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TABLES IN THE THEORY OF NUMBERS

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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

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4 n C »

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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]

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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]

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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]

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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]

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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

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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

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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 = OCR for page R1
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