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Theorems on factorization and primality testing

  • J. M. Pollard (a1)
Abstract

1. Introduction. This paper is concerned with the problem of obtaining theoretical estimates for the number of arithmetical operations required to factorize a large integer n or test it for primality. One way of making these problems precise uses a multi-tape Turing machine (e.g. (1), although we require a version with an input tape). At the start of the calculation n is written in radix notation on one of the tapes, and the machine is to stop after writing out the factors in radix notation or after writing one of two symbols denoting ‘prime’ or ‘composite’. There are, of course, other definitions which could be used; but the differences between these are unimportant for our purpose.

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(1)Hartmanis J. and Stearns R. E. On the computational complexity of algorithms. Trans. Amer. Math. Soc. 117 (1965), 285306.
(2)Lehmer D. H. Computer technology applied to the theory of numbers. Studies in Number Theory (LeVeque W. J., Editor) (Prentice-Hall, Englewood Cliffs N.J. 1969), 117151.
(3)Burgess D. A. On character sums and primitive roots. Proc. London Math. Soc. 12 (1962), 179192.
(4)Berlekamp E. R. Factoring polynomials over large finite fields. Math. Comp. 24 (1970), 713735.
(5)Shanks D. Class number, a theory of factorisation, and genera. 1969 Number Theory Institute, Proc. Sympos. Pure Math. 20, Amer. Math. Soc., Providence, Rhode Island (1970), 415440.
(6)Lehman R. S. Factoring large integers. To appear in Math. Comp.
(7)Pollard J. M. An algorithm for testing the primality of any integer. Bull. London Math. Soc. 3 (1971), 337340.
(8)Good I. J. The interaction algorithm and practical Fourier analysis. J. Roy. Statist. Soc. Ser. B 20 (1958), 361372.
(9)Cooley J. W. and Tukey J. W. An algorithm for the machine computation of complex Fourier series. Math. Comp. 19 (1965), 297301.
(10)Schonnage A. and Strassen V. Schnelle Multiplikation grosser Zahlen. Computing(Arch. Elektron Rechnen), 7 (1971), 281292.
(11)Pollard J. M. The fast Fourier transform in a finite field. Math. Comp. 25 (1971), 365374.
(12)Knuth D. E. The art of computer programming, Volume 2 (revised edition 1971), Semi-numerical algorithms. Addison-Wesley, New York, 1971.
(13)Norton Karl K. Numbers with small prime factors and the least kth power non-residue. Memoirs of the Amer. Math. Soc. 106, Amer. Math. Soc., Providence, Rhode Island, 1971.
(14)Brillhart John and Selfridge J. L. Some factorizations of n ± 1 and related results. Math. Comp. 21 (1967), 8796.
(15)Borning Alan. Some results for k! ± 1 and 2.3.5 … xp + 1. Math. Comp. 26 (1972), 567570.
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Mathematical Proceedings of the Cambridge Philosophical Society
  • ISSN: 0305-0041
  • EISSN: 1469-8064
  • URL: /core/journals/mathematical-proceedings-of-the-cambridge-philosophical-society
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