NUMBERS, THEORY OF. The theory of numbers is in fact the science of whole or integer numbers, and its moat general problem is : " Given any equation whatsoever involving two or more unknown quantities, or any number of tsluatious between a greater number of unknown quantities, to determine every possible solution in which the values of the unknown letters are whole numbers." It may also be considered that the science extends to the determination of all solo. Lions which contain nothing but rational or commensurable fractions, all surd quantities or incommensurablee being excluded. lf, for example, the equation .e+ were to be solved, x and y being whole numbers or rational fractions, let the rational fractions reduced to a common denominator be : z and q :2; then the equation becomes and if all possible whole values of p, e, and z be found, all the fractional solutions of the former equation can be exhibite(L Connected with the science before us is a very large quantity of properties of uumbera, of which it must be said that they can be proved easily enough, but cannot be explained. Usually, in retracing the steps of an algebraical demonstration, WO can easily connect the result with common and self-evident notions, which seem both to justify the conclusion, to render it natural, and destroy much of the curiosity, and even interest, with which it is looked at by a person used to algebra, who hears of the conclusion for the first time. In the theory of numbers it seems to us that the curious character of the conclusions is not so much lessened by the demonstrations, and perhaps this may be the reason why the science becomes a sort of passion, as Legendre remarks, with most of those who take it up. The instances given by the writer just cited, in his preface, will show the sort of properties which we speak of. If c be any prime number, and 5 any other number not divisible by r, then is always divisible by r. Then 2°-1, or 63, is divisible by 7. Again, if any prime number divided by 4 leave a remainder 1, it is the sum of two square numbers : thus 13 is the sum of 9 and 4, 17 of 16 and 1, 29 of 25 and 4, de.
The theory of numbers is not of much immediate practical utility in the applications of mathematics, which generally involve continuously increasing magnitude, and in which therefore the introduction of whole numbers is matter of convenience, and not of necessity. Again, the data of such applications are usually only approximate, so that an answer in whole numbers, should such a thing occur, is not exact, and possesses no particular interest. Hence this theory is little studied by
a very large class of mathematicians, among whom it is not uncommon to meet with a person deeply versed in the higher analysis, who does not even know the principal results obtained by Gauss or Legendre. The subject is, in fact, an isolated part of mathematics, which may be taken up or not, at the choice of the student. It may possibly at some future time be connected with ordinary analysis, that is to say, the determination of the integer solutions of a set of equations may not he so distinct a thing from that of a mere solution, integer or not, as it is at present. In fact, a hint given by M. Libri, in a tract presently to be cited, does give completely the means of assimilating the expression of a problem in this theory to that of one in ordinary analysis. Suppose, for example, it is required to solve in whole numbers the equation xl Let r represent two right angles; then it is well known that sin w x= 0 when x is a whole number, and never else ; so that " required a solution of in whole numbers" is precisely the same problem as "required any solution of the three equations yl= sin 2. 0, sin r y= O." The earliest consideration of the theory of numbers may have been made in India [Vials Galena, in lhioa. Div.]; but the earliest treatise is probably that of Diophantus, which consists of nothing else hut problems of this science, insomuch that the theory itself has been sometimes called the Diophantine analysis. The subject then rested, without making any progress, until the time of Bachet do 3lcziriac and Fermat, the editor and commentator of Diophantus. The subject rested again until the time of the man who literally left no part whatever of mathematics unaugmented, Euler. After him, Lagrange, Legendre, and Gauss applied themselves contemporaneously to this theory. The works of the two latter are the separate treatises on this isarticular science, in which the advanced mathematical student must look to know its present state. Various Memoirs of MM. Cauchy and Libri may also be mentioned ; one in particular by the latter (in the Memoires de Mathemetique et de Physique,' vol. i., Florence, 1629), in which the subject is made to have more resemblance than usual to ordinary analysis. An elementary treatise on the theory of numbers, suited to the present state of the subject, is much wanted. The only one in English, that of Barlow, was published in 1811: it can he well recommended for all that it includes.