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Sums of divisors, perfect numbers, and factoring

Authors:

Jeffrey ShallitAuthors Info & Claims

STOC '84: Proceedings of the sixteenth annual ACM symposium on Theory of computing

Pages 183 - 190

https://doi.org/10.1145/800057.808680

Published: 01 December 1984 Publication History

Abstract

Let N be a positive integer, and let σ(N) denote the sum of the positive integral divisors of N. We show computing σ(N) is equivalent to factoring N in the following sense: there is a random polynomial time algorithm that, given σ(N), produces the prime factorization of N, and σ(N) can be easily computed given the factorization of N.

We show that the same sort of result holds for σ_k(N), the sum of the k-th powers of divisors of N.

We give three new examples of problems that are in Gill's complexity class BPP: {perfect numbers}, {multiply perfect numbers}, and {amicable pairs}. These are the first “natural” candidates for BPP - RP.

References

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Leonard M. Adleman, Carl Pomerance, and Robert S. Rumely, "On distinguishing prime numbers from composite numbers", Ann. Math.117 (1983) 173-206.

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Eric Bach, "Discrete logarithms and factoring", to appear.

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Leonard Carlitz, "The arithmetic of polynomials in a Galois field", Amer. Journ. Math.,54 (1932) 39-50.

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R. D. Carmichael, "A table of multiply perfect numbers", Bull. Amer. Math. Soc.13 (1907) 383-386.

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J. D. Dixon, "Asymptotically fast factorization of integers", Math. Comp.36 (1981) 255-260.

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Genesis, xxxii, 14.

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John Gill, "Computational complexity of probabilistic Turing machines", Siam J. Comput.6 (1977) 675-695.

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Richard K. Guy, "How to factor a number", Proc. Fifth Manitoba Conf. on Numerical Math., Winnipeg, 1976, 49-89.

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Bernhard Hornfeck and Eduard Wirsing, "Uber die Haufigkeit vollkommener Zahlen", Math. Annalen133 (1957) 431-438.

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Elvin J. Lee and Joseph S. Madachy, "The history and discovery of amicable numbers", Journ. Rec. Math.5 (1972) 77-93, 153-173, 231-249.

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Douglas L. Long, "Random equivalence of factorization and computation of orders", to appear, Theoretical Computer Science.

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Gary Miller, "Riemann's hypothesis and tests for primality", J. Comp. System Sci.13 (1976) 300-317.

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Daniel Shanks, "Class number, a theory of factorization, and genera", Proceedings of Symposia in Pure Mathematics, V. 20 (1969 Number Theory Institute), American Mathematical Society (1971) 415-440.

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H. J. J. te Riele, "Perfect numbers and aliquot sequences", in Computational Methods in Number Theory, Amsterdam Math. Centre Tracts 154 (1982) 141-157.

Cited By

Hinman PZachos S(2006)Probabilistic machines, oracles, and quantifiersRecursion Theory Week10.1007/BFb0076220(159-192)Online publication date: 17-Sep-2006
https://doi.org/10.1007/BFb0076220
Bernstein D(2005)Factoring into coprimes in essentially linear timeJournal of Algorithms10.1016/j.jalgor.2004.04.00954:1(1-30)Online publication date: 1-Jan-2005
https://dl.acm.org/doi/10.1016/j.jalgor.2004.04.009
Zachos S(2005)Probabilistic quantifiers, adversaries, and complexity classes : An overviewStructure in Complexity Theory10.1007/3-540-16486-3_112(383-400)Online publication date: 2-Jun-2005
https://doi.org/10.1007/3-540-16486-3_112
Show More Cited By

Index Terms

Sums of divisors, perfect numbers, and factoring
1. Mathematics of computing
  1. Mathematical analysis
    1. Numerical analysis
      1. Number-theoretic computations
  2. Probability and statistics
    1. Probabilistic algorithms
    2. Probabilistic reasoning algorithms
      1. Markov-chain Monte Carlo methods
      2. Sequential Monte Carlo methods
2. Theory of computation
  1. Models of computation
    1. Computability
      1. Turing machines

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cover image ACM Conferences

STOC '84: Proceedings of the sixteenth annual ACM symposium on Theory of computing

December 1984

547 pages

ISBN:0897911334

DOI:10.1145/800057

Chairman:
Richard DeMillo

Copyright © 1984 ACM.

Permission to make digital or hard copies of all or part of this work for personal or classroom use is granted without fee provided that copies are not made or distributed for profit or commercial advantage and that copies bear this notice and the full citation on the first page. Copyrights for components of this work owned by others than ACM must be honored. Abstracting with credit is permitted. To copy otherwise, or republish, to post on servers or to redistribute to lists, requires prior specific permission and/or a fee. Request permissions from [email protected]

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Association for Computing Machinery

New York, NY, United States

Publication History

Published: 01 December 1984

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Cited By

Hinman PZachos S(2006)Probabilistic machines, oracles, and quantifiersRecursion Theory Week10.1007/BFb0076220(159-192)Online publication date: 17-Sep-2006
https://doi.org/10.1007/BFb0076220
Bernstein D(2005)Factoring into coprimes in essentially linear timeJournal of Algorithms10.1016/j.jalgor.2004.04.00954:1(1-30)Online publication date: 1-Jan-2005
https://dl.acm.org/doi/10.1016/j.jalgor.2004.04.009
Zachos S(2005)Probabilistic quantifiers, adversaries, and complexity classes : An overviewStructure in Complexity Theory10.1007/3-540-16486-3_112(383-400)Online publication date: 2-Jun-2005
https://doi.org/10.1007/3-540-16486-3_112
Adleman L(1994)Algorithmic number theory-the complexity contributionProceedings of the 35th Annual Symposium on Foundations of Computer Science10.1109/SFCS.1994.365702(88-113)Online publication date: 20-Nov-1994
https://dl.acm.org/doi/10.1109/SFCS.1994.365702
Adleman LMcCurley K(1987)Open Problems in Number Theoretic ComplexityDiscrete Algorithms and Complexity10.1016/B978-0-12-386870-1.50020-4(237-262)Online publication date: 1987
https://doi.org/10.1016/B978-0-12-386870-1.50020-4
Bach EShallit J(1985)Factoring with cyclotomic polynomialsProceedings of the 26th Annual Symposium on Foundations of Computer Science10.1109/SFCS.1985.24(443-450)Online publication date: 21-Oct-1985
https://dl.acm.org/doi/10.1109/SFCS.1985.24

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