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  • Mathematical Proceedings of the Cambridge Philosophical Society, Volume 76, Issue 3
  • November 1974, pp. 521-528

Theorems on factorization and primality testing

  • J. M. Pollard (a1)
  • DOI:
  • Published online: 24 October 2008

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)J. Hartmanis and R. E. Stearns On the computational complexity of algorithms. Trans. Amer. Math. Soc. 117 (1965), 285306.

(4)E. R. Berlekamp Factoring polynomials over large finite fields. Math. Comp. 24 (1970), 713735.

(7)J. M. Pollard An algorithm for testing the primality of any integer. Bull. London Math. Soc. 3 (1971), 337340.

(9)J. W. Cooley and J. W. Tukey An algorithm for the machine computation of complex Fourier series. Math. Comp. 19 (1965), 297301.

(10)A. Schonnage and V. Strassen Schnelle Multiplikation grosser Zahlen. Computing(Arch. Elektron Rechnen), 7 (1971), 281292.

(11)J. M. Pollard The fast Fourier transform in a finite field. Math. Comp. 25 (1971), 365374.

(14)John Brillhart and J. L. Selfridge Some factorizations of n ± 1 and related results. Math. Comp. 21 (1967), 8796.

(15)Alan Borning . Some results for k! ± 1 and 2.3.5 … xp + 1. Math. Comp. 26 (1972), 567570.

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