Numbers

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; but its principal field is the theory of equations, in as far as equations are soluble in whole numbers or rational fractions, and more 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 which 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, &c. (2) Find three square numbers in arithmetical progression; Answer, 1, 25, and 49; 4, 100, 196, &c.

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 without exception—e.g. by giving to $m$ the successive values 0, 1, 2, 3, &c., in any of the following groups of formulas, $2m$, $2m\; +\; 1$; $3m$, $3m\; +\; 1$, $3m\; +\; 2$; $4m$, $4m\; +\; 1$, $4m\; +\; 2$, $4m\; +\; 3$, 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 without 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 $2m(2m\; \backslash pm\; 1)$ 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 $3m(9m^2\; -\; 1)$, where 3 is a factor, and it is necessary to show that $m(9m^2\; -\; 1)$ is divisible by 2: if $m$ be even, the thing is proved; but if odd, then $m^2$ is odd, $9m^2$ is odd, and $9m^2\; -\; 1$ is even; hence in this case also the proposition is true. It can similarly be proved that the product of four consecutive 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 properties 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, $41\; =\; 25\; +\; 16\; =\; 5^2\; +\; 4^2$, $233\; =\; 169\; +\; 64\; =\; 13^2\; +\; 8^2$, &c.

Besides these there are a great many interesting properties of numbers which defy classification; such as that the sum of the odd numbers beginning with unity is a square number (the square of the number of terms added)—i.e. $1\; +\; 3\; +\; 5\; =\; 9\; =\; 3^2$, $1\; +\; 3\; +\; 5\; +\; 7\; +\; 9\; =\; 25\; =\; 5^2$, &c.; and the sum of the cubes of the natural numbers is the square of the sum of the numbers—i.e. $1^3\; +\; 2^3\; +\; 3^3\; =\; 1\; +\; 8\; +\; 27\; =\; 36\; =\; (1\; +\; 2\; +\; 3)^2$, $1^3\; +\; 2^3\; +\; 3^3\; +\; 4^3\; =\; 100\; =\; (1\; +\; 2\; +\; 3\; +\; 4)^2$, &c.

Numbers 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 consequently 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, are odd. Every composite number can be put under the form of a product of powers of numbers; thus, $144\; =\; 2^4\; \backslash times\; 3^2$, or generally, $n\; =\; a^x\; b^y\; c^z$, where $a$, $b$, and $c$ are prime numbers, and the number of the divisors of such a composite number is equal to the product $(p\; +\; 1)(q\; +\; 1)(r\; +\; 1)$, unity and the number itself being included. In the case of 144 the number of divisors would be $(4\; +\; 1)(2\; +\; 1)$, or $5\; \backslash times\; 3$, or 15, which we find by trial to be the case. Perfect numbers are those which are equal to the sum of their divisors (the number itself being of course excepted); thus, $6\; =\; 1\; +\; 2\; +\; 3$, $28\; =\; 1\; +\; 2\; +\; 4\; +\; 7\; +\; 14$, and 496 are perfect numbers. Amicable numbers are pairs of numbers, either one of the pair being equal to the sum of the divisors of the other; thus, $220\; (=\; 1\; +\; 2\; +\; 4\; +\; 5\; +\; 10\; +\; 11\; +\; 20\; +\; 22\; +\; 44\; +\; 55\; +\; 110\; +\; 284)$ and $284\; (=\; 1\; +\; 2\; +\; 4\; +\; 71\; +\; 142\; =\; 220)$ are amicable numbers. For other series of numbers, see FIGURATE NUMBERS.

The most ancient writer on the theory of numbers was Diophantus, who flourished in the 3d century, and the subject received no further development till the time of Vieta and Fermat (q.v.), who greatly extended it. Euler next added his quota, and was followed by Lagrange, Legendre, and Gauss, who in turn successfully 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 success.

See Barlow's Theory of Numbers (1811); Legendre's

Essai sur la Théorie des Nombres (3d ed. Paris, 1830); and Gauss's Disquisitiones Arithmetice (1801; new ed. 1860; Fr. trans. 1807); H. J. S. Smith, in Brit. Ass. Reports (1859-65); Cayley, in Brit. Ass. Reports (1875).