Page:Encyclopædia Britannica, Ninth Edition, v. 17.djvu/682

Rh 624 N U M N U M 41. The congruence - imaginaries of Galois. If F(x) be an irreducible function to a given prime modulus &amp;gt;, this implies that there is no integer value of x satisfying the congruence F(x) = (mod. p) ; we assume such a value and call it i, that is, we assume F(i] = (mod. p) ; the step is exactly analogous to that by which, starting from the notion of a real root, we introduce into algebra the ordinary imaginary i= N/ 1. For instance, x 2 - x + 3 is an irreducible function to the modulus 7, there is no integer solution of the congruence x 2 x + 3 = (mod. 7). Assuming a solution i such that i 2 - i + 3 = (mod. 7), we have, always to this modulus, i 2 = i - 3, and thence i*, i &c., each of them equal to a linear function of i. We consider the numbers of the form a + bi, where a and b are ordinary integers which may be regarded as having each of them the values 0, 1, 2, 3, 4, 5, or 6 ; there are thus 7 2, = 49, such numbers, or, excluding zero, 48 numbers ; and it is easy to verify that these are in fact the numbers i, i 2 , .... i 4 &quot;, t 48 , = 1, that is, we have i a prime root of the congruence x* s 1=0 (mod. 7). The irreducible func tion may be of the third or any higher degree ; thus for the same modulus 7 there is the cubic function x 3 - x + 2, giving rise to a theory of the numbers of the form a + bi + ct 2, where i is a congruence- imaginary such that i 3 - i + 2 = (mod. 7); and instead of 7 the modulus may be any other odd prime p. Ordinary Theory, Third Part. 42. In what precedes no mention has been made of the so-called Pellian equation x 2 - Dy z = 1 (where D is a given positive number), and of the allied equations x 2 D// 2 = - 1, or = 4. The equations with the sign + have always a series of solutions, those &quot;with the sign - only for certain values of D ; in every case where the solutions exist a least solution is obtainable by a process depending on the expression of /Z&amp;gt; as a continued fraction, and from this least solution the whole series of solutions can be obtained without difficulty. The equations are very interesting, as well for their own sakes as in connexion with the theory of the binary quadratic forms of a positive non- square determinant. 43. The theory of the expression of a number as a sum of squares or polygonal numbers has been developed apart from the general theory of the binary, ternary, and other quadratic forms to which it might be considered as belonging. The theorem for two squares, that every prime number of the form 4 + 1 is, and that in one way only, a sum of two squares, is a fundamental theorem in relation to the com plex numbers a + bi. A sum of two squares multiplied by a sum of two squares is always a sum of two squares, and hence it appears that every number of the form 2 a (4?i + 1) is (in general in a variety of ways) a sum of two squares. Every number of the form 4?i + 2 or 8n + 3 is a sum of three squares ; even in the case of a prime number Sn + 3 there is in general more than one decomposition, thus 59 = 25 + 25 + 9 and = 49 + 9 + 1. Since a sum of three squares into a sum of three squares is not a sum of three squares, it is not enough to prove the theorem in regard to the primes of the form 8n + 3. Every prime number is (in general in more than one way) a sum of four squares ; and therefore every number is (in general in more than one way) a sum of four squares, for a sum of four squares into a sum of four squares is always a sum of four squares. Every number is (in general in several ways) a sum of m + 2 (m + 2)gonal numbers, that is, of numbers of the form fyn(x i - x) + x ; and of these m - 2 may be at pleasure equal to or 1 ; in particular, every number is a sum of three triangular numbers (a theorem of Fermat s). The theorems in regard to three triangular numbers and to four square numbers are exhibited by certain remark able identities in the Theory of Elliptic Functions ; and generally there is in this subject a great mass of formulae connected with the theory of the representation of numbers by quadratic forms. The various theorems in regard to the number of representations of a number as the sum of a definite number of squares cannot be here referred to. 44. The equation x^ + y^ = z where A is any positive integer greater than 2, is not resoluble in whole numbers (a theorem of Fermat s). The general proof depends on the theory of the complex numbers composed of the Ath roots of unity, and presents very great difficulty ; in parti cular, distinctions arise according as the number A does or does not divide certain of Bernoulli s numbers. 45. Lejeune-Dirichlet employs for the determination of the number of quadratic forms of a given positive or nega tive determinant a remarkable method depending on the summation of a series 2/~ s, where the index s is greater than but indefinitely near to unity. 46. Very remarkable formulas have been given by Legendre, Tchebycheff, and Eiemann for the approximate determination of the number of prime numbers less than a given large number x. Factor tables have been formed for the first nine million numbers, and the number of primes counted for successive intervals of 50,000 ; and these are found to agree very closely with the numbers calculated from the approximate formulas. Legendre s expression oc is of the form. -, where A is a constant not very log x-A different from unity; Tchebycheff s depends on the log arithm-integral (x) ; and Eiemann s, which is the most accurate, but is of a much more complicated form, con tains a series of terms depending on the same integral. The classical works on the Theory of Numbers are Legendre, Theorie des Nombres, 1st ed. 1798. 3d ed. 1830 ; Gauss, Disquisi- tiones Arithmetics?., Brunswick, 1801 (reprinted in the collected works, vol. i., Gottingen, 1863 ; French trans., under the title Rcchcrchcs Arithmetiqucs, by Poullet - Delisle, Paris, 1807) ; and Lejeune-Dirichlet, Vorlesungcn iibcr Zahlcntheorie, 3d ed., with extensive and valuable additions by Dedekind, Brunswick, 1879-81. We have by the late Prof. H. J. S. Smith the extremely valuable series of &quot;Reports on the Theory of Numbers,&quot; Parts I. to VI., British Association Reports, 1859-62, 1864-65, which, with his own original researches, will be printed in the collected works now in course of publication by the Clarendon Press. See also Cayley, &quot;Report of the Mathematical Tables Committee,&quot; Brit. Assoc. Re port, 1875, pp. 305-306, for list of tables relating to the Theory of Numbers, and Mr. J. W. L. Glaisher s introduction to the Factor Table for tJic Sixth Million, London, 1883, in regard to the approxi mate formulfe for the number of prime numbers. (A. C. ) NUMENIUS, one of the so-called Neo-Pythagoreans, and a forerunner of the Neo-Platonists, was a native of Apamea in Syria, and flourished during the latter half of the second Christian century. He was a somewhat volu minous writer in philosophy and philosophical biography, but all that is known of his opinions is found in passing references by Clement of Alexandria, Origen, Eusebius, and one or two of the Neo-Platonists. He seems to have taken Pythagoras as his highest authority, and at the same time to have been unaware of any discrepancy be tween his own views and those of Plato, whom he further described as an &quot;Atticizing Moses,&quot; and as deriving all his knowledge, like Pythagoras, from the East. He held a kind of trinity, the members of which he designated as TraTrTros, e/cyovos, and d/royovos respectively, the first being the supreme deity, or pure vou?, the second the demiurge, and the third the world. NUMERALS. The use of visible signs to represent numbers and aid reckoning is not only older than writing but older than the development of numerical language on the denary system ; we count by tens because our ancestors counted on their fingers and named numbers accordingly. So used, the fingers are really numerals, that is, visible