SEE (mil + S Z E (sp)  = hk =1(p -1) (4- 1) r=I 4 s=I q the truth of which is obvious, if we

rule a rectangle p"Xq" into'unit squares, and draw its diagonal . This formula is Gauss's, but the geometrical proof is due to Eisenstein . Another useful formula is ,=m-I rl E (x +-) =E(mx) - E(x), which is due to Hermite. m Various other arithmetical functions have been devised for particular purposes; two that deserve mention (both due to Kronecker) are Ohl, which means o or i according as h, k are unequal or equal, and sgn x, which means x= Ix~ . 76 . Transcendental Numbers.-It has been proved by Cantor that the aggregate of all algebraic numbers is countable . Hence immediately follows the proposition (first proved by Liouville) that there are numbers, both real and complex, which cannot be de-fined by any combination of a finite number of equations with rational integral coefficients . Such numbers are said to be transcendental . Hermite first completely proved the transcendent character of e ; and Lindemann, by a similar method, proved the transcendence of Sr . Thus it is now finally established that the quadrature of the circle is impossible, not only by rule and compass, but even with the help of any number of algebraic curves of any order when the coefficients in their equations are rational (see Hermite, C.R. lxxvii., 1873, and Lindemann, Math . Ann. xx., 1882) . Another number which is almost certainly transcendent is Euler's constant C . It may be convenient to give here the following numerical values:- 4.bf (KK')2— I13+(25+13J 5)2(KK )4=0 one root of which is given by (2KK )2=9-41/.5=T-UJ5 which is right, because in this case h(D) = I .

74 . Frequency of Primes.-The

distribution of primes in a finite interval (a, a+b) is very irregular, if we change a and keep b constant . Thus if we put nl =µ, the numbers o+2, u+3, . . . (o+n- I) are all composite, so that we can form a run of consecutive composite numbers as extensive as we please; on the other hand, there is possibly no limit to the number of cases in which p and p+2 are both primes . Legendre was the first to find an approximate formula for F(x), the number of primes not exceeding x . He found by induction F(x) =x= (log,x- I.08366) which answers fairly well when x lies between ioo and 1,000,000, but becomes more and more inaccurate as x increases . Gauss found, by theoretical considerations (which, however, he does not explain), the approximate formula x =3.14159 26535 89793 23846 ... e =2.71828 18284 ° 59045 23536 ... C =0.59921 56649 01532 8606065 ... (Gauss-Nicolai) login = (a logioe) =o• 13493 41840 ... (Weber), the last of which is useful in calculating class-invariants .

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 18or 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 NUMBERS, PARTITION OF, 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 Ic;x,=m1 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 aiscovered by P . A .

MacMahon . The author last named has also considered Diophantine in-equalities, a simple problem in which is " to enumerate all the solutions of 7x13Y 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 con- jecture (he did not affirm it) that 2'"+1 is a prime whenever in is a power of 2 . The first case of failure is when m=32; in fact 232+1 (mod 641) . Other known cases of failura are m=2", with n=6, 12, 23, 26 respectively; at the same time, Eisenstein asserted that he had proved that the formula 2m-F1 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—I, 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—I ) 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 (numbri 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 arithmetic (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 FUNCTIOx .