16. Shew that the coefficient of an in the expansion of 17. If n is a positive integer, find the value of (n-2) (n − 3) (n − 3) (n − 4) (n − 5) 2n-6+..............; 12 2n-4 18. If n is a positive integer greater than 3, shew that 20. Sum to infinity the series whose nth term is (-1)+12m n(n+1)(n+2)* +C, n being a positive 7 17 31 49 (2) 71 + 23. Prove that, if a<1, (1+ax)(1 + a3x)(1+a3x)...... 24. If A, is the coefficient of x in the expansion of 25. If n is a multiple of 6, shew that each of the series n 3 n(n-1)(n-2) 1 n (n-1)(n-2) (n-3) (n-4) 1 + 15 32 27. If P, (n− r) (n−r+1) (n−r+2)......(n-r+p-1), CHAPTER XXX. THEORY OF NUMBERS. 407. In this chapter we shall use the word number as equivalent in meaning to positive integer. A number which is not exactly divisible by any number except itself and unity is called a prime number, or a prime; a number which is divisible by other numbers besides itself and unity is called a composite number; thus 53 is a prime number, and 35 is a composite number. Two numbers which have no common factor except unity are said to be prime to each other; thus 24 is prime to 77. 408. We shall make frequent use of the following elementary propositions, some of which arise so naturally out of the definition of a prime that they may be regarded as axioms. (i) If a number a divides a product be and is prime to one factor b, it must divide the other factor c. For since a divides bc, every factor of a is found in be; but since a is prime to b, no factor of a is found in b; therefore all the factors of a are found in c; that is, a divides c. (ii) If a prime number a divides a product bed..., it must divide one of the factors of that product; and therefore if a prime number a divides b", where n is any positive integer, it must divide b. (iii) If a is prime to each of the numbers b and c, it is prime to the product bc. For no factor of a can divide b or c; therefore the product bc is not divisible by any factor of a, that is, a is prime to bc. Conversely if a is prime to bc, it is prime to each of the numbers b and c. Also if a is prime to each of the numbers b, c, d, it is prime to the product bcd... ; and conversely if a is prime to any number, it is prime to every factor of that number. (iv) If a and b are prime to each other, every positive integral power of a is prime to every positive integral power of b. This follows at once from (iii). a (v) If a is prime to b, the fractions and α lowest terms, n and m being any positive integers. Also if and a b are any two equal fractions, and is in its lowest terms, then b c and d must be equimultiples of a and b respectively. 409. The number of primes is infinite. For if not, let p be the greatest prime number; then the product 2.3.5.7.11...p, in which each factor is a prime number, is divisible by each of the factors 2, 3, 5,...p; and therefore the number formed by adding unity to their product is not divisible by any of these factors; hence it is either a prime number itself or is divisible by some prime number greater than p: in either case p is not the greatest prime number, and therefore the number of primes is not limited. 410. No rational algebraical formula can represent prime numbers only. ... If possible, let the formula a + bx + cx2 + dx3 + represent prime numbers only, and suppose that when x = m the value of the expression is p, so that thus the expression is divisible by p, and is therefore not a prime number. 411. A number can be resolved into prime factors in only one way. Let N denote the number; suppose N = abcd..., where a, b, c, d, ... are prime numbers. Suppose also that N = aßyd..., where a, B, y, d, are other prime numbers. Then ... abcd... aẞyd... ; = |