Skip to main content
SpringerLink
Log in

Paul Erdős and Egyptian Fractions

  • Chapter
Book cover Erdős Centennial

Part of the book series: Bolyai Society Mathematical Studies ((BSMS,volume 25))

Abstract

One of Paul Erdős’ earliest mathematical interests was the study of so-called Egyptian fractions, that is, finite sums of distinct fractions having numerator 1. In this note we survey various results in this subject, many of which were motivated by Erdős’ problems and conjectures on such sums. This note complements the excellent treatment of this topic given by A. Schinzel in 2002.1

This is a preview of subscription content, log in via an institution to check access.

Access this chapter

Log in via an institution
Chapter
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD 149.00
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
Hardcover Book
USD 199.99
Price excludes VAT (USA)
  • Durable hardcover edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. M. H. Ahmadi and M. N. Bleicher, On the conjectures of Erdős and Straus, and Sierpiński on Egyptian fractions, J. Math. Stat. Sci., 7 (1998), 169–185.

    MATH  MathSciNet  Google Scholar 

  2. A. V. Aho and N. J. A. Sloane, Some doubly exponential sequences, Fib. Quart. 11 (1973), 429–438.

    Google Scholar 

  3. P. J. van Albada and J. H. van Lint, Reciprocal bases for the integers, Amer. Math. Monthly 70 (1963), 170–174.

    Article  MATH  MathSciNet  Google Scholar 

  4. E. J. Barbeau, Compute challenge corner: Problem 477: A brute force program, J. Rec. Math, 9 (1976/77), p. 30.

    Google Scholar 

  5. E. J. Barbeau, Expressing one as a sum of odd reciprocals: comments and a bibliography, Crux Mathematicorum 3 (1977), 178–181.

    Google Scholar 

  6. M. N. Bleicher and P. Erdős, The number of distinct subsums of Σ Ni=1 1/i, Math. Comp 29 (1975), 29–42.

    MATH  MathSciNet  Google Scholar 

  7. M. N. Bleicher and P. Erdős, Denominators of unit fractions, J. Number Th, 8 (1976), 157–168.

    Article  MATH  Google Scholar 

  8. M. N. Bleicher and P. Erdős, Denominators of unit fractions II, Illinois J. of Math. 20 (1976), 598–613.

    MATH  Google Scholar 

  9. Robert Breusch, A special case of Egyptian fractions, solution to Advanced Problem 4512, Amer. Math. Monthly 61, (1954), 200–201.

    Article  MathSciNet  Google Scholar 

  10. T. C. Brown and V. Rödl, Monochromatic solutions to equations with unit fractions, Bull. Australian Math. Soc. 43 (1991), 387–392.

    Article  MATH  Google Scholar 

  11. N. Burshtein, On distinct unit fractions whose sum equals 1, Discrete Math. 5 (1973), 201–206.

    Article  MATH  MathSciNet  Google Scholar 

  12. N. Burshtein, Improving solutions of Σ ki=1 1/xi=1 with restrictions as required by Barbeau respectively by Johnson, Discrete Math. 306 (2006), 1438–1439.

    Article  MATH  MathSciNet  Google Scholar 

  13. N. Burshtein, An improved solution of Σ ki=1 1/xi=1 with restrictions as required by Barbeau respectively by Johnson, Discrete Math. 306 (2006), 1438–1439.

    Article  MATH  MathSciNet  Google Scholar 

  14. J. W. S. Cassels, On the representation of integers as the sums of distinct summands taken from a fixed set, Acta Sci. Math. Szeged 21 (1960), 111–124.

    MATH  MathSciNet  Google Scholar 

  15. Y.-G. Chen, On a conjecture of Erdős, Graham and Spencer, J. Number Th. 119 (2006), 307–314.

    Article  MATH  Google Scholar 

  16. Y.-G. Chen and M. Tang, On the elementary symmetric functions of 1, … 1/n, Amer. Math. Monthly 119 (2012), 862–867.

    Article  MATH  MathSciNet  Google Scholar 

  17. E. S. Croot III, On unit fractions with denominators from short intervals, Acta Arith. 99, (2001), 99–114.

    Article  MATH  MathSciNet  Google Scholar 

  18. E. S. Croot III, On some questions of Erdős and Graham about Egyptian fractions, Mathematika 46 (1999), 359–372.

    Article  MATH  MathSciNet  Google Scholar 

  19. E. S. Croot III, On a coloring conjecture about unit fractions, Ann. of Math. 157 (2003), 545–556.

    Article  MATH  MathSciNet  Google Scholar 

  20. E. S. Croot III, Sums of the form 1/x k1 modulo a prime, Integers 4, (2004), A20 6.

    MathSciNet  Google Scholar 

  21. D. R. Curtiss, On Kellogg’s Diophantine problem, Amer. Math. Monthly 29 (1922), 380–387.

    Article  MathSciNet  Google Scholar 

  22. C. Elsholtz, Sums of k unit fractions, Trans. Amer. Math. Soc. 353 (2001), 3209–3227.

    Article  MATH  MathSciNet  Google Scholar 

  23. P. Erdős, Beweis eines Satzes von Tschebyschef (in German), Acta Litt. Sci. Szeged 5, (1932), 194–198.

    Google Scholar 

  24. P. Erdős, Egy Kürschák-féle elemi számelméleti tétel általánosítása (Generalization of an elementary number-theoretic theorem of Kürschák, in Hungarian), Mat. Fiz, Lapok 39 (1932), 17–24.

    Google Scholar 

  25. P. Erdős, Az 1/x1 + 1/x2 + … + 1/xn=a/b egyenlet egész számú megoldásairól (On a Diophantine equation, in Hungarian), Mat. Lapok 1 (1950) 192–210.

    MathSciNet  Google Scholar 

  26. P. Erdős and R. L. Graham, Old and New Problems and Results in Combinatorial Number Theory, Mono. No. 28 de L’Enseignement Math., Univ. Geneva (1980) 128 pp.

    Google Scholar 

  27. P. Erdős and I. Niven, On certain variations of the harmonic series, Bull. Amer. Math. Soc., 51 (1945), 433–436.

    Article  MathSciNet  Google Scholar 

  28. P. Erdős and R. Rado, Intersection theorems for systems of sets, J. London Math. Soc. 35 (1960), 85–90.

    Article  MathSciNet  Google Scholar 

  29. J.-H. Fang and Y.-G. Chen, On a conjecture of Erdős, Graham and Spencer II, Disc. Appl. Math. 156 (2008), 2950–2958.

    Article  MATH  MathSciNet  Google Scholar 

  30. C. Friedman, Sums of divisors and Egyptian fractions, J. Num. Th 44 (1993), 328–339.

    Article  MATH  Google Scholar 

  31. Richard J. Gillings, Mathematics in the Time of the Pharaohs, 1972, MIT Press, Dover reprint ISBN 0-486-24315-X.

    Google Scholar 

  32. R. L. Graham, A theorem on partitions, J. Australian Math. Soc. 4 (1963), 435–441.

    Article  Google Scholar 

  33. R. L. Graham, On finite sums of unit fractions, Proc. London Math. Soc. 14 (1964), 193–207.

    Article  MATH  MathSciNet  Google Scholar 

  34. R. L. Graham, On finite sums of reciprocals of distinct n thpowers, Pacific J. Math. 14 (1964), 85–92.

    Article  MATH  MathSciNet  Google Scholar 

  35. S. Guo, A counterexample to a conjecture of Erdős, Graham and Spencer, The Electronic Journal of Combinatorics 15.43 (2008): 1.

    Google Scholar 

  36. R. K. Guy, Unsolved Problems in Number Theory, Third edition, Springer, New York, 2004, 437 pp.

    Book  Google Scholar 

  37. Chao Ko, Chi Sun and S. J. Chang, On equations 4/n = 1/x + 1/y + 1/z, Acta Sci. Natur. Szechuanensis, 2 (1964), 21–35.

    Google Scholar 

  38. J. Kürschák. A harmonikus sorról (On the harmonic series, in Hungarian), Mat. Fiz, Lapok 27 (1918), 288–300.

    Google Scholar 

  39. Greg Martin, Dense Egyptian fractions, Trans. Amer. Math. Soc. 351 (1999), 3641–3657.

    Article  MATH  MathSciNet  Google Scholar 

  40. Greg Martin, Denser Egyptian fractions, Acta Arith. 95 (2000), 231–260.

    MATH  MathSciNet  Google Scholar 

  41. R. Nowakowski, Unsolved Problems, 1989–1999, Amer. Math. Monthly, 106, 959–962.

    Google Scholar 

  42. M. R. Obláth, Sur l’equation diophantienne 4/n = 1/x1 + 1/x2 + 1/x3, Mathesis 59 (1950), 308–316.

    MATH  MathSciNet  Google Scholar 

  43. Gay Robins and Charles Shute, The Rhind Mathematical Papyrus: an ancient Egyptian text, Dover, New York, 1987, 88pp.

    MATH  Google Scholar 

  44. J. Sander, On 4/n = 1/x + 1/y + 1/z and Rosser’s sieve, Acta Arith., 59 (1991), 183–204.

    MATH  MathSciNet  Google Scholar 

  45. C. Sándor, On a problem of Erdős, J. Number Th. 63 (1997), 203–210.

    Article  MATH  Google Scholar 

  46. C. Sándor, On the number of solutions of the Diophantine equation Σ ni=1 1/xi=1 , Period. Math. Hungar. 47 (2003), no. 1–2, 215–219.

    Article  MATH  MathSciNet  Google Scholar 

  47. A. Schinzel, Erdős’ work on finite sums of unit fractions, in Paul Erdős and His Mathematics, vol. I, Springer, Berlin, 2002, pp. 629–636.

    Google Scholar 

  48. I. Shparlinski, On a question of Erdős and Graham, Archiv der Math. 78 (2002), 445–448.

    Article  MATH  MathSciNet  Google Scholar 

  49. W. Sierpiński, Sur les décompositions de nombres rationnels en fractions primaires, Mathesis, 65 (1956), 16–32.

    MATH  MathSciNet  Google Scholar 

  50. B. M. Stewart, Sums of distinct divisors, Amer. J. Math. 76 (1954), 779–785.

    Article  MATH  MathSciNet  Google Scholar 

  51. R. C. Vaughan, On a problem of Erdős, Straus and Schinzel, Mathematica. 17 (1970), 193–198.

    MATH  MathSciNet  Google Scholar 

  52. M. D. Vose, Egyptian fractions, Bull. London Math. Spc., 17, (1985), 21–24.

    Article  MATH  MathSciNet  Google Scholar 

  53. W. A. Webb, Rationals not expressible as the sum of three unit fractions, Elemente der Math. 29, (1974), 1–6.

    MATH  Google Scholar 

  54. W. A. Webb, On 4/n = 1/x + 1/y + 1/z, Proc. Amer. Math. Soc. 25 (1970), 578–584.

    MATH  MathSciNet  Google Scholar 

  55. H. S. Wilf, Reciprocal bases for the integers, Bull. Amer. Math. Soc. 67 (1961), p. 456.

    Article  MathSciNet  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Computer Science and Engineering, University of California, San Diego, La Jolla, CA, 92093-0404, USA

    Ronald L. Graham

Authors
  1. Ronald L. Graham
    View author publications

    You can also search for this author in PubMed Google Scholar

Editor information

Editors and Affiliations

  1. Department of Computer Science, Eötvös Lóránd University, Pázmány P. sétány 1/c, Budapest, 1117, Hungary

    László Lovász

  2. Alfréd Rényi Institute of Mathematics, Hungarian Academy of Sciences, Reáltanoda u. 13–15, Budapest, 1053, Hungary

    Imre Z. Ruzsa  & Vera T. Sós  & 

Rights and permissions

Reprints and permissions

Copyright information

© 2013 János Bolyai Mathematical Society and Springer-Verlag

About this chapter

Cite this chapter

Graham, R.L. (2013). Paul Erdős and Egyptian Fractions. In: Lovász, L., Ruzsa, I.Z., Sós, V.T. (eds) Erdős Centennial. Bolyai Society Mathematical Studies, vol 25. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-39286-3_9

Download citation

Publish with us

Policies and ethics

Search

Navigation