## A Monte Carlo method for factorization.(English)Zbl 0312.10006

The following simple method will find small prime factors $$p$$ of a number $$n$$ in $$O(p^{1/2})$$ arithmetical operations as opposed to the $$O(p)$$ operations required by trial division. Let $$x_0=2$$, $$x_{i+1}\equiv x_i^2-1\pmod n$$ (other similar sequences may be used). Generate in turn the pairs $$(x_j,x_{2j})$$, accumulating the product (mod $$n$$) of the numbers $$x_{2j}.x_j$$. At intervals, compute the g.c.d. of this product and $$n$$, which may give a factor, not necessarily prime. Probabilistic arguments predict that on average one should take about $$1.03\,p^{1/2}$$ steps to find a prime $$p$$; this is confirmed by experiment. The method has been further discussed by R. K. Guy [Proc. 5th Manitoba Conf. numer. Math., Winnipeg 1975, 49–89 (1976; Zbl 0338.10001)].

### MSC:

 11Y05 Factorization 11Y11 Primality 65C05 Monte Carlo methods

Zbl 0338.10001
Full Text:

### References:

 [1] M. C. Wunderlich and J. L. Selfridge,A Design for a Number Theory Package with an Optimized Trial Division Routine, Comm. A.C.M. 17,5 (May 1974), 272–276. · Zbl 0276.68025 [2] J. M. Pollard,Theorems on Factorization and Primality Testing, Proc. Camb. Phil. Soc. 76 (1974), 521–528. · Zbl 0294.10005 [3] D. E. Knuth,Seminumerical Algorithms: the Art of Computer Programming, Vol. 2, Addison-Wesley, 1969. · Zbl 0191.18001 [4] Michael A. Morrison and John Brillhart,A Method of Factoring and the Factorization of F 7, Math. Comp. 29,129 (1975), 183–206. · Zbl 0302.10010
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