math., vol. 85, Birkhäuser Boston, Boston, MA, 1990, pp. 165–204.CrossRefGoogle Scholar
  23. [23]
    P. Erdős and R. R. Hall, On the values of Euler’s φ-function, Acta Arith. 22 (1973), 201–206.MathSciNetGoogle Scholar
  24. [24]
    -, Distinct values of Euler’s φ-function, Mathematika 23 (1976), 1–3.CrossRefMathSciNetGoogle Scholar
  25. [25]
    P. Erdős, P. Kiss, and C. Pomerance, On the prime divisors of Mersenne numbers, Acta Arith. 57 (1991), 267–281.MathSciNetGoogle Scholar
  26. [26]
    P. Erdős and C. Pomerance, On the number of false witnesses for a composite number, Math. Comp. 46 (1986), 259–279.MathSciNetGoogle Scholar
  27. [27]
    P. Erdős, C. Pomerance, and E. Schmutz, Carmichael’s lambda function, Acta Arith. 58 (1991), 363–385.MathSciNetGoogle Scholar
  28. [28]
    P. Erdős and G. J. Rieger, Ein Nachtrag über befreundete Zahlen, J. Reine Angew. Math. 273 (1975), 220.MathSciNetGoogle Scholar
  29. [29]
    P. Erdős and S. S. Wagstaff, Jr., The fractional parts of the Bernoulli numbers, Illinois J. Math. 24 (1980), 104–112.MathSciNetGoogle Scholar
  30. [30]
    P. Erdös and A. Wintner, Additive arithmetical functions and statistical independence, Amer. J. Math. 61 (1939), 713–721.CrossRefMathSciNetGoogle Scholar
  31. [31]
    K. Ford, The distribution of totients, Ramanujan J. 2 (1998), 67–151. (Updated version on the author’s web page.)CrossRefzbMATHMathSciNetGoogle Scholar
  32. [32]
    K. Ford, Sieving by very thin sets of primes and Pratt trees with missing primes, preprint, 2012, arXiv:1212.3498 au][math.NT], IMRN, to appear.Google Scholar
  33. [33]
    K. Ford, F. Luca, and C. Pomerance, Common values of the arithmetic functions ϕ and σ, Bull. Lond. Math. Soc. 42 (2010), 478–488.CrossRefzbMATHMathSciNetGoogle Scholar
  34. [34]
    K. Ford and P. Pollack, On common values of φ(n) and σ(m), I, Acta Math. Hungarica 133 (2011), 251–271.CrossRefzbMATHMathSciNetGoogle Scholar
  35. [35]
    -, On common values of φ(n) and σ(m), II, Algebra Number Theory 6 (2012), 1669–1696.CrossRefzbMATHMathSciNetGoogle Scholar
  36. [36]
    A. Granville and C. Pomerance, Two contradictory conjectures concerning Carmichael numbers, Math. Comp. 71 (2001), 883–908.CrossRefMathSciNetGoogle Scholar
  37. [37]
    N. Harland, The number of iterates of the Carmichael lambda function required to reach 1, preprint, 2012, arXiv:1203.4791 [math.NT].Google Scholar
  38. [38]
    B. Hornfeck, Zur Dichte der Menge der vollkommenen Zahlen, Arch. Math. (Basel) 6 (1955), 442–443.CrossRefzbMATHMathSciNetGoogle Scholar
  39. [39]
    B. Hornfeck and E. Wirsing, Über die Häufigkeit vollkommener Zahlen, Math. Ann. 133 (1957), 431–438.CrossRefzbMATHMathSciNetGoogle Scholar
  40. [40]
    H. J. Kanold, Über die Dichten der Mengen der vollkommenen und der befreundeten Zahlen, Math Z. 61 (1954), 180–185.CrossRefzbMATHMathSciNetGoogle Scholar
  41. [41]
    -, Über die Verteilung der vollkommenen Zahlen und allgemeinerer Zahlenmengen, Math. Ann. 132 (1957), 442–450.CrossRefzbMATHMathSciNetGoogle Scholar
  42. [42]
    M. Kobayashi, On the density of abundant numbers, Ph.D. thesis, Dartmouth College, 2010.Google Scholar
  43. [43]
    M. Kobayashi, P. Pollack, and C. Pomerance, On the distribution of sociable numbers, J. Number Theory 129 (2009), 1990–2009.CrossRefzbMATHMathSciNetGoogle Scholar
  44. [46]
    H. Maier and C. Pomerance, On the number of distinct values of Euler’s ϕ-function, Acta Arith. 49 (1988), 263–275.zbMATHMathSciNetGoogle Scholar
  45. [47]
    J. Perrott, Sur une proposition empirique énoncée au Bulletin, Bull. Soc. Math. France 17 (1889), 155–156.zbMATHMathSciNetGoogle Scholar
  46. [48]
    S. S. Pillai, On some functions connected with φ(n), Bull. Amer. Math. Soc. 35 (1929), 832–836.CrossRefzbMATHMathSciNetGoogle Scholar
  47. [49]
    P. Pollack, A remark on sociable numbers of odd order, J. Number Theory 130 (2010), no. 8, 1732–1736.CrossRefzbMATHMathSciNetGoogle Scholar
  48. [50]
    -, On the greatest common divisor of a number and its sum of divisors, Michigan Math. J. 60 (2011), no. 1, 199–214.CrossRefzbMATHMathSciNetGoogle Scholar
  49. [51]
    -, Quasi-amicable numbers are rare, J. Integer Seq. 14 (2011), no. 5, Article 11.5.2, 13 pages.Google Scholar
  50. [52]
    P. Pollack and C. Pomerance, Prime-perfect numbers, Integers 12A (2012), article A14, 19 pages.Google Scholar
  51. [53]
    C. Pomerance, On the distribution of amicable numbers, J. Reine Angew. Math. 293/294 (1977), 217–222.MathSciNetGoogle Scholar
  52. [54]
    -, On the distribution of amicable numbers. II, J. Reine Angew. Math. 325 (1981), 183–188.zbMATHMathSciNetGoogle Scholar
  53. [55]
    -, On the distribution of pseudoprimes, Math. Comp. 37 (1981), 587–593.Google Scholar
  54. [57]
    G. J. Rieger, Bemerkung zu einem Ergebnis von Erdős über befreundete Zahlen, J. Reine Angew. Math. 261 (1973), 157–163.zbMATHMathSciNetGoogle Scholar
  55. [58]
    H. Salie, Über die Dichte abundanter Zahlen, Math. Nachr. 14 (1955), 39–46.CrossRefzbMATHMathSciNetGoogle Scholar
  56. [59]
    I. J. Schoenberg, Über die asymptotische Verteilung reeller Zahlen mod 1, Math. Z. 28 (1928), 171–199.CrossRefzbMATHMathSciNetGoogle Scholar
  57. [60]
    -, On asymptotic distributions of arithmetical functions, Trans. Amer. Math. Soc. 39 (1936), 315–330.CrossRefMathSciNetGoogle Scholar
  58. [61]
    S. S. Wagstaff, Jr., Divisors of Mersenne numbers, Math. Comp. 83 (1983), 385–397.CrossRefMathSciNetGoogle Scholar
  59. [62]
    C. R. Wall, Density bounds for the sum of divisors function, The theory of arithmetic functions (Proc. Conf., Western Michigan Univ., Kalamazoo, Mich., 1971), Lecture Notes in Math., vol. 251, Springer, Berlin, 1972, pp. 283–287.CrossRefGoogle Scholar
  60. [63]
    E. Wirsing, Bemerkung zu der Arbeit über vollkommene Zahlen, Math. Ann. 137 (1959), 316–318.CrossRefzbMATHMathSciNetGoogle Scholar
  61. [64]
    P. Zimmerman, Aliquot sequences, internet resource, http://www.loria.fr/ zimmerma/records/aliquot.html.Google Scholar

Copyright information

© János Bolyai Mathematical Society and Springer-Verlag 2013

Authors and Affiliations

  • Paul Pollack
    • 1
  • Carl Pomerance
    • 2
  1. 1.Department of MathematicsUniversity of GeorgiaAthensUSA
  2. 2.Department of MathematicsDartmouth CollegeHanoverUSA

Personalised recommendations