A sieve method for factoring numbers of the form
Author:
Daniel Shanks
Journal:
Math. Comp. 13 (1959), 7886
MSC:
Primary 65.00; Secondary 10.00
MathSciNet review:
0105784
Fulltext PDF Free Access
References 
Similar Articles 
Additional Information
 [1]
L. E. Dickson, History of the Theory of Numbers, Stechert, New York, 1934, v. 1, Ch. XVI. For example, Euler (1752) gave See also D.H. Lehmer, Guide to Tables in the Theory of Numbers, National Research Council, Washington, D. C., 1941, p. 3132 and p. 45.
 [2]
The most extensive table of all the prime factors of ( up to ) is the unpublished table of J. W. Wrench, Jr. See UMT 1, MTAC, v. I, 1943, p. 26. Recently a 704 program by the author in collaboration with Dr. Wrench raised this limit to 50,000 for a table of the greatest prime factor. However, we now consider that type of program (with trial divisions) to be superseded by the present sieve method.
 [3]
G. H. Hardy & J. E. Littlewood, ``Partitio numerorum III: On the expression of a number as a sum of primes,'' Acta Math., v. XLIV, 1923, p. 48.
 [4]
A. E. Western, ``Note on the number of primes of the form ,'' Cambridge Phil. Soc., Proc., v. XXI, 1922, p. 108109. Western assumes following Cunningham, who omits . The correct value of is 1200.
 [5]
Fortune, June, 1958, p. 140.
 [6]
John
Todd, A problem on arc tangent relations, Amer. Math. Monthly
56 (1949), 517–528. MR 0031496
(11,159d)
 [7]
S.
D. Chowla and John
Todd, The density of reducible integers, Canadian J. Math.
1 (1949), 297–299. MR 0030558
(11,14d)
 [8]
John
Todd, Table of Arctangents of Rational Numbers, National
Bureau of Standards, Applied Mathematics Series, No. 11, United States
Government Printing Office, Washington, D. C., 1951. MR 0040796
(12,750e)
 [10]
Cyrus
Colton MacDuffee, An Introduction to Abstract Algebra, John
Wiley & Sons, Inc., New York, 1940. MR 0003591
(2,241b)
 [1]
 L. E. Dickson, History of the Theory of Numbers, Stechert, New York, 1934, v. 1, Ch. XVI. For example, Euler (1752) gave See also D.H. Lehmer, Guide to Tables in the Theory of Numbers, National Research Council, Washington, D. C., 1941, p. 3132 and p. 45.
 [2]
 The most extensive table of all the prime factors of ( up to ) is the unpublished table of J. W. Wrench, Jr. See UMT 1, MTAC, v. I, 1943, p. 26. Recently a 704 program by the author in collaboration with Dr. Wrench raised this limit to 50,000 for a table of the greatest prime factor. However, we now consider that type of program (with trial divisions) to be superseded by the present sieve method.
 [3]
 G. H. Hardy & J. E. Littlewood, ``Partitio numerorum III: On the expression of a number as a sum of primes,'' Acta Math., v. XLIV, 1923, p. 48.
 [4]
 A. E. Western, ``Note on the number of primes of the form ,'' Cambridge Phil. Soc., Proc., v. XXI, 1922, p. 108109. Western assumes following Cunningham, who omits . The correct value of is 1200.
 [5]
 Fortune, June, 1958, p. 140.
 [6]
 John Todd, ``A problem on arc tangent relations,'' American Math Monthly, v. LVI, 1949, p. 517528. MR 0031496 (11:159d)
 [7]
 S. D. Chowla & J. Todd, ``The density of reducible integers,'' Canadian Jour. of Math, v. I, 1949, p. 297299. The table of to has many errors. It indicates . A mimeographed errata sheet later circulated stated , but the correct value is 1467. MR 0030558 (11:14d)
 [8]
 John Todd, Table of Arctangents of Rational Numbers, NBS, Applied Math. Series 11, Washington, D. C., 1951. MR 0040796 (12:750e)
 [10]
 C. C. MacDuffee, An Introduction to Abstract Algebra, Wiley, New York, 1940, p. 193202. MR 0003591 (2:241b)
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Additional Information
DOI:
http://dx.doi.org/10.1090/S00255718195901057842
PII:
S 00255718(1959)01057842
Article copyright:
© Copyright 1959
American Mathematical Society
