# Lehmer sieve by dsppro

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```									Lehmer sieve                                                                                                                      1

Lehmer sieve
Lehmer sieves are mechanical devices that implement sieves in
number theory. Lehmer sieves are named for Derrick Norman
Lehmer and his son Derrick Henry Lehmer. The father was a
professor of mathematics at the University of California, Berkeley at
the time, and his son who followed in his footsteps, as a number
theorist and professor at Berkeley.

A sieve in general is intended to find the numbers which are
remainders when a set of numbers are divided by a second set.
Generally, they are used in finding solutions of diophantine equations
or to factor numbers. A Lehmer sieve will signal that such solutions
are found in a variety of ways depending on the particular
construction.

Construction
The first Lehmer sieve in 1926 was made using bicycle chains of
varying length, with rods at appropriate points in the chains. As the
chains turned, the rods would close electrical switches, and when all    A Lehmer sieve - a primitive digital computer once
the switches were closed simultaneously, creating a complete                used for finding primes and solving simple
diophantine equations.
electrical circuit, a solution had been found. This version could test
60 number combinations a second.

Built in 1932, a device using gears was shown at the Century of Progress Exposition in Chicago. These had gears
representing numbers, just as the chains had before, with holes. Holes left open were the remainders sought. When
the holes lined up, a light at one end of the device shone on a photocell at the other, which could stop the machine
allowing for the observation of a solution. This incarnation allowed checking of 5000 combinations a second.
In 1936, a version was built using 16 mm film instead of chains, with holes in the film instead of rods. Brushes
against the rollers would make electrical contact when the hole reached the top. Again, a full sequence of holes
created a complete circuit, indicating a solution.
Several Lehmer sieves are on display at the Computer History Museum. Since then, the same basic design has been
used to design sieves in integrated circuits or software.

References
• Lehmer, D. N. (1932), "Hunting big game in the theory of numbers" [1], Scripta Mathematica 1: 229–235.
• Lehmer, D. H. (1928), "The mechanical combination of linear forms", American Mathematical Monthly
(Mathematical Association of America) 35 (3): 114–121, doi:10.2307/2299504, JSTOR 2299504. Also online [2]
• Beiler, Albert H. (1964), Recreations in the Theory of Numbers, Dover, chap.XX,XXI.
Lehmer sieve                                                                                                    2

• Lehmer Sieves, by Dr. Michael R. Williams, Head Curator of The Computer History Museum [3]
• The Computer History Museum page about Lehmer Sieves [4]
• A modern Lehmer sieve [5]

References
[1]   http:/ / ed-thelen. org/ comp-hist/ Lehmer-NS03. html
[2]   http:/ / ed-thelen. org/ comp-hist/ Lehmer-NS-01. html
[3]   http:/ / ed-thelen. org/ comp-hist/ Mike-Williams-Lehmer. html
[4]   http:/ / www. computerhistory. org/ virtualvisiblestorage/ artifact_main. php?tax_id=01. 01. 06. 00
[5]   http:/ / groups. google. com/ group/ sci. math/ msg/ bbe87db07b43cf2f
Derrick Henry Lehmer                                                                                                     3

Derrick Henry Lehmer
Derrick Henry Lehmer

Born               February 23, 1905
Berkeley, California

Died               May 22, 1991 (aged 86)
Berkeley, California

Nationality        United States

Fields             Mathematics

Institutions       UC Berkeley

Alma mater         Brown University

Doctoral students Tom Apostol
Ronald Graham
Harold Stark
Peter J. Weinberger

Known for          Lehmer's polynomial
Lehmer matrix
Lehmer sieve
Lehmer–Schur algorithm
Lehmer's GCD algorithm
Lehmer code
Lehmer's conjecture
Lehmer number
Lehmer five
Lucas–Lehmer test
Lucas–Lehmer test for Mersenne
numbers
Lucas–Lehmer–Riesel test
Pocklington–Lehmer test
Lehmer random number generator
Lehmer mean

Derrick Henry "Dick" Lehmer (February 23, 1905 – May 22, 1991) was an American mathematician who refined
Édouard Lucas' work in the 1930s and devised the Lucas–Lehmer test for Mersenne primes. Lehmer's peripatetic
career as a number theorist, with he and his wife taking numerous types of work in the United States and abroad to
support themselves during the Great Depression, fortuitously brought him into the center of research into early
electronic computing.
Derrick Henry Lehmer                                                                                                         4

Early life
Lehmer was born in Berkeley, California, to Derrick Norman Lehmer, a professor of mathematics at the University
of California, Berkeley, and Clara Eunice Mitchell.
He studied physics and earned a Bachelor degree from UC Berkeley, and continued with graduate studies at the
University of Chicago.
He and his father worked together on Lehmer sieves.

Marriage
During his studies at Berkeley, Lehmer met Emma Markovna Trotskaia, a Russian student of his father's, who had
begun with work toward an engineering degree but had subsequently switched focus to mathematics, earning her
B.A. in 1928. Later that same year, Lehmer married Emma and, following a tour of Northern California and a trip to
Japan to meet Emma's family, they moved by car to Providence, Rhode Island, after Brown University offered him
an instructorship.

Career
Lehmer received a Master's degree and a Ph.D., both from Brown University, in 1929 and 1930, respectively; his
wife obtained a Master's degree in 1930 as well, coaching mathematics to supplement the family income, while also
helping her husband type his Ph.D. thesis, An Extended Theory of Lucas' Functions, which he wrote under Jacob
Tamarkin.

Movements during the Depression
Lehmer became a National Research Fellow, allowing him to take positions at the California Institute of Technology
from 1930 to 1931 and at Stanford University from 1931 to 1932. In the latter year, the couple's first child Laura was
born.
After being awarded a second National Research Fellowship, the Lehmers moved on to Princeton, New Jersey
between 1932 and 1934, where Dick spent a short time at the Institute for Advanced Study.
He worked at Lehigh University in Pennsylvania from 1934 until 1938. Their son Donald was born in 1934 while
Dick and Emma were at Lehigh.
The year 1938-1939 was spent in England on a Guggenheim Fellowship visiting both the University of Cambridge
and the University of Manchester, meeting G. H. Hardy, John Edensor Littlewood, Harold Davenport, Kurt Mahler,
Louis Mordell, and Paul Erdős. The Lehmers returned to America by ship with second child Donald just before the
beginning of the Battle of the Atlantic.
Lehmer continued at Lehigh University for the 1939-1940 academic year.

Settling down
In 1940, Lehmer accepted a position back at the mathematics department of UC Berkeley. At some point in his
career there, he developed the Linear congruential generator (pseudorandom number generator), which is frequently
referred to as a Lehmer random number generator. The Lehmers also assisted Harry Vandiver with his work on
Fermat's Last Theorem, computing many Bernoulli numbers required.
Lehmer was chairman of the Department of Mathematics at University of California, Berkeley from 1954 until 1957.
He continued working at UC Berkeley until 1972, the year he became professor emeritus.
Derrick Henry Lehmer                                                                                                       5

ENIAC involvement
From 1945-1946, Lehmer served on the Computations Committee at Aberdeen Proving Grounds in Maryland, a
group established as part of the Ballistics Research Laboratory to prepare the ENIAC for utilization following its
completion at the University of Pennsylvania's Moore School of Electrical Engineering; the other Computations
Committee members were Haskell Curry, Leland Cunningham, and Franz Alt. It was during this short tenure that the
Lehmers ran some of the first test programs on the ENIAC—according to their academic interests, these tests
involved number theory, especially sieve methods, but also pseudorandom number generation. When they could
arrange child care, the Lehmers spent weekends staying up all night running such problems, the first over the
Thanksgiving weekend of 1945. (Such tests were run without cost, since the ENIAC would have been left powered
on anyway in the interest of minimizing vacuum tube failures.) The problem run during the 3-day Independence Day
weekend of July 4, 1946, with John Mauchly serving as computer operator, ran around the clock without interruption
or failure. The following Tuesday, July 9, 1946, Lehmer delivered the talk "Computing Machines for Pure
Mathematics" as part of the Moore School Lectures, in which he introduced computing as an experimental science,
and demonstrated the wit and humor typical of his teaching lectures.
Lehmer would remain active in computing developments for the remainder of his career. Upon his return to
Berkeley, he made plans for building the California Digital Computer (CALDIC) with Paul Morton and Leland
Cunningham.

McCarthy era
In 1950, Lehmer was one of 31 University of California faculty fired after refusing to sign a loyalty oath, a policy
initiated by the Board of Regents of the State of California in 1950 during the Communist scare personified by
Senator Joseph McCarthy. Lehmer took a post as Director of the National Bureau of Standards' Institute for
Numerical Analysis (INA), working with the Standards Western Automatic Computer (SWAC). On October 17,
1952, the State Supreme Court proclaimed the oath unconstitutional, and Lehmer returned to Berkeley shortly
thereafter.

Later years
Lehmer continued to be active for many years and would certainly qualify as a dotagy, Paul Erdos's term for
someone active in their dotage. When John Selfridge was at Northern Illinois University he twice invited Lehmer
and Emma to spend a semester there. One year Selfridge arranged that Erdos and Lehmer taught a course together on
Research Problems in the Theory of Numbers. Lehmer taught the first eight weeks and then Erdos taught the
remainder. Erdos didn't often teach a course, and he said "You know it wasn't that difficult. The only problem was
being there."
Lehmer had quite a wit. On the occasion of the first Asilomar number theory conference, which became an annual
event (now called West Coast Number Theory), Lehmer, as the organizer, was inspecting the facilities of the
Asilomar Conference Grounds—basically a wooden building on the beach. Someone said they couldn't find a
blackboard and Lehmer spotted some curtains in the middle of the wall. Moving the curtains aside revealed a very
small blackboard, whereupon Lehmer said "Well, I guess we won't be doing any analytic number theory!"
Derrick Henry Lehmer                                                                                                            6

Lasting impact
In addition to his significant contributions to number theory algorithms for multiprecision integers, such as factoring,
Euclid's algorithm, long division, and proof of primality, he also formulated Lehmer's conjecture and participated in
the Cunningham project.

Combinatorics
His father Derrick Norman Lehmer, known mainly as a pioneer in number theory computing, also made major
contributions to combinatorial computing, having devised algorithms for efficiently generating all the permutations
on n elements He also developed two algorithms, rank(p) and unrank(k). Given a permutation p, if k = rank(p), then
p is the kth permutation. Given an integer k, unrank(k) is the kth permutation. D N Lehmer's method uses his
so-called factorial representation of the integer k. See section 5.1 in Permutation.
D. H. Lehmer continued his father's interest in combinatorial computing and in fact wrote the article "Machine tools
of Computation," which is chapter one in the book "Applied Combinatorial Mathematics," by Edwin Beckenbach,
1964. It describes methods for producing permutations, combinations etc. This was a uniquely valuable resource and
has only been rivaled recently by Volume 4 of Donald Knuth's series.

Death
Lehmer died in Berkeley on May 22, 1991.

• Brillhart, J. (1992). "Derrick Henry Lehmer" [1]. Acta Arithmetica 62: 207–213.
• Photo of Derrick Henry Lehmer [2]
• The Lehmers at Berkeley [3]
• Timeline: Summary of events of the Loyalty Oath Controversy 1949-54 [4]
• O'Connor, John J.; Robertson, Edmund F., "Derrick Henry Lehmer" [5], MacTutor History of Mathematics
archive, University of St Andrews.
• Interview with the Lehmers and others about their experiences at Princeton [6]
• Derrick Henry Lehmer [7] at the Mathematics Genealogy Project
• Alt, Franz L. (1972-07). "Archaeology of Computers—Reminiscences, 1945-1947" [8] (PDF). Communications of
the ACM (ACM) 15 (7): 693–694. doi:10.1145/361454.361528. ISSN 0001-0782.

References
[1]   http:/ / matwbn. icm. edu. pl/ ksiazki/ aa/ aa62/ aa6231. pdf
[2]   http:/ / www. numbertheory. org/ obituaries/ AA/ lehmer/ page0. html
[3]   http:/ / bancroft. berkeley. edu/ Exhibits/ Math/ intro. html
[4]   http:/ / sunsite. berkeley. edu/ uchistory/ archives_exhibits/ loyaltyoath/ symposium/ timeline/ short. html
[5]   http:/ / www-history. mcs. st-andrews. ac. uk/ Biographies/ Lehmer_Derrick. html
[6]   http:/ / infoshare1. princeton. edu/ libraries/ firestone/ rbsc/ finding_aids/ mathoral/ pmc12. htm
[7]   http:/ / genealogy. math. ndsu. nodak. edu/ id. php?id=4283
[8]   http:/ / delivery. acm. org/ 10. 1145/ 370000/ 361528/ p693-alt. pdf?key1=361528& key2=2909455511& coll=& dl=GUIDE&
CFID=15151515& CFTOKEN=6184618
Derrick Norman Lehmer                                                                                                          7

Derrick Norman Lehmer
Derrick Norman Lehmer

Born        July 27, 1867
Somerset, Indiana, United States

Died        September 8, 1938
Berkeley, California, United States

University of Chicago

Occupation Mathematician

Spouse(s)   Clara Eunice Mitchell

Children    Eunice (b. 1903), Helen (b. 1904), Derrick Henry (b. 1905), Stephen (b. 1907), Alice (b. 1911)

Parents     Derrick Fernstück Lehmer; Isabel Smith Perry

Derrick Norman Lehmer (27 July 1867 – 8 September 1938) was an American mathematician and number
theorist.
He was educated at the University of Nebraska, obtaining a bachelor's degree in 1893 and master's in 1896. Lehmer
was awarded his Ph.D. from the University of Chicago in 1900 for a thesis Asymptotic Evaluation of Certain
Totient-Sums under the supervision of E. H. Moore. He was appointed instructor in mathematics at the University of
California at Berkeley in 1900 and married Clara Eunice Mitchell on 12 July 1900 in Decatur, Illinois. He was
promoted to professor at Berkeley in 1918 and continued to teach there until retiring in 1937.
In 1903, he presented a factorization of Jevons' number (8,616,460,799) at the San Francisco Section of the
American Mathematical Society, December 19, 1903.[1][2]
He published tables of prime numbers and prime factorizations, reaching 10,017,000 by 1909.[3] He developed a
variety of mechanical and electro-mechanical factoring and computational devices, such as the Lehmer sieve, built
with his son Derrick Henry Lehmer.
He is also known for a reversible algorithm that assigns a Lehmer code to every permutation of size n. See section
5.1 in Permutation.
Derrick Norman Lehmer                                                                                                                                      8

Selected works
• "Arithmetical theory of certain Hurwitzian continued fractions". Proc Natl Acad Sci U S A 4 (8): 214-218. 1918.
PMC 1091449.
• "On Jacobi's extension of the continued fraction algorithm". Proc Natl Acad Sci U S A 4 (12): 360-364. 1918.
PMC 1091496.
• "On a new method of factorization". Proc Natl Acad Sci U S A 11 (1): 97-98. 1925. PMC 1085844.
• "A theorem on factorization". Bull. Amer. Math. Soc. 33 (1): 35-36. 1927. MR1561316.
• "Inverse ternary continued fractions". Bull. Amer. Math. Soc. 37 (8): 565-569. 1931. MR1562198.
• "On the enumeration of magic cubes". Bull. Amer. Math. Soc. 40 (12): 833-837. 1934. MR1562983.

Notes
[1] Lehmer, D.N., "A Theorem in the Theory of Numbers" (http:/ / projecteuclid. org/ DPubS/ Repository/ 1. 0/ Disseminate?view=body&
id=pdf_1& handle=euclid. bams/ 1183419373), read before the San Francisco Section of the American Mathematical Society, December 19,
1903.
[2] William Stanley Jevons had written in his Principles of Science, p. 123, "Can the reader say what two numbers multiplied together will
produce the number 8616460799 ? I think it unlikely that anyone but myself will ever know." Lehmer added "I think that the number has been
resolved before, but I do not know by whom."
[3] Lehmer, D. N., Factor table for the first ten millions containing the smallest factor of every number not divisible by 2, 3, 5, or 7 between the
limits 0 and 10017000, Carnegie institution of Washington. Publication no. 105, 1909.

References
• Albert H. Beiler, Recreations in the theory of numbers, Dover, 1964; chap.XX

• O'Connor, John J.; Robertson, Edmund F., "Derrick Norman Lehmer" (http://www-history.mcs.st-andrews.ac.
uk/Biographies/Lehmer_Derrick_N.html), MacTutor History of Mathematics archive, University of St
Andrews.
• Derrick Norman Lehmer (http://genealogy.math.ndsu.nodak.edu/id.php?id=5865) at the Mathematics
Genealogy Project
Article Sources and Contributors                                                                                                                                                            9

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Image Sources, Licenses and Contributors                                                                                                                                                    10

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