The number of representations as a sum of any even number of squares was found by elliptic functions by Boulyguine in 1914-15 and by elliptic modular functions by Mordell in 192o. The same year, Hardy applied his powerful analytic methods to sums of k squares, k < 8. In 1924 Kloosterman gave exact and asymptotic formulas for the number of representations as a sum of r squares. In 1922 Siegel found an asymptotic expression for the number of representations as a sum of 5 or more squares of integers of a real quadratic field. See EQUATIONS; DETERMI gression. A finite or a simply infinite sequence of numbers is said to form a geometric progression if for every pair of consecutive elements A and B (A preceding B) the quotient B/A is one and the same fixed number r. If the first term is a then the numbers of the sequence are Simple Infinite Sequences.—A simply infinite sequence a2, a3, . . . of numbers is said to be convergent and to form a regular sequence if for every positive number e, however small, there exists a positive integer n such that the numerical value of is less than E for every integer m greater than r. Regular sequences play a fundamental role in the development of mathe matical analysis. To begin with, they furnish one of the means by which the number system may be extended from the rational domain to the domain of real numbers, and thus serve to lay the foundations on which an adequate theory of functions may be built. If, having developed the system of rational numbers (see NUMBERS), we proceed to form simply infinite sequences of rational numbers, it will be found that some such sequences have rational numbers as limits (see LIMIT), and that all such se quences having rational numbers as limits are regular in accord ance with the foregoing definition, the number e in the definition being taken rational in this case. It becomes desirable to extend the number system so that every regular sequence of rational numbers shall have a limit in the extended number system. This is done by taking the regular sequence of rational numbers itself to represent a definite number defined by the sequence, in accord ance with a method introduced by G. Cantor (Math. Annalen, vol. v., 1872, and vol. xxi., 1883).
Before presenting the method of Cantor it is convenient to note some properties of regular sequences of rational numbers. Let be a symbol to denote the sequence If { and lb„1 are two regular sequences of rational numbers, then it may be shown that the sequences are also regular sequences, with suitable restrictions in the last case—for instance, that all the elements of { b„} shall be greater than some given positive number. It is natural to take as the sum, difference, product, and quotient, of two sequences { and the sequences respectively. Then from the foregoing theorem it follows that the processes of addition, subtraction, multiplication and division may be carried out on regular sequences of rational numbers (with suitable restrictions in the case of division) and that the sequences which result from any finite number of applications of these oper ations are themselves regular sequences of rational numbers.
Furthermore, it may be shown that the ordinary fundamental laws of algebra hold for operations with sequences, namely, the asso ciative and commutative laws of addition and multiplication and the distributive law of multiplication (qq.v.) with respect to addi tion. (See ALGEBRA.) Since these regular sequences of rational numbers combine ac cording to the same formal laws as rational numbers themselves, it is natural to take regular sequences of rational numbers as them selves defining a new sort of number ; and this is what Cantor does. The new numbers are called real numbers. A rational number a may then be denoted by the sequence a, a, a, . . . each element of which is the rational number a. But this is not a unique repre sentation of a. In fact, any sequence of rational numbers al, a2, . . . having the rational limit a may be used to denote the number a. In general, the real numbers a and b defined by two regular sequences and respectively, are said to be equal if for every positive rational number E however small, a number n exists such that the numerical value of is less than E for every positive integer m. The real number represented by { is said to be of higher rank than the real number represented by 1 b„ 1 if a value of n exists such that is numeri cally greater than some given positive number 6 for all positive integers m. These definitions are sufficient to establish the rela tions of order among the real numbers.
The real numbers thus postulated satisfy the usual laws of algebra and possess an order relation analogous to that of rational numbers. In the domain of real numbers every regular sequence of rational numbers has a limit ; namely, the real number defined by the sequence itself. The question naturally arises, whether every regular sequence of real numbers has a limit in the domain of real numbers. The answer is affirmative; that is, if definitions are introduced in connection with sequences of real numbers, in all respects analogous to those already given for sequences of rational numbers, then every regular sequence of real numbers has a limit in the domain of real numbers. Therefore regular sequences of real numbers lead to no further extension of the system of num bers. The fact that a regular sequence of real numbers always has a limit in the domain of real numbers renders the set of real num bers suitable to be the field of the real variable in a general theory of functions of real variables.
Any given simply infinite sequence al, 02, . . . may be employed to define a new sequence s2, s3, . . . by writing The problems connected with the convergence of the sequence then identical with the problems of the convergence of the infinite series . . . .
Therefore a part of the theory of limits (see LIMIT) and the whole of the general theory of simply infinite series (see SERIES) are aspects of the theory of simply infinite number sequences. In a similar way the definition of a definite integral as the limit of a sum presents the theory of integration as another one of the funda mental applications of the theory of simply infinite sequences.