NUMBERS, THEORY OF, the most subtle and intricate, and at the same time one of the most extensive, branches of mathematical analysis. It treats primarily of the forms of numbers, and of the properties at once deducible from these forms; hut its principal field is the theory of equations, in as far as equations are soluble in whole numbers or rational fractions, and inure particularly that branch known as indeterminate equations. Closely allied to this•branch are those problems which are usually grouped under the diophantine analysis (q.v.), a class of problems alike interesting and difficult; and of winch the following are examples: 1. Find the numbers the sum of whose squares shall be a square number; a condition satisfied by 5 and 12, 8 and 15, 9 and 40, etc. 2. Find three square numbers in arithmetical progression; Answer, 1, 25, and 49; 4, 100, 196, etc.
Forms of numbers are certain algebraic formulas, which, by assigning to the letters successive numerical values from 0 upwards, are capable of producing all numbers with out exception, e.g., by giving to m the successive values 0, 1. 2, 3, etc., in any of the groups of formulas: 2m, 2m + 1; 3m, 3m ± 1, 3m + 2; 4m, 4m + 1, 4m+ 2, 47n -1-3, we can produce the natural series of numbers. These formulas are based on the self-evident principle, that the remainder after division is less than the divisor, and that, consequently, every number can be represented in the form of the product of two factors + a number less than the smaller factor.
By means of these formulas, many properties of numbers can be demonstrated with out difficulty. To give a few examples. (1.) The product of two-consecutive numbers is divisible by 2: Let 2m be one number, then the other is either 2m + 1 or 2m — 1, and the product 2m (2m ± 1) contains 2 as a factor, and is thus divisible by 2. (2.) The product of three consecutive numbers is divisible by 6: Let 3m be one of the numbers (as in every triad of consecutive numbers one must be a multiple of 3). then the others are either 3m — 2, 3m — 1; 3m — 1, 3m + 1; or 3m + 1, 3m + 2. In the first and third cases, the proposition is manifest, as (3m — 2) (3m — 1), and (3m + 1) (3m + 2), are each divisible by 2, and therefore their product into 3m is divisible by 6 (= 1.2.3). In the second case the product is 3m (3m — 1) (3m + 1), or 3m — 1), where 3 is a factor, and it is necessary to show that m — 1) is divisible by 2: if m be even, the thing is proved; but if odd, then is odd, 9m* is odd, and — 1 is even; hence, in this case also the proposition is true. It can similarly be proved that the product of four consecu tive numbers is divisible by 24 (= 1.2.3.4). of five consecutive numbers by 120 (= 1.2.3. 4.5), and so on generally. These propositions form the basis for proof of many proper ties of numbers, such as that the-difference of the squares of any two odd numbers is divisible by 8. The difference between a number and its cube is the product of three consecutive numbers, and is consequently (see above) always divisible by 6. Any prime
number which, when divided by 4, leaves a remainder unity, is the sum of two square numbers: thus, 41 = 25 + 16 = 5g + 233 = 169 + 64 = + etc.
Besides these, there are a great many interesting properties of numbers which defy classification; such as, that the suns of the odd numbers beginning with unity is a square number (the square of the numbe'r of terms added), i.e., 1 + 3 + 5 = 9 1 + 3 + 5 + 7 + 9 = 25 = etc.; and, the sum of the cubes of the natural numbers is the square of the sum of the numbers, i.e.. + = 1 + 8 + 27 = 36 = (1 + 2 + + X = 100 = (1 + 2 + 3 etc.
We shall close this article with a few general remarks on numbers themselves. Num bers are divided into prime and composite—prime numbers being those which contain no factor greater than unity; composite numbers those which are the product of two (not reckoning unity) or more factors. The number of primes is unlimited, and so conse quently are the others. The product of any number of consecutive numbers is even, as also are the squares of all even numbers; while the product of two odd numbers, or the squares of odd numbers, arc odd. Every composite number can be put under the form of a product of powers of numbers; thus, 144 = X or, generally, is = aP./A.cr, where a, b, and c are prime numbers, and the number of the divisors of such a com posite number is equal to the product (p 1)(r 1), unity and the number itself hieing included. In the case of 144, the number of divisors would be (4 + 1X2 + 1), or 5 or 15, which we find by trial to be the case. Pofect numbers are those which are equal to the sum of their divisors (the number itself being of course excepted); thus, = 1 + 2 + 3, 28 = 1 + 2 + 4 + 7 + 14, and 496. are perfect numbers. Amicable numbers are pairs of numbers, either one of the pair being equal to the sum of the ivisors of the other; thus, 220 (= 1+ 2 + 4 + 5 + 10 + 11 + 20 + 22 + 44+ 55 + 110 = 284). and 284 (= 1 + 2 + 4 + 71 + 192 = 220), are amicable numbers. For other serifs of numbers see FIGURATE NUMBERS.
The most ancient writer on the theory of numbers was Diophantus, who flourished in the 3d c., and the subject received no further development till the time of Vieta and Fermat (the latter being the author of several celebrated theorems, a discussion of which, however, is quite unsuited to this work), who greatly extended it. Euler next added his quota, and was followed by Lagrange, Legendre, and Gauss, who in turn success fully applied themselves to the study of numbers, and brought the theory to its present state. Cauchy, Libri, and Gill (in America) have also devoted themselves to it with suc cess. The chief authorities down to the present century are Barlow's Theory of Num bers (1811); Legendre's Essai surlaTheorie des Nombres (3d ed., Paris, 1830): and Gauss's Disquisitiones Arithmetica (Brunswick, 1801; Fr. translation, 1807); and for the latest dis coveries, the transactions of the various learned societies may be consulted.