Below are the first 10 and last 10 pages of uncorrected machine-read text (when available) of this chapter, followed by the top 30 algorithmically extracted key phrases from the chapter as a whole.

Intended to provide our own search engines and external engines with highly rich, chapter-representative searchable text on the opening pages of each chapter.
Because it is UNCORRECTED material, please consider the following text as a useful but insufficient proxy for the authoritative book pages.

Do not use for reproduction, copying, pasting, or reading; exclusively for search engines.

OCR for page R1

TABLES IN THE
THEORY OF NUMBERS

OCR for page R1

OCR for page R1

BULLETIN
US£ DULY
OF THE
NATIONAL RESEARCH COUNCIL
NUMBER 105 FEBRUARY, 1941
DIVISION OF PHYSICAL SCIENCES
COMMITTEE ON MATHEMATICAL TABLES AND AIDS TO COMPUTATION
RAYMOND CLARE ARCHIBALD, Chairman
REPORT 1
Report of the Subcommittee on Section F: Theory of Numbers
GUIDE TO
TABLES IN THE
THEORY OF NUMBERS
BY
DERRICK HENRY LEHMER
\\
PROPERTY OF
SEfflH nrPi
NRC LIBRARY
PUBLISHED BY THE NATIONAL RESEARCH COUNCIL
NATIONAL ACADEMY OF SCIENCES
WASHINGTON, D. C.
1941

OCR for page R1

4
n
C »

OCR for page R1

COMMITTEE ON MATHEMATICAL TABLES AND AIDS
TO COMPUTATION
December 1940
Executive Committee
Professor R. C. ARCHIBALD, chairman, Brown University, Providence, Rhode
Island
Professor H. BATEMAN, California Institute of Technology, Pasadena, Cali-
fornia
Doctor L. J. COMRIE, Scientific Computing Service Ltd., 23 Bedford Square,
London, W.C.I, England
Professor H. T. DAVIS, Department of Mathematics, Northwestern Univer-
sity, Evanston, Illinois
Doctor W. J. ECKERT, director Nautical Almanac, U.S. Naval Observatory,
Washington
Mister J. S. ELSTON, Travelers Life Insurance Co., Hartford, Connecticut
Professor D. H. LEHMER, Department of Mathematics, University of Cali-
fornia, Berkeley, California
Professor S. S. WILKS, Department of Mathematics, Princeton University,
Princeton, New Jersey
Professor S. H. CALDWELL, Department of Electrical Engineering, Massa-
chusetts Institute of Technology, Cambridge, Massachusetts
Professor W. G. COCHRAN, Iowa State College of Agriculture and Mechanic
Arts, Ames, Iowa
Professor A. T. CRAIG, State University of Iowa, Iowa City, Iowa
Professor C. EISENHART, Agronomy Building, University of Wisconsin, Madi-
son, Wisconsin
Professor J. D. ELDER, Department of Mathematics, University of Michi-
gan, Ann Arbor, Michigan
Doctor L. GOLDBERG, Harvard College Observatory, Cambridge, Massa-
chusetts
Professor P. W. KETCHTJM, Department of Mathematics, University of Illi-
nois, Urbana, Illinois
Miss C. M. KRAMPE, U.S. Naval Observatory, Washington
Professor T. KUBOTA, T6hoku Imperial University, Sendai, Japan. Repre-
sentative of the National Research Council of Japan
Doctor A. N. LOWAN, Work Projects Administration, 35 E. 58th St.,
New York City
Doctor J. C. P. MILLER, Department of Applied Mathematics, University of
Liverpool, Liverpool, England
Doctor W. A. SHEWHART, Bell Telephone Laboratories, 463 West Street,
New York City
[v]

OCR for page R1

COMMITTEE
Doctor G. R. STIBITZ, Bell Telephone Laboratories, 463 West Street, New
York City
Mister J. S. THOMPSON, Mutual Benefit Life Insurance Co., Newark, New
Jersey
Professor I. A. TRAVIS, Moore School of Electrical Engineering, University
of Pennsylvania, Philadelphia, Pennsylvania
Classification of Tables,
A. Arithmetic. Mathematical Con-
stants
B. Powers
C. Logarithms
D. Circular Functions
E. Hyperbolic and Exponential
Functions
Professor DAVIS, chairman
Professor ELDER
Professor KETCHUM
Doctor LOWAN
F. Theory of Numbers
Professor LEHMER
G. Higher Algebra
Professor LEHMER
and Subcommittees
L. Higher Mathematical Functions
M. Integrals
Professor BATEMAN
N. Interest and Investment
O. Life Insurance
Mister ELSTON, chairman
Mister THOMPSON
•
P. Engineering
Q. Astronomy
Doctor ECKERT, chairman
Doctor GOLDBERG
Miss KRAMPE
H. Numerical Solution of Equations
J. Summation of Series
L
K.
Finite Differences
Statistics
Professor WILKS, chairman
Professor COCHRAN
Professor CRAIG
Professor EISENHART
Doctor SHEWHART
!R. Geodesy
S. Physics
T. Chemistry
CT. Navigation
Z.
Calculating Machines and Me-
chanical Computation
Doctor COMRIE, chairman
Professor CALDWELL
Professor LEHMER
Doctor MILLER
Doctor STIBITZ
Professor TRAVIS
[vi]

OCR for page R1

FOREWORD
This Report of the Subcommittee on the Theory of Numbers is the first
one to be published by the Committee. In broad outline it exhibits the general
plan for all Reports in the series. In adopting this plan the Committee desires
to make clear that the Reports are being prepared primarily for scholars and
others active in scientific work throughout the world.
It is recognized however that, even in the United States, those using this
and later Reports may often be greatly hampered through lack of library facili-
ties. Because of this fact the bibliographic section of our present Report is more
extended than it might otherwise have been. Information is there given con-
cerning the holdings, in libraries of the United States and Canada, of the books
and pamphlets to which reference has been made. It may thus frequently be
found that a desired publication is near at hand. The Union List of Serials
furnishes similar information concerning serials containing tables and errata
in the tables discussed. But these errata are often in periodicals and books
somewhat difficult of access. Hence it was finally decided, as a matter of
policy, to list all known errata in tables surveyed. It seemed desirable in this
Report to group all errata together in a special section; in later Reports,
however, they may be included in the bibliographic section.
Authorities for all errata are indicated, and in the case of errata previously
printed the sources are given. Professor Lehmer's personal contributions in
this connection are very notable; where no authority is mentioned it is to be
assumed that the discovery of the errata was due to him. The reader who
makes checks will find that the reprinting in this Report of all known pub-
lished errata has two other great advantages over giving mere references to
sources, namely, that they are combined with other known unpublished errata,
and that source notations (often difficult of comprehension, except by the ex-
pert) have been made to conform with those of this Report.
It is a pleasure to acknowledge notable courtesies extended to us. Doctor
Arthur Beer, of the University of London Observatory, placed at our disposal
for this Report the late Doctor JIri Kavan's manuscript lists of errata in the
tables of Chernac, Goldberg, and Inghirami, discovered while preparing his re-
markable Factor Tables. Hence it may well be assumed that our lists of errata
in the cases of the two latter are complete. The same may be said of the Gifford
tables errata supplied by Doctor L. J. Comrie of London, the great authority
on all that pertains to table making.
The directions for the use of this Report in the contents and index ought to
render all of its material readily available.
The undersigned will be happy to hear from anyone who may notice in
this Report any omission, inaccuracy, or misstatement. It is not expected that
another Report will be ready for publication before 1942.
R. C. ARCHIBALD
December 1940 Chairman of the Committee
[vii]

OCR for page R1

OCR for page R1

STYLE, NOTATIONS, AND ABBREVIATIONS
In the series of Reports of this Committee there will be references to Serials,
Books and Pamphlets, and Manuscripts. It seemed desirable to be able readily
to determine where such material might be consulted. The serial holdings of
libraries of the United States and Canada are indicated in the Union List of
Serials and its Supplements, of which a new and enlarged edition, in a single
alphabet, is now in an advanced stage of preparation. The present custodian
of all manuscripts is stated. From the hundreds of Libraries listed in the Union
List of Serials the following 37 were selected, representing Canada and 22
states. These Libraries are as follows:
CPT California Institute of Technology, Pasadena
CU University of California, Berkeley
CaM McGill University, Montreal
CaTU University of Toronto
CoU University of Colorado, Boulder
CtY Yale University, New Haven, Conn.
DLC Library of Congress, Washington
ICJ John Crerar Library, Chicago, I11.
ICU University of Chicago
IEN Northwestern University, Evanston, I11.
IU University of Illinois, Urbana
InU University of Indiana, Bloomington
laAS Iowa State College, Ames
laU University of Iowa, Iowa City
KyU University of Kentucky, Lexington
MdBJ The Johns Hopkins University, Baltimore, Md.
MB Boston Public Library
MCM Massachusetts Institute of Technology, Cambridge, Mass.
MH Harvard University, Cambridge, Mass.
MiU University of Michigan, Ann Arbor
MnU University of Minnesota, Minneapolis
MoU University of Missouri, Columbia, Mo.
NhD Dartmouth College, Hanover, N. H.
NjP Princeton University, Princeton, N. J.
NIC Cornell University, Ithaca, N. Y.
NN New York Public Library
NNC Columbia University, New York, N. Y.
NRU University of Rochester, Rochester, N. Y.
NcD Duke University, Durham, N. C.
OCU University of Cincinnati
[ix]

OCR for page R1

STYLE, NOTATIONS, AND ABBREVIATIONS
OU Ohio State University, Columbus
PEL Lehigh University, Bethlehem, Pa.
PU University of Pennsylvania, Philadelphia, Pa.
RPB Brown University, Providence, R. I.
TxU University of Texas, Austin
WvU West Virginia University, Morgantown
WU University of Wisconsin, Madison
In the case of all Books and Pamphlets mentioned in our Reports, the hold-
ings of each of these Libraries are indicated in the Bibliographies. It may be
noted that the forms of titles of Serials in our Bibliographies follow the forms
in the newest Union List. Transliterations of Russian and Ukrainian names,
and titles of articles and periodicals, are in accordance with Manual of Foreign
Languages, third edition, Washington, 1936.
A few of the Abbreviations used in the Reports are as follows:
Abt. = Abteilung
Acad. = Academy, Academie, etc.
Akad. = Akademiia, Akademija, Akademie, etc.
Am. = America, American
App. = Appendix
Ass. = Association
Ast. = Astronomy, Astronomische, etc.
Biog. = Biography
Br. = British
Bull. = Bulletin
Cambridge = Cambridge, England
col. = column
d. = der, die, di, etc.
Dept. = Department
ed. = edited, edition
f. = for, fur
Fis. = Fisiche
Gesell. = Gesellschaft
heraus. = herausgegeben
Inst. = Institute (English or French)
Int. = International
1st. = Istituto (Italian)
Jahresb. = Jahresbericht
Jn. = Journal
Kl. = Klasse
Mat. = Matematica, Matematica, Matematica, etc.
Math. = Mathematics, Mathematical, Mathematische, etc.
Mo. = Monthly
n.s. = new series

OCR for page R1

STYLE, NOTATIONS, AND ABBREVIATIONS
Nach. = Nachrichten
Nat. = National
Natw. = Naturwissenschaften
no. = number
nos. = numbers
opp. = opposite
p. = page, pages
Phil. = Philosophical
Phys. = Physical, Physics, Physik, Physikalische
Proc. = Proceedings
Rev. = Review
s. = series
Sci. = Science, scientifique
Sitzungsb. = Sitzungsberichte
So. = Society
Sup. = Superiore, Superieure, etc.
Trans. = Transactions
transl. = translated, translation
u. = und
Univ. = University, Universidade, Universite, Universita, etc.
v. = volume, volumes, voor
Wiss. = Wissenschaf ten
z. = zur
Z. = Zeitschrift

OCR for page R1

OCR for page R1

CONTENTS
FOREWORD vii
STYLE, NOTATIONS, AND ABBREVIATIONS ix
INTRODUCTION 1
I—DESCRIPTIVE SURVEY: F. Theory of Numbers 5
a. Perfect and amicable numbers and their generalizations 5
b. Numerical functions 6
1. Euler's totient function and its inverse, sum, and generaliza-
tions 6
2. Sum and number of divisors, and allied functions 8
3. Mobius' inversion function and its sum 9
4. The quotients of Fermat and Wilson 10
5. Sums of products of consecutive integers 10
6. Numerical recurring series 10
7. Triangular numbers 11
c. Periodic decimals 11
d. The binomial congruence 12
1. Primitive roots 13
2. Exponents and residue-indices 15
3. Powers and indices 17
4. Solutions of special binomial congruences 19
5. Higher residues 22
6. Converse of Fermat's theorem 23
e. Factor tables 24
1. Ordinary factor tables 24
2. Tables of factors of numbers of special form 27
f. Lists of primes and tables of their distribution 37
1. Consecutive primes 37
2. Primes of special form 43
g. Tables for facilitating factoring and identifying primes 47
h. Tables of solutions of linear Diophantine equations and congru-
ences 49
i. Congruences of the second degree 50
1. Solutions of quadratic congruences 50
2. Quadratic residues and characters and their distribution 52
3. Linear forms dividing x* — D-f 53
j. Diophantine equations of the second degree 54
1. The Pell equations. x?-Df =

CONTENTS
k. Non-binomial congruences of degree ^3 62
1. Diophantine equations of degree > 2 63
m. Diophantine continued fractions 65
n. Non-linear forms, their classes and class numbers 68
0. Tables related to cyclotomy 72
p. Tables related to algebraic number theory 75
q. Tables related to additive number theory 77
1. Theory of partitions 77
2. Goldbach's problem 79
3. Waring's problem 81
II—BIBLIOGRAPHY 85
III—ERRATA 127
INDEX 173