616 
NUMBERS. 
[795 
The theorems in regard to three triangular numbers and to four square numbers 
are exhibited by certain remarkable 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 K + y K — z K , 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 A,th roots of unity, and pre 
sents very great difficulty; in particular, 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 negative determinant, a remarkable method depending on 
the summation of a series X/ -s , where the index s is greater than but indefinitely near 
to unity. 
46. Very remarkable formulae have been given by Legendre, Tchebycheff, and 
Riemann 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 
formulae. Legendre’s expression is of the form j——^—-j, where A is a constant not 
very different from unity; Tchebycheff’s depends on the logarithm-integral li (x); and 
Riemann’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, Théorie des Nombres, 
1st ed. 1798, 3rd ed. 1830 ; Gauss, Disquisitiones Arithmeticae, Brunswick, 1801 (reprinted 
in the collected works, vol. I., Gottingen, 1863; French translation, under the title Recherches 
Arithmétiques, by Poullet-Delisle, Paris, 1807); and Lejeune-Dirichlet, Vorlesungen ilber 
Zahlentheorie, 3rd 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 “ Reports on the Theory of Numbers,” Parts I. to VI., British Association Reports, 
1859—62, 1864—65, which, with his own original researches, [are] printed in the [first 
volume of the] collected works [published in 1894] by the Clarendon Press. See also Cayley, 
“Report of the Mathematical Tables Committee,” Brit. Assoc. Report, 1875, pp. 305—336, 
[611], for a list of tables relating to the Theory of Numbers, and Mr J. W. L. Glaisher’s 
introduction to the Factor Table for the Sixth Million, London, 1883, in regard to the 
approximate formulae for the number of prime numbers.