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INTRODUCTION The theory of numbers is a peculiar subject, being at once a purely deduc- tive and a largely experimental science. Nearly every classical theorem of im- portance (proved or unproved) has been discovered by experiment, and it is safe to say that man will never cease to experiment with numbers. The results of a great many experiments have been recorded in the form of tables, a large number of which have been published. The theory suggested by these experi- ments, when once established, has often made desirable the production of fur- ther tables of a more fundamental sort, either to facilitate the application of the theory or to make possible further experiments. It is not surprising that there exists today a great variety of tables concerned with the theory of num- bers. Most of these are scattered widely through the extensive literature on the subject, comparatively few being "tables" in the usual sense of the word, i.e., appearing as separately published volumes. This report is intended to pre- sent a useful account of such tables. It is written from the point of view of the research worker rather than that of the historian, biographer, or biblio- phile. Another peculiarity of the theory of numbers is the fact that many of its devotees are not professional mathematicians but amateurs with widely vary- ing familiarity with the terminology and the symbolism of the subject. In describing tables dealing with those subjects most apt to attract the amateur, some care has been taken to minimize technical nomenclature and notation, and to explain the terminology actually used, while for subjects of the more advanced type no attempt has been made to explain anything except the con- tents of the table, since no one unfamiliar with the rudiments of the subject would have any use for such a table. There are three main parts of the report: I. A descriptive account of existing tables, arranged according to the topi- cal classification of tables in the theory of numbers indicated in the Contents. II. A bibliography arranged alphabetically by authors giving exact refer- ences to the source of the tables referred to in Part I. III. Lists of errata in the tables. Brief comment on each Part may be given here. Part I is not so much a description of tables as a description of what each table contains. It is assumed that the research worker is not interested in the size of page or type, or the exact title of column headings, or even the notation or arrangement of the table in so far as these features do not affect the practi- cal use of the table. Since there is very little duplication of tables the user is seldom in a position to choose this or that table on such grounds as one does with tables of logarithms, for example. However, it is a well known fact that [1]

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INTRODUCTION many tables in the theory of numbers have uses not contemplated by the au- thor of the table. A particular table is mentioned as many times and in as many places as there are, to the writer's best knowledge, practical uses to which it may be put. The practical viewpoint was taken in deciding what constitutes a table in the theory of numbers, and what tables are worthy of inclusion. Tables vary a great deal in the difficulty of their construction, from completely trivial tables of the natural numbers to such tables as those of the factors of 22"+l, one ad- ditional entry in which may require months of heavy computing. In general, old obscure tables, which have been superseded by more extensive and more easily available modern tables, have been omitted. Short tables, every entry in which is easily computed, merely illustrating some universal theorem and with no other conceivable use, have also been omitted. The present century with its improved mechanical computing devices has seen the development of many practical methods for finding isolated entries in number theory tables. In spite of this, many old tables, any single entry of which is now almost easier to compute than to consult, have been included in the report since they serve as sources of statistical information about the function or the problem con- sidered. Most of those tables prior to 1918 which have not been included here are mentioned in Dickson's exhaustive three-volume History of the Theory of Num- bers. Under DICKSON 14 of the Bibliography in the present report will be found supplementary references to the exact places in this history where these tables are cited, arranged according to our classification of tables in the theory of numbers. For example the entry d4 v. 1, ch. I, no. 54: ch. Ill, no. 235. means that two tables of class di (solutions of special binomial congruences) are cited in vol. 1, chapter I, paper 54, and chapter III, paper 235. For a fuller description of many of the older tables cited in this report the reader is referred to Cayley's valuable and interesting report on tables in the theory of numbers, CAYLEY 7. The writer has tried to include practically all tables appearing since 1918, and on the whole has probably erred on the side of inclusion rather than ex- clusion. A few remarks about nomenclature in Part I may be made here. The unqualified word "number" in this report means a positive integer and is denoted generally by n. The majority of tables have numbers for arguments. In saying that a table gives values of /(«) for «^1000 it is meant that /(l),/(2), • • • ,/(1000) are tabulated. If the table extends from 500 to 10000 at intervals of 100 we write « = 500(100) 10 000. A great many tables have prime numbers as arguments, however. Throughout the report the letter p designates a prime which may be ^1, >1, or >2 according to the context. [2]

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INTRODUCTION To say that the function / is tabulated for each prime of the first million as argument, we write "/(/>) is given for /><10*." Occasionally it is convenient to use the words decade, century, chiliad, or myriad to indicate an interval of 10,100,1000, or 10 000 numbers. Frequently the arguments of a table are num- bers (or primes) of some special form, such as a multiple of 6 plus 1. In cases of this sort we use such notations as n = 6* +1< 1000, or 1013 g p = 6x-1 ^ 10 007. In Part I, tables are described as though entirely free from errors, with the exception of an occasional remark on the reliability of certain general utility tables where the user has some choice in his selection. The uninterrupted description of tables in Part I is made possible by Part II, where one may find complete bibliographic references, arranged by au- thors, to the one or more places in which each of the tables mentioned in Part I appears. The various reprints, editions, or reproductions of a table are distinguished by subscripts on the number following the author's name. Thus, for example, CAYLEY 6i refers to the original table, while CAYLEY 62 refers to the same table as reprinted in his Collected Mathematical Papers. In Part I these distinctions are rarely used, but in Part III they are convenient. Following each reference in Part II (except CAYLEY 7, CUNNINGHAM 40-42, DICKSON 14, and D. H. LEHMER 11) there appears in square brackets, [ ], an indication of the kind (or kinds) of tables contained in the work referred to, together with their location. The small boldface letters, with or without sub- scripts, refer to the classification of tables given in the Contents. The page numbers following any particular classification letter not only locate the table for the reader in possession of the publication, but give an idea of the extent of the table to the reader who may not have it, and will be of help in ordering photostats or a microfilm of the table from a distant library. In further expla- nation of the notation used, it should be noted that the absence of page num- bers after a particular letter indicates that practically the whole work is devoted to a table, or tables, of this particular class. An asterisk placed on a classification letter indicates that errors in the corresponding table are cited in Part III. When a publication has tables capable of several classifications and errors are cited in all tables, an asterisk is placed after the closing bracket. The following examples with explanations should make these notations clear. [fi] A list of consecutive primes occupying practically the whole work re- ferred to. [d,, 14-29: df, 30-35: fi] Tables of primitive roots on pages 14-29. Solu- tions of special binomial congruences on pages 30-35, with errors cited in Part III. Lists of consecutive primes on practically every page. As already mentioned, Part III gives errata in certain tables mentioned in Part I and is arranged alphabetically according to authors. The list of errors given for any particular table is not necessarily complete. Tables mentioned in Part I but not in Part III, so that no asterisk appears after the reference in Part II, may contain errors, either unknown to the writer or too trivial to be [3]

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INTRODUCTION of any practical interest. In cases where errors have been found by others, the authority for the corrections, together with a reference to their source in case they have been published, is generally given in parentheses after the errors in question. In no case has an error been listed which was printed in connection with the table itself. The writer has seen nearly all the tables mentioned in this report in at least one of the following libraries: Brown University Mathematical Library, Providence, R. I. Princeton University Mathematical Library, Princeton, N. J. University of California Library, Berkeley, California Cambridge University Library, Cambridge, England The Science Library, London, England. The writer's best thanks are due to the chairman of this Committee, Pro- fessor Archibald, whose unceasing efforts and expert knowledge have added greatly to the accuracy and reliability of Part II, and to Mr. S. A. Joffe, who has read all the manuscript and proof with great care, and has given many valuable suggestions. The writer also wishes to acknowledge the frequent assistance of Miss M. C. Shields of the Princeton University Mathematical Library. Dr. N. G. W. H. Beeger has kindly supplied information about lists of primes, Mr. H. J. Woodall, information about works of Cunningham, and Dr. S. Perlis, information concerning the tables in the University of Chicago dissertations. The part of the work on this report which was done abroad was made possible by a fellowship of the John Simon Guggenheim Memorial Foundation. [4]