PRIMATES

The name given by Linnaeus to the first order of animals in his System a NaivreB, which a ociated man with the monkeys and bats, and corresponded to the Bimana, Quadruman a, and Cheiroptera of Cuvier. PRIME NUMBERS, in Arithmetic, are numbers which have no divisors, or which cannot be divided into 974 PRIMINE. any le number of equal integral parts than the number of units of which they are composed; such are 2, 3, 5, 7, II, 13. . A general method of finding prime numbers, beyond a certain limit, by a direct proce , is one of the most difficult problems in the theory of numbers; and, though often sought after, has not yet been discovered. Many remarkable properties of numbers have, however, been detected, by means of which it is in most cases not diflScult to determine whether an a igned number is prime or not. Some of these properties are the following:— 1. Every prime number above 3 is comprehended in one of these forms, 6n-(-l,or6« — 1 (w being any whole number); that is to say, if a prime number be increased or diminished by unity, the result is a multiple of G. In order to prove this, it will be sufficient to remark that every whole number is nece arily comprised in one or other of these six forms (where n is succe ively 0, 1, 2, 3, .), "^ 6» 4- I, 6«-l-2, 6» + 3, 6n + 4, 6m-|- 5, 6n-l-6. Now the second, fourth, and sixth of these forms, being divisible by 2, cannot give prime numbers. The third gives only numbers divisible by 3; therefore the primes can only be of the form 6 n -I- 1, or 6 w -1- 5. But 6 w -I- 5 = 6w+6 — 1 = 6 (m-J-1) — l=Gn' — l,w' being any whole number; it follows therefore that all prime numbers are comprehended in one of the forms 6?i -(- 1, or 6 7J-1. 2. In like manner, every prime number above 2 is of one of the forms 4«+ l,or4H — 1; and every prime number, excepting 2, of one of the forms 8 « -|- 1, 8 n + 3, 8 w -1- 5, 8 » -l- 7. In fact, prime numbers may be divided in this manner into cla es, according to any modulus, at pleasure; the last four forms, however, are those which po e the most distinctive properties. Although every prime number is comprehended in one of these forms, the converse proposition is not true; namely, that every number in one of these forms is a prime number. No direct rule has yet been given by which it can be determined d. prior i whether a given number be prime or not. 3. If a number cannot be divided by another number le than the square root of itself, that number is a prime. 4. If « denote any prime number, the product, 1 • 2 • 3 • 4 • • • • (7J — 1), increased by unity, is divisible by n. It is frequently of use in arithmetical investigations to know whether a number is prime or not; tables of them have accordingly been formed to a certain extent, and are given in various works. Vega's tables contain the prime numbers under 400,000. The largest prime number which has yet been verified is 2^1 — I =2147483647. This was found by Euler. For properties of prime numbers, see Fermat's edition of Diophantus; Euler' s Algebra, and Analysis In jin i to rum; Legendre, E ai sur la Theorie da, Notnbres; Barlow's Elementary Investigations, SjC.j and especially the Dis qui sit i ones Arithmeticce of Gau , of which there is a French translation by Delisle. Prime and Ultimate Ratios — A method of calculation invented by Newton, and employed in the Principia, being an extension and simplification of the ancient method of exhaustions. It may be thus explained: — Let there be two variable quantities constantly approaching each other in value, so that their ratio or quotient continually approaches to unity, and at last differs from unity by le than any a ignable quantity; the tUtimate ratio of these two quantities, is said to be a ratio of equality. In general, when different variable quantities respectively and simultaneously approach other quantities considered as invariable, so that the differences between the variable and invariable quantities become at the same time le than any a ignable quantity, the ultimate ratios of the variables are the ratios of the invariable quantities or limits, to which they continually and simultaneously approach. They are called prime ratios, or ultimate ratios, according as the ratios of the variables are considered as receding from, or approaching to, the ratios of the limits. (See Principia, book i.) PRl'MER, signified anciently a religious work employed in the Roman Catholic service; but it is now generally used to denote the first book for children. [s. 987]