7. Find two numbers, such that their sum and difference shall be each, square numbers. The sum The difference m2 + 2 m n + n2 = (m + n)2 =a square. m2 - 2 m n + n2: = (m -n)2 = a square. If m = 2, n= 1; what would the numbers be? SECTION LXIV. Diophantine Analysis. 1. Find a number, such that its third power added to four times its square, may be a square number. 2. Can you find any other number that will answer the conditions of this question? 3. Find a number, such that its third power added to four times its second power, may make a cube number; that is, one of which you can get the cube root in rational numbers. x= the number, Let Then Assume x2+4x2 a cube number. 3+ 4x2 = 27 x3 x+4=27x: 26x=4 x= 2 power, and 4. Find a number, such that six times its third eight times its second power, may make a square number. 5. Find a number, such that twelve times its third power, and fifteen times its second power, may be a square number. 6. Find a number, such that twenty times its third power, and sixteen times its second power, may be a cube number. SECTION LXV. Diophantine Analysis. This subject might be continued; but few general rules however can be given, and an infinite multitude of questions may occur, which cannot be solved by any general rule, and which must be left to the ingenuity of the Analyst. We will give some examples and the mode of solving them, which may be of use to the student in suggesting the mode of solving other questions which may occur. 1. It is required to find two numbers, such that if each of them be added to their product, the two sums may be, each squares. be Then, if the second be added to their product, the sum will which is a square number, whatever x may be; for, it is the second power of x+1. If the first be added to their product, the sum will be x2+2x We have, then, only to find such a value for x, as will make this a square number. 2. It is required to find three square numbers in arithmetical progression. By the conditions, the three numbers must have a common difference. The two first are made square numbers by the assumption; for, they are square formulæ. One is the square of x, and the other is the square of x + 1, whatever x may be. We have only, then, to find such a value for x, as will make the third, x2+4x+2 a square number. Assume 3. It is required to find two numbers, such that if their product be added to the sum of their squares, the result will be a square number. 10-x= the other, x2= the square of one, 100-20x+x2 = the square of the other, 100-20x+2x2: the sum of their squares, 10x- their product, 100-10x+x2 = a square number. Assume 100 - 10x + x2 = (2 + x)2 100-10x+x2 = 4 + 4 x + x2 x= 61 = one of the numbers, 10—x=32= the other. 4. Find three square numbers whose sum shall be equal to a given square number (100.) Find first, two square numbers equal to the given square number (100.) Take one of these for one of the squares required, and divide the other into two squares, for the two other squares required. Otherwise, thus: 5. Find three square numbers, whose sum shall be a square number. Here it is evident that the first, second, and sum of the three, are each squares, whatever x and y may be. It only remains to make the third, 2 x y a square number. This can be done by trial. 1. Divide 49 into two square numbers. 2. Divide 121 into two square numbers. 3. Find a number, such that if it be increased by 4, and diminished by 4, the product of the sum and difference will be a square number. 4. Find a number, such that if it be increased by 4, or diminished by 4, the sum and remainder will be each squares. 5. Find three square numbers equal to 144. 6. Find four square numbers, the sum of which shall be a square number. SECTION LXVII, On Exponents. If we wish to represent the product of two numbers, represented by two letters, we write the letters one after the other. Thus: ab, cd, a g, &c. signify the result of a multiplied by b, of c by d, of a by g. a multiplied by b, is the same as b multiplied by a; and, hence we may represent the product of a and b, either by ab or ba. 1. If a 4 and b will b a be equal to ? = = 6; what will ab be equal to? What 2. If c = 1 and d 3; what will cd or dc be equal to? 3. If ab be divided by b; what will the quotient be? 4 If ab be divided by as what will the quotient be? If we wish to represent the continued product of three or more letters, we may place the letters in the same manner. a multiplied by b, and their product by c, may be represented by abc, or a cb, or b c a. 5. If a = 2, b 3 and c = 4; what number will abc represent? What number will a cb represent? What number will be a represent? 6. If a = 2, b = 3, c = 4 and d=5; what number will abcd, ac db, or ad bc, or cb ad represent? 7. What will a b c d divided by bc, represent? If we wish to find the continued product of a, b, c and d, it is immaterial in what order the multiplication is performed; as the result of multiplying is the same, whatever number we commence with, and in whatever order the process is continued. 9. If a, b, c, d, f are equal to 4, 5, 6, 7, 8, respectively; what numbers will the following combinations of these letters represent ? ab abf α (5) = cf? |