૪ and for p and ≈ substituting 0, or any whole numbers, integral values of x and y are obtained. If x=3, and p=1, then x=2 1; if x = 4, and p = 0, then x = 7 and y == - 6; &c. [Indeterminate equations may also be solved as follows: and y = 337. : To find the integral values of x and y which satisfy an equation of the form ax + by = c. Suppose a and b prime to each other; for, if they are not, a, b and c must have a common measure, and by division the equation is reduced to one in which the coefficients of r and y are prime to each other. Hence it is only necessary to consider the case when a and b are prime. Let x = a, y = ẞ be one solution; then and since a and b are prime to each other, y - ẞ must be some multiple of a; or y− ß= at, ..x-α = bt, or x = a - bt, and y = ß + at ; which values of x and y, upon substitution in the original equation, are found to satisfy the equation, whatever be the value of t, positive or negative. Hence all the integral values of x and y are found by giving different integral values to t in the equations The question, then, is reduced to finding one solution x = a, y = B. This may be done by Art. 335, or as follows: Find the series of fractions converging to and let α b' a = ±1, + or according as > or < b Р q . a.cq-b.cp = c; 00 Ex. 5x + 7y = 29; required the values of x and y in posi Lett 12, then = 3, and y = 2: which is the only solu tion in positive integers. 338. To find the number of solutions in positive integers of the equation ax + by = C. α Let the series of fractions converging to be found, and let a be that which immediately precedes, then (Art. 331). or a (cq - bt) + b (at − cp) = c ; therefore, comparing this with the original equation, and the several solutions will be found by giving t such values, that cq-bt and at - cp may be positive integral quantities, ср that is, t may be less than and greater than cq but no a other number. Hence the number of solutions will be expressed by the greatest integer contained in shewn, as before, that the number of solutions is the greatest integer contained in Ex. In how many different ways may £1000. be paid in crowns and guineas? Let a be the number of guineas, and y the number of crowns, then 21x+5y=20000; and it is required to find how many solutions this equation admits of in positive integers. .. the number of solutions required is 190. For further information on the subject of unlimited problems and continued fractions the Student is referred to Barlow's Theory of Numbers, Arts. 40, 41, and Part 11. Chap. I. and II.] pro 339. In the solution of different kinds of unlimited blems different expedients must be made use of, which expedients, and their application, are chiefly to be learned by practice. Ex. 1. To find a "perfect number," that is, one which is equal to the sum of all the numbers which divide it without remainder. Suppose yr to be a "perfect number;" its divisors are 1, y, y2... y", x, xy, xy2... xy"-1; and, that a may be a whole number, let y+1-2y" — 0, Also, let n be so assumed that 2"+1-1 may have no divisor but unity, which was supposed in taking the divisors of y"; then y"x, or 2′′ × (2"+1-1) is a "perfect number." Thus, if n = 1, the number is 2 × 3 or 6, which is equal to 1 + 2 + 3 the sum of its divisors: If n = 2, the number is 23x (231) = 4 x 7 28. = Ex. 2. To find two square numbers, whose sum is a square. Let x and y be the two square numbers; Assume x2 + y2 = (n x − y)2 (nx − y)2 = n2x2 - 2nxy + y2, then 2= n2x2 - 2nxy And if n and y be assumed at pleasure, such a value of x is obtained that a2 + y2 is a square number. But if it be required to find integers of this description, let y = n - 1, then a = 2n, and n being taken at pleasure, integral values of x and y, and consequently of a2 and y2, will be found. Thus, if n = 2, then y = 3, and a = 4, and the two squares are 9 and 16, whose sum is 25, a square number. Ex. 3. To find two square numbers, whose difference is a [340. To explain the different systems of notation. In the common system of notation each figure increases its value in a tenfold proportion in proceeding from right to left. Thus 3256 may be expressed by 6 + 5 × 10 + 2 × 10 + 3 × 103. The figures 3, 2, 5, 6, by which the number is formed, are called its digits, and the number 10, according to whose powers their values proceed, is called the radix of the scale. It is purely conventional that 10 should be the radix; and therefore there may be any number of different scales, each of which has its own radix. * On this subject see the Edinburgh Transactions, Vol. II, p. 193, |