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Mathematics of Computation

ISSN 1088-6842(online) ISSN 0025-5718(print)



A new factorization technique using quadratic forms

Authors: D. H. Lehmer and Emma Lehmer
Journal: Math. Comp. 28 (1974), 625-635
MSC: Primary 10A25; Secondary 10-04, 10B05
MathSciNet review: 0342458
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Abstract: The paper presents a practical method for factoring an arbitrary N by representing N or $\lambda N$ by one of at most three quadratic forms: $\lambda N = {x^2} - D{y^2},\lambda = 1, - 1,2,D = - 1, \pm 2, \pm 3, \pm 6$. These three forms appropriate to N, together with inequalities for y, are given for all N prime to 6. Presently available sieving facilities make the method quite effective and economical for numbers N having 20 to 25 digits. Four examples arising from aliquot series are discussed in detail.

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Keywords: Factorization, primality, binary quadratic forms, representation
Article copyright: © Copyright 1974 American Mathematical Society