numbers. He also succeeded in showing that in the field R(en) the equation a+B+yo has no integral solutions whenever his not divisible by p. What is known as the "last" theorem of Fermat is his assertion that if m is any natural number exceeding 2, the equation x+y=2" has no rational solutions, except the obvious ones for which xyz=o. It would be sufficient to prove Fermat's theorem for all prime values of m, and whenever Kummer's theorem last quoted applies, Fermat's theorem will hold. Fermat's theorem is true for al values of m such that 2 <m<101, but no complete proof of it has yet been obtained. Hilbert has studied in considerable detail what he calls Kummer fields, which are obtained by taking x, a primitive pth root of unity, and y any root of yao, where a is any number in the field R(x) which is not a perfect pth power in that field. The Kummer field is then R (x, y), consisting of all rational functions of x and y. Other fields that have been discussed more or less are general cubic fields, some special biquadratic and a few Abelian fields not cyclic. Among the applications of cyclotomy may be mentioned the proof which it affords of the theorem, first proved by Dirichlet, that if m, n are any two rational integers prime to each other, the linear form mx+ is capable of representing an infinite number of primes. 62. Gauss's Sums.-Let m be any positive real integer; then This remarkable formula, when m is prime, contains results which were first obtained by Gauss, and thence known as Gauss's sums. The casiest method of proof is Kronecker's, which consists in finding the value of fleis2/mdz/(1-eawiz)}, taken round an appropriate contour. It will be noticed that one result of the formula is that the square root of any integer can be expressed as a rational function of roots of unity. The most important application of the formula is the deduction from it of the law of quadratic reciprocity for real primes: this was done by Gauss. 63. One example may be given of some remarkable formulae giving explicit solutions of representations of numbers by certain quadratic forms. Let p be any odd prime of the form 7n+2; then we shall have p=7n+2=x2+7y, where x is determined by the congruences (3n)! 2x ¡(mod p); x=3 (mod 7). '(n)! (2n)!' This formula was obtained by Eisenstein, who proved it by investigating properties of integers in the field generated by 3-217-7=0, which is a component of the field generated by seventh roots of unity. The first formula of this kind was given by Gauss, and relates to the case p=4n+1=x+y; he conceals its connexion with complex numbers. Probably there are many others which have not yet been stated. 64. Higher Congruences. Functional Moduli.-Suppose that p is. a real prime, and that f(x), (x) are polynomials in x with rational integral coefficient. The congruence f(x)=(x) (mod p) is identical when each coefficient of f is congruent, mod p, to the corresponding coefficient of 4. It will be convenient to write, under these circumstances, f-(mod p) and to say that f. are equivalent, mod p. Every polynomial of degree h is equivalent to another of equal or lower degree, which has none of its coefficients negative, and each of them less than p. Such a polynomial, with unity for the coefficient of the highest power of x contained in it, may be called a reduced polynomial with respect to p. There are, in all, p reduced polynomials of degree h. A polynomial may or may not be equivalent to the product of two others of lower degree than itself; in the latter case it is said to be prime. In every case, F being any polynomial, there is an equivalence F-cfiff where is an integer and fff are prime functions; this resolution is unique. Moreover, it follows from Fermat's theorem that {F(x)}P-F(x2), {F(x)}P2_F(x2), and so on. As in the case of equations, it may be proved that, when the modulus is prime, a congruence f(x) o (mod p) cannot have more incongruent roots than the index of the highest power of x in f(x), and that if xis a solution, f(x)~(x-E)f(x), where f(x) is another polynomial. The solutions of xx are all the residues of p; hence x-x-x(x+1) (x+2)...(x+p-1), where the right-hand expression is the product of all the linear functions which are prime to p. A generalization of this is contained in the formula x(x»"'-L1)~[ƒ(x) (mod p) where the product includes every prime function f(x) of which the degree is a factor of m. By a process similar to that employed in finding the equation satisfied by primitive mth roots of unity, we can find an expression for the product of all prime functions of a given degree m, and prove that their number is (m> 1) m where a, b, c... are the different prime factors of m. Moreover, if F is any given function, we can find a resolution F-cFF... F(mod p) | | where e is numerical, F, is the product of all prime linear functions which divide F, F, is the product of all the prime quadratic factors, and so on: 65. By the functional congruence (x)=(x) (mod p, f(x)) is meant that polynomials U, V can be found such that o(x)=(x)+pU+ Vf(a) identically. We might also write(x)~(x) (mod p, f(x)); but this is not so necessary here as in the preceding case of a simple modulus. Let m be the degree of f(x); without loss of generality we may suppose that the coefficient cf x is unity, and it will be further assumed that f(x) is a prime function, mod p. Whatever the dimensions of (x), there will be definite functions x(x), (x) such that o(x)=f(x)x(x)+1(x) where o1(x) is of lower dimension than f(x); moreover, we may suppose (x) replaced by the equivalent reduced function 2(x) mod p. Finally then, = (mod p, (x)) where 2 is a reduced function, mod p, of order not greater than (m-1). If we put p=n, there will be in all (including zero) n residues to the compound modulus (p, f); let us denote these by R, R.,... R. Then (cf. § 28) if we reject the one zero residue (R2, suppose) and take any function of which the residue is not zero, the residues of R1, R2,...R will all be different, and we conclude that - (mod p.). Every function therefore satisfies O (mod p, f); by putting ø=x we obtain the principal theorem stated in § 64 A still more comprehensive theory of compound moduli is due to Kronecker; it will be sufficiently illustrated by a particular case. Let m be a fixed natural number; X, Y, Z, T assigned polynomials, with rational integral coefficients, in the independent variables x, y, z; and let U be any polynomial of the same nature as X, Y, Z, T. We may write U-o (mod m, X, Y, Z, T) to express the fact that there are integral polynomials M, X', Y', Z', T' such that U=mM+X'X+Y′'Y+Z'Z+T'T identically. In this notation U-V means that U-V-o. The number of independent variables and the number of functions in the modulus are unrestricted; there may be no number m in the modulus, and there need not be more than one. This theory of Kronecker's is admirably adapted for the discussion of all algebraic problems of an arithmetical character, and is certain to attain a high degree of development. It is worth mentioning that one of Gauss's proofs of the law of quadratic reciprocity (Göll. Nachr. 1818) involves the principle of a compound modulus. 66. Forms of Higher Degree.-Except for the case alluded to at the end of § 55, the theory of forms of the third and higher degree is still quite fragmentary. C. Jordan has proved that the class number is finite. H. Poincaré has discussed the classification of ternary and quaternary cubics. With regard to the ternary cubic it is known that from any rational solution of fo we can deduce another by a process which is equivalent to finding the tangential of a point (x, y, z) on the curve, that is, the point where the tangent at (x, y, z) meets the curve again. We thus obtain a series of solutions (x1,y1, 1), (x2, y2, 22), &c., which may or may not be periodic. E. Lucas and J. J. Sylvester have proved that for certain cubics f=0 has ne rational solutions; for instance x+y-Aso has rational solutions only if A=ab(a+b)/c, where a, b, c are rational integers. Waring asserted that every natural number can be expressed as the sum of not more than 9 cubes, and also as the sum of not more than 19 fourth powers; these propositions have been neither proved nor disproved. 67. Results derived from Elliptic and Theta Functions.-For the sake of reference it will be convenient to give the expressions for the four Jacobian theta functions. Let w be any complex quantity such that the real part of iw is negative; and let geriw. Then +∞ O‰x (v) = Σg" e2i =1+29 cos 2v+2g cos 4v+2qp cos 6πv+... Fuchsian or automorphic function. It is an analytical-function of g, and may be expanded in the form J=172819+744+cig°+cgʻ+ . . .} where 1, 2, &c., are rational integers. the nation being that which is now usual for the elliptic functions. dv(a, b, c, d) = 1 and ad-ben, a positive integer. Let (a+b) It is found that = 2K 1+4 •=0002 (1+2g+2g1+2ge+...)2, and hence, by expanding both sides in ascending powers of g, and equating the coefficients of q", we arrive at a formula for the number of ways of expressing n as the sum of two squares. If & is any odd divisor of n, including 1 and n itself if n is odd, we find as the coefficient of q* in the expansion of the left-hand side 42(-1)(-1); on the right-hand side the coefficient enumerates all the solutions n = (x)2+(y)?, taking account of the different signs (except for o2) and of the order in which the terms are written (except when 2=y). Thus if n is an odd prime of the form 4k+1, Σ(−1)1(6-1) = 2, and the coefficient of q is 8, which is right, because the one possible composition na may be written n = ( − a)2 + ( + b)2 = (+b)2+(±a)2, giving eight representations. 69. Suppose, now, that a, b, c, d are rational integers. such that (cw+d)=w; then the equation (w') = J(w) is satisfied if and only if w-w, that is, if there are integers a,ß, y, & such that aō-By = 1, and (aw+b) (rw+8)-— (cw+d) (aw+B)=0. If we write(n) =nII (1+p1), where the product extends to all prime factors (p) of ", it is found that the values of a fall into (n) equiva lent sets, so that when w is given there are not more than (n) different values of J(w'). Putting J(w')=J', J(w)=J, we have a modular equation symmetrical in J. J', with integral coefficients and of degree (n). Similarly when dv(a, b, c, d) = we have an equation f,(J', }) = 0 of order (/); hence the complete modular equation for transforma tions of the nth order is F(J'.J) = IIƒ.(J', J) =0, the degree of which is (n), the sum of the divisors of ". alone, which we may call G). To every linear factor of G corre Now if in F(J',J) we put II, the result is a polynomial in J sponds a class of quadratic forms of determinant (2-4) where <4 and is an integer or zero: conversely from every such form we can derive a linear factor (J-a) of G. Moreover, if with each form we associate its weight (8 41) we find that with the notation of § 39 the degree of G is precisely H(4n—k2)—e,, where 1 when is a square, and is zero in other cases. But this degree may be found in another way as follows. A complete representative set of transformations of order n is given by w'=(aw+b)/d, with ad=n, ob<d; hence By methods of a similar character formulae can be found for the number of representations of a number as the sum of 4, 6, 8 squares respectively. The four-square theorem has been stated in § 41; the eight-square theorem is that the number of representations of a number as the sum of eight squares is sixteen times the sum of the and by substituting for J(w) and j (a+b) their values in terms of cubes of its factors, if the given number is odd, while for an even number it is sixteen times the excess of the cubes of the even factors above the cubes of the odd factors. The five-square and sevensquare theorems have not been derived from q-series, but from the general theory of quadratic forms. 68. Still more remarkable results are deducible from the theory of the transformation of the theta functions. The elementary formulae For convenience let (w): then the substitutions (w.w+1) and (w,-) convert a into o/(-1) and (1-0) respectively. Now if 4. B. Y. & are any real integers such that a5-By =1, the substitution [w, (aw+B)/(yw+8)] can be compounded of (w, w+1) and (w, -w-1); the effect on a will be the same as if we apply a corresponding substitution compounded of [o, o/(-1)] and [o, 1-6]. But these are periodic and of order 3, 2 respectively]; therefore we cannot get more than six values of a, namely q. we find that the lowest term in the factor expressed above is either 9/1728 or gt/d/1728, or a constant, according as a<d, a> or a=d. Hence if is the order of G(J), so that its expansion in begins with a term in q-2 we must have extending to all divisors of n which exceed √n. Comparing this with the other value, we have EH (4n-x3)=22d+en=4(n)+¥(n), as stated in § 39. 70. Each of the singular moduli which are the roots of G(J) =o corresponds to exactly one primitive class of definite quadratic forms, and conversely. Corresponding to every given negative determinant -A there is an irreducible equation)=0, where j=1728J, the coefficients of which are rational integers, and the degree of which is h(-3). metical integer, and its conjugate values belong one to each primitive The coefficient of the highest power of j is unity, so that j is an arith class of determinant -A. By adjoining the square roots of the prime factors of A the function(j) may be resolved into the product of as many factors as there are genera of primitive classes, and the degree of each factor is equal to the number of classes in each genus. In particular, if {1, 1, (A+1)} is the only reduced form for the determinant -A, the value of j is a real negative rational cube. At the same time its approximate value is exp [24] +7+4= 744-ev A, so that, approximately, e-VA=m3+744 where m is a rational integer. For instance ev "884736743-9997775: 960 +744 very nearly, and for the class (1, 1, 11) the exact value of jis-960. Four and only four other similar determinants are known to exist, namely 11, 19, -67, 163, although thousands have been classified. According to Hermite the decimal part of e*v 15 begins with twelve nines; in this case Weber has shown that the exact value of jis - 21.3.5.23-29. 71. The function j(w) is the most fundamental i 2 set of quantities called class-invariants. Let (a, b, c) be the representative of any class of definite quadratic forms, and let w be the root of ax + bx+c=0 which has a positive imaginary part; then F (w) is said to be a classlaw invariant for (a, b, c) if F (+5)= F(w) for all real integers a hence (o2 —o +1)3 +s2(☛ − 1)a has the same value at equivalent places. 8, 7, 8 such that as-By-1. This is true for j(w) whatever may F. Klein writes be, and it is for this reason that jis so fundamental. But, as will be seen from the above examples, the value of j soon becomes so lange that its calculation is impracticable. Moreover, there is the d which is a special case of a culty of constructing the modular equation ƒ¡U, J') =o (§69) which this is a transcendental function of has only been done in the cases when n=2, 3 (the latter by Smith in extended over all the representations m=+4no. In a similar way Proc. Lond. Math. Soc. ix. p. 242). For moderate values of ▲ the difficulty can generally be removed by constructing algebraic functions of j. Suppose we have an irreducible equation the coefficients of which are rational functions of j(w). If we apply jɩ(w) = x(−1) 1(m-1)x(m) qmiz, ja(w)=22(−1)!(m-1)x(m) qm14, we shall have ja(w) =2&X(m)qm/4 di:dja:dja 01:001:000 m any modular substitution w'S(w), this leaves the equation un- As one of the simplest examples, let n=2, x3-j(w)=y-j (w)=0. Then the equation connecting x, y in its complete form is of the ninth degree in each variable; but it can be proved that it has a rational factor, namely y3-x2y2+495xy+x3-2′ . 33 . 53 =0, and if in this we put x=y=u, the result is u2u3-495u2+24.3.53=0, which represents, in homogeneous co-ordinates, a quartic curve of deficiency 3. For this curve, or any equivalent algebraic figure, ji(w), ja(w) and ja(w) supply an independent set of Abelian integrals of the first kind. If we put x=√,,y=√x', it is found that Sdx=djo(w), Sex=dir(w), fxdx = kji(w). so that the integrals which the algebraic theory gives in connexion with x+y=0 are directly identified with (w), ja(w), js(w), provided that we put x=√x(w). Other functions occur in this theory analogous to ji(w), but such that in the g-series which are the expansions of them the coefficients and exponents depend on representations of numbers by quaternary quadratic forms. 73. In the Berliner Sitzungsberichte for the period 1883-1890, L. Kronecker published a very important series of articles on elliptic functions, which contain many arithmetical results of extreme elegance; some of these Kronecker had announced without proof many years before. A few will be quoted here, without any attempt at demonstration; but in order to understand them, it will be necessary to bear in mind two definitions. The first relates to the Putting (a, B. v. 8)=(0, -1, 1, 0) the conditions are satisfied, and Legendre-Jacobi symbol (2). If a, b have a common factor we put 2w=i√2. Now j(i)=1728, so that (i) = 12; and since j(w) is positive for a pure imaginary, (i√2)=20. The remaining case is settled, by putting 72. Another powerful method, developed by C. F. Klein and K. E. R. Fricke, proceeds by discussing the deficiency of fij, j') =o considered as representing a curve. If this deficiency is zero, j and j' may be expressed as rational functions of the same parameter, and this replaces the modular equation in the most convenient manner. For instance, when n=7, we may put j = (x2 + 13r+49) (r2 +5r + 1)2 = p(t), j′ = p(r'), T () =0; while if a is odd and b=2^c, where c is odd, we put (1) - (2) (2). The other definition relates to the classification of discriminants of quadratic forms. If D is any number that can be such a discriminant, we must have Do or 1 (mod. 4), and in every case we can write D=DoQ, where Q is a square factor of D, and Do satisfies one of the following conditions, in which P denotes a product of different odd primes: Do P, with P=1 (mod 4) Numbers such as Do are called fundamental discriminants; every discriminant is uniquely expressible as the product of a fundamental discriminant and a positive integral square. Now let D1, D2 be any two discriminants, then D.D, is also a discriminant, and we may put D,D,D=DoQ, where Do is fundamental: this being done, we shall have ΤΣ Σ =} Σ a, b, c where we are to take h, k=1, 2, 3,... +∞; m, n = 0, 1, ±2....±0 except that, if D<o, the case m=n=0 is excluded, and that, if D>0, (zam+bn)TnU where (T, U) is the least positive solution of T-DU-4. The sum applies to a system of representative primitive forms (a, b, c) for the determinant D, chose so that a is prime to Q, and b, c are cach divisible by all the prime factors of Q. A is any number prime to 2D and representable by (a, b, c); and finally =2, 4, 6, I according as D<-4, D=-4. D=-3 or D30. The function F is quite arbitrary, subject only to the conditions that F(xy) = F(x) F(y), and that the sums on both sides are convergent. By putting F(x)=x-1-, where is a real positive quantity, it can be deduced from the foregoing that, if D, is not a square, and if D, is different from 1, TH(DQ3)H(D2Q=Lt Z where the function H(d) is defined as follows for any discriminant d:→→ () () -Σ (尚筒) [() + where, on the left-hand side, we are to sum for s, 1, 2, 3 ... [D]; and on the right we are to take a complete set of representative primitive forms (a, b, c) for the determinant D, D2, and give to m, n all positive and negative integral values such that am2+bmn+cn2 is odd. The quantity is 2, if D,D<−4, 7=4 if D1D2=-4, 7-6 if DD-3. By putting D2-1, we obtain, after some easy transformations, where the product on the right extends to a certain sixth part of those values of 2KK' which are singular, and correspond to the field (-D), or in other words are connected with the class invariant j(-D). For instance, if D=5, the equation to find (««')3 is 4.8{ (kx')2 — 1}3+(25+13√5)3 (kx′)* =0 one root of which is given by (2kk')2=9-4√5=T-U√5 which is right, because in this case h(D) = 1. 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 nμ, the numbers μ+2, μ+3,... (u+n−1) 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 pand 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-1.08366) (where, as in all that follows, log x is taken to the base e). This value is ultimately too large, but when x exceeds a million it is nearer the truth than the value given by Legendre's formula. By a singularly profound and original analysis, Riemann succeeded in finding a formula, of the same type as Gauss's, but more exact for very large values of x. In its complete form it is very complicated; but, by omitting terms which ultimately vanish (for sufficiently large values of x) in comparison with those retained, the formula reduces to F(x)=A+2(−1)~_—_L(x1/m) (m = 1, 2, 3, 5, 6, 7, 11, . . .) this may be inferred the truth of Bertrand's conjecture that there is always at least one prime between a and (2a-2) if za >7. Tchébichev's results were generalized and made more precise by Sylvester. The actual calculation of the number of primes in a given interval may be effected by a formula constructed and used by D. F. E Meissel. The following table gives the values of F(n) tor various values of n, according to Meissel's determinations:- Riemann's analysis mainly depends upon the properties of the function (n=1, 2, 3, .). $(s) = considered as a function of the complex variable s. The above definition is only valid when the real part of s exceeds 1; but it can be generalized by writing where the integral is taken from x =+ along the axis of real quantities to x=e, where e is a very small positive quantity, then round a circle of radius e and centre at the origin, and finally from x= to x + along the axis of real quantities. This function (e) is of great importance, and has been recently studied by von Mangoldt Landau and others. Reference has already been made to the fact that if 1, m are coprimes the lincar form lx+m includes an infinite number of primes. Now let (a, b, c) be any primitive quadratic form with a total generic character C; and let lx+m be a primitive linear form chosen so that all its values have the character C. Then it has been proved by Weber and Meyer that (a, b, c) is capable of representing an infinity of primes all of the linear form lx+m. 75. Arithmetical Functions.-This term is applied to symbols such as (n), (n), &c, which are associated with n by an intrinsic arithmetical definition. The function (n) was written fn by Euler, who investigated its properties, and by proving the formula Sn = S(n−1)+S(n−2) −S(n−5)- .=Σ(-1)'-' (n-35±5) that n-(35+) is not negative, and to interpret fo as n, if this term where on the right hand we are to take all positive values of s such occurs. J. Liouville makes frequent use of this function in his papers, but denotes it by 5(n) If the quantity is positive and not integral, the symbol E(x) or omitting the fractional part of x; thus E(v2)=1, E(0.7)=0, and so [x] is used to denote the integer (including zero) which is obtained by putting E(-x)=-E(x), and agreeing that when x is a positive on. For some purposes it is convenient to extend the definition by integer, E(x)=x-, it is then possible to find a Fourier sine-series representing x-E(x) for all real values of x. The function E(x) has many curious and important properties, which have been investigated by Gauss Hermite, Hacks, Pringsheim, Stern and others. What is perhaps the simplest proof of the law of quadratic reciprocity depends upon the fact that if p, q are two odd primes, and we put p=2h+1, g=2k+1 = = 'Σ E (2) + 2 E (52) - hk - t (p − 1) (9 − 1) Various other arithmetical functions have been devised for par ticular purposes; two that deserve mention (both due to Kronecker) are 8, which means o or I according as k, k are unequal or equal, and sgn x, which means x÷x. 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. 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. Ixxvii., 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 fellowing numerical values: -3-14159 26535 89793 e-2-71828 18284 59045 C=0.57721 23846... 23536... 56649 01532 8606065... (Gauss-Nicolai) logo=(logie)=0.13493 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 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 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 2c,x=m, 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 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 conjecture (he did not affirm it) that 2+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=2", with n=6, 12, 23, 26 respectively; at the same time, Eisenstein asserted that he had proved that the formula 2+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 2-1, with 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 (2-1) is perfect when (2-1) 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 arithmetic (qv) 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 FUNCTION of AUTHORITIES.-W. H. and G. E. Young, The Theory of Sets 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 (Bril. 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, 1968) and Encycl. d. math. Wissenschaften, vol. i. (Leipzig, 1898). (G. B. M.) NUMBERS, BOOK OF, the fourth book of the Bible, which takes its title from the Latin equivalent of the Septuagint 'Apiμoi. 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 cleven 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 |