PRIME. A number is said to be prime when it is not divisible without remainder by any less number than itself, except unity. Thus 1, 2, 3, are of necessity prime ; 4 is not, being divisible by 2 ; 5 is prime, and so are 7, 11, 13, 17, 19, 23, 29, 31, &c.
Large lists of prime numbers have been published [TABLES], but they are seldom possessed by the elementary student. As it is, how ever, frequently desirable to know whether a number not exceeding 10,000 is prime or not, we shall give a table to that extent, the manner of using which is as follows :If we wish to know whether 2897 be a prime number, under the heading 2 and in the column 8 we look for 97, which we find there : whence the table shows that 2897 is a prime number. Again, by the same means we find that 1457 is not a prime number, the adjacent prime numbers being 1453 and 1459.
The distribution of the prime numbers does not follow any discover able law, but it begins to be evident from the preceding table, that in a given interval the number of primes is generally leas, the higher the beginning of the interval is taken. The following table will set this in a clearer light : the numbers in the first column mean thousands, and in the second column are found the numbers of primes which lie in the interval specified in the first column. Thus, between 10,000 and 20,000 lie 1033 primes.
In the first 10,000 numbers, upwards of 12per cent. are primes; but between 900 thousand and a million, only 71 per cent. are primes. The annexed enumerations are taken from Legendre'e Theory of Numbers, and were made from the large tables of primes given by Vega, Chernac, and Burkhardt. The only thing known relative to the proportions of prime numbers to others is that if x be a very largo number, the number of primes contained between 0 and x is nearly (log x 1'08366), log x being the Naperean logarithm. This very curious theorem was discovered empirically, that is, by looking for a formula which should nearly represent the results of tables. Legendre, in the work cited, gave proof that such a formula must have the form (A log a-- al, but no reason has yet been given why A is 1, and a is 1-030. Using the common logarithm, we find that the number of primes less than x is such a proportion of x As .4342945 is of log x .47062S, nearly. Thus, of all numbers lees than a million of million of millions, only one out of 40 is prime, while the number of primes under the square of that number is one out of 82.
It thus appears that we might name a succession of numbers begin ning with one so high, that a million, or any other number however great, of numbers should pass without containing one prime number. Nevertheless there cannot be an end to the prime numbers ; for if so, let p be the last prime number, and let x be the product of all the prime numbers, 2, 3, 5, .. , p. Now every number is either prime or divisible by a prime ; but x +1 is not divisible by 2,3, 5, ... . or p, since it leaves a remainder I in every such division. It 19 therefore prime, or there is a prime number N+1, greater than the greatest prime number p, which is absurd. The following are among the pro perties of prime numbers :(1.) Every prime number (except 2) is odd, or of the form 2r+1. (2.) Every prime is of the form 4r+1 or 4.r+ 3, and a prime of the form 4x + 1 is always the sum of two squares. (3.) Every prime is of the form 6x+1 or 6x+5. (4.) No algebraical formula can always represent a prime number; but some funnul:c show a long succession of primes : thus xs+x+41 is prime from x=0 to x=39, both inclusive. (5.) If 2r+1 be a prime num ber, and N any number which it does not divide, either or + I must be divisible by 2-r +1. (6.) If xi and re be two prime numbers, and if re =2x +1, x= 2y +1, then if r and y be both odd numbers, either (U7-1) : N and (s 1) : M are both whole num bers, or (ss +1) : N and (s +1) : M are both whole numbers. But if r and y be not both odd, then either (II 3 1) : N and +1) : M are both whole numbers, or else (us +1j : x and (x 1) : M are both whole numbers. This theorem In of considerable importance in the theory of numbers, and has been termed the law of reciprocity of prime numbers.
Two numbers are said to be prime to one another, when they have not any common measure except unity: as 36 and 55.
The prime factors of it number are those prime numbers which divideit. Thus, 360 being 5, its prime factors are 2, 3, 6, of which the first enters three times, the second twice, and the third once. If A, n, c, . be the prime factors of a number, and a, b, c, . the number of times which they severally enter, the number is x x x , and the number of divisors which it admits of ( unity and itself included) is (a+1) x (b+1) x (e+1) x Let a .... be s ; then the number of numbers less than N, and prime to N, is a-1 b-1 c 1 xxx b :a