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77. Miscellaneous Investigations.—The foregoing articles (§§ 24-76) give an outline of what may be called the analytical theory of numbers, which is mainly the work of the 19th century, though many of the researches of Lagrange, Legendre and Gauss, as well as all those of Euler, fall within the 18th. But after all, the germ of this remarkable development is contained in what is only a part of the original Diophantine analysis, of which, beyond question, Fermat was the greatest master. The spirit of this method is still vigorous in Euler; but the appearance of Gauss’s Disquisitiones arithmeticae in 1801 transformed the whole subject, and gave it a new tendency which was strengthened by the discoveries of Cauchy, Jacobi, Eisenstein and Dirichlet. In recent times Edouard Lucas revived something of the old doctrine, and it can hardly be denied that the Diophantine method is the one that is really germane to the subject. Even the strange results obtained from elliptic and modular functions must somehow be capable of purely arithmetical proof without the use of infinite series. Besides this, the older arithmeticians have announced various theorems which have not been proved or disproved, and made a beginning of theories which are still in a more or less rudimentary stage. As examples of the latter may be mentioned the partition of numbers (see, below), and the resolution of large numbers into their prime factors.

The general problem of partitions is to find all the integral solutions of a set of linear equations 𝑐𝑖𝑥𝑖＝𝑚𝑖 with integral coefficients, and fewer equations than there are variables. The solutions may be further restricted by other conditions—for instance, that all the variables are to be positive. This theory was begun by Euler: Sylvester gave lectures on the subject, of which some portions have been preserved; and various results of great generality have been discovered by P. A. MacMahon. The author last named has also considered Diophantine inequalities, a simple problem in which is “to enumerate all the solutions of 7𝑥⪖13𝑦 in positive integers.”

The resolution of a given large number into its prime factors is still a problem of great difficulty, and tentative methods have to be applied. But a good deal has been done by Seelhoff, Lucas, Landry, A. J. C. Cunningham and Lawrence to shorten the calculation, especially when the number is given in, or can be reduced to, some particular form.

It is well known that Fermat was led to the erroneous conjecture (he did not affirm it) that 2𝑚+1 is a prime whenever 𝑚 is a power of 2. The first case of failure is when 𝑚＝32; in fact 232+1 ≡ 0 (mod 641). Other known cases of failure are 𝑚＝2𝑛, with 𝑛＝6, 12, 23, 26 respectively; at the same time, Eisenstein asserted that he had proved that the formula 2𝑚+1 included an infinite number of primes. His proof is not extant; and no other has yet been supplied. Similar difficulties are encountered when We examine Mersenne’s numbers, which are those of the form 2𝑝−1, with 𝑝 a prime; the known cases for which a Mersenne number is prime correspond to 𝑝＝2, 3, 5, 7, 13, 17, 19, 31, 61.

A perfect number is one which, like 6 or 28, is the sum of its aliquot parts. Euclid proved that 2𝑝−1 (2𝑝−1) is perfect when (2𝑝−1) is a prime: and it has been shown that this formula includes all perfect numbers which are even. It is not known whether any odd perfect numbers exist or not.

Friendly numbers (numeri amicabiles) are pairs such as 220, 284, each of which is the sum of the aliquot parts of the other. No general rules for constructing them appear to be known, but several have been found, in a more or less methodical way.

78. In conclusion it may be remarked that the science of (q.v.) has now reached a stage when all its definitions, processes and results are demonstrably independent of any theory of variable or measurable quantities such as those postulated in geometry and mathematical physics; even the notion of a limit may be dispensed with, although this idea, as well as that of a variable, is often convenient. For the application of arithmetic to geometry and analysis, see.

.—W. H. and G. E. Young, The Theory of Sets of Points (Cambridge, 1906; contains bibliography of theory of aggregates); P. Bachmann, Zahlentheorie (Leipzig, 1892; the most complete treatise extant); Dirichlet-Dedekind, Vorlesungen über Zahlentheorie (Braunschweig, 3rd and 4th ed., 1879, 1894); K. Hensel, Theorie der algebraischen Zahlen (Leipzig, 1908); H. J. S. Smith, Report on the Theory of Numbers (Brit. Ass. Rep., 1859–1863, 1865, or Coll. Math. Papers, vol. i.); D. Hilbert, “Bericht über die Theorie der algebraischen Zahlkörper” (in Jahresber. d. deutschen Math.-Vereinig., vol. iv., Berlin, 1897); Klein-Fricke, Elliptische Modulfunctionen (Leipzig, 1890–1892); H. Weber, Elliptische Functionen u. algebraische Zahlen (Braunschweig, 1891). Extensive bibliographies will be found in the Royal Society’s Subject Index, vol. i. (Cambridge, 1908) and ''Encycl. d. math. Wissenschaften'', vol. i. (Leipzig, 1898).

NUMBERS, BOOK OF, the fourth book of the Bible, which takes its title from the Latin equivalent of the Septuagint . While the English version follows the Septuagint directly in speaking of Genesis, Exodus, Leviticus and Deuteronomy, it follows the Vulgate in speaking of Numbers. Since this book describes the way in which an elaborate census of Israel was taken on two separate occasions, the first at Sinai at the beginning of the desert wanderings and the second just before their close on the plains of Moab, the title is quite appropriate. The name given to it in modern Hebrew Bibles from its fourth word Bemidhbar (“In the desert”) is at least equally appropriate. The other title in use among the Jews, Vayyidhabber (“And he said”), is simply the first word of the book and has no reference to its contents.

Numbers is the first part of the second great division of the Hexateuch. In the first three books we are shown how God raised up for Himself a chosen people and how the descendants of Israel on entering at Sinai into a solemn league and covenant with Yahweh (Jehovah) became a separate nation, a peculiar people. In the last three books we are told what happened to Israel between the time it entered into this solemn covenant and its settlement in the Promised Land under the successor of Moses. Yet, though thus part of a larger whole, the book of Numbers has been so constructed by the Redactor as to form a self-contained division of that whole.

The truth of this statement is seen by comparing the first verse of the book with the last. The first is as evidently meant to serve as an introduction to the book as the last is to serve as its conclusion. This is not to say, however, that the book is all of a piece, or written on a systematic plan. On the contrary, no book in the Hexateuch gives such an impression of incoherence, and in none are the different strata which compose the Hexateuch more distinctly discernible.

It is noteworthy that the problems of Hexateuchal criticism are gradually changing their character, as one after another of the main contentions of Biblical scholars regarding the date and authorship of the Hexateuch passes out of the list of debatable questions into that of acknowledged facts. No competent scholars now question the existence, hardly any one the relative dates, of J, E, and P. In Numbers one can tell almost at a glance which parts belong to P, the Priestly Code, and which to JE, the narrative resulting from the combination of the Judaic work of the Yahwist with the Ephraimitic work of the Elohist. The main difficulty in Numbers is to determine to which stratum of P certain sections should be assigned.

The first large section (i.—x. 10) is wholly P, and the last eleven chapters are also P with the exception of two or three paragraphs in chap. xxxii., While the intervening portion is mainly P with the exception of three important episodes and two or three others of less importance. The three main episodes are those of the twelve spies, the rebellion of Korah, Dathan and Abiram, and Balaam’s mission to Balak. The last is the only one even of these three in which there is nothing belonging to P. Another passage which we may here mention is one where the elements of JE can be readily separated and assigned to their respective authors, viz. chaps. xi. and xii. It is generally agreed that to E belongs the passage describing the outpouring of the Spirit on Eldad and Medad and the remarkable prayer of Moses in xi. 29, “Would God that all the Lord’s people were prophets that the Lord