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

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Erdős Centennial

Part of the book series: Bolyai Society Mathematical Studies ((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 d │ n 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.

The authors would like to thank Enrique Treviño and the anonymous referee for their helpful suggestions. The second author was supported in part by NSF grant DMS-1001180.

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Authors and Affiliations

  1. Department of Mathematics, University of Georgia, Athens, GA, 30602, USA

    Paul Pollack

  2. Department of Mathematics, Dartmouth College, Hanover, NH, 03755, USA

    Carl Pomerance

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  1. Paul Pollack
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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  & 

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Pollack, P., Pomerance, C. (2013). Paul Erdős and the Rise of Statistical Thinking in Elementary Number Theory. 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_19

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