SECTION LVII. On Indeterminate Equations. 1. Given, 5x+8y+9x=314, to find all the values of x, y and z in positive whole numbers. Making p = 0 will give the least value of y = 2 z—2. 5x+8y+9x=314 that when x and y are the least possible, z will be the greatest possible; but, the least possible values of x and y cannot be less than a unit. Suppose, therefore, x = 1, and y = 1, and we will find the greatest value of z. 5+8+9x=314 9 z=301 z= 33 Hence z cannot be assumed greater than 33. 2. Find the values of x, y and z in the following equations: If we have two equations and three letters, we can eliminate one of the letters, and we shall have an equation with two letters; and the values of these two letters may be found, as in the preceding section. Let there be given x-2y+x=5 Multiplying the first equation by 2, and subtracting the second from it, we have 1. Let it be required to find a whole number, such that if it be divided by 5, 6 and 7, it will give the remainders 4, 5 and 6, respectively. Let x, y and z be the whole numbers in the respective quo tients. 7z+6=209: = the number required. 2. What number is that, which being divided by 3, 4 and 5, will give the remainders 2, 3, 4, respectively? 3. A market woman carrying some eggs to market, was met by a rude fellow, who broke them all. After considering what he had done, he determined to make reparation; finding the woman, he asked her how many eggs he had broken. She said she could not tell; she had between one and two dozen; and, when she told them out, 2 at a time, there was one left; and when she told them out, 3 at a time, there were two left; and, when she told them out, 4 at a time, there were three left. How many eggs had she? SECTION LIX. Diophantine Analysis. There are comparatively few numbers, of which we can obtain the exact second root. We have seen [SEC. 50.] that the approximate root of 2 is 1.4%, or 1. By continuing the process explained in [SEC. 50.,] we may approximate nearer and nearer the root of 2; but, we can never find a number, which multiplied by itself, will produce exactly 2; because there is, in fact, no such number. In order to demonstrate this, we shall use the following propositions, the truth of which, can be seen by a little reflection: 1st. The second power of an odd number, is an odd number. 2nd. The second power of an even number, is an even number. 3rd. Twice an odd number, gives an even number. 4th. The second power of an even number, can always be divided by 4, without any remainder; for, any even number may be represented by 2n, where n is some whole number; but, the second power of 2n = 4 n2, this divided by 4, gives n2, a whole number. Now, if there were any number, which multiplied by itself, would make 2, it might be represented by; where m and m n n are some whole numbers. For example, if it were 1., or 1, m would be 141, and ʼn 100. Let us suppose, then, 141 m that is equal to the second root of 2; and, that the fraction n is reduced to its lowest terms. Then, m and n cannot be both even numbers; for, then the numerator and denominator might each be divided by 2, and the fraction reduced to lower terms, contrary to the supposition. As m and n cannot be both even, m2 and n2 cannot be both even [PROP. 1 and 2.] There are, then, but three possible suppositions, respecting m2 and n2, in equation 3rd. Sup. 1st. Either m2 and n2 are both odd numbers, or 2nd, m2 is odd and n2 even, or 3rd, n2 is odd and m2 even. Now, if we can show that neither of these suppositions can m be true, then cannot express the root of 2; and, conse n quently, the root of 2 cannot be expressed in numbers. The first and second suppositions cannot be true; for, then we should have m2 an odd number, equal to 2 n2, which would in both cases, be an even number. If m2 is an even number, and n2 odd, which is the 3rd. supposition, then divide both members of equation (3) by 4, and we have The first member of this equation would be a whole number [PROP. 4;] but, the second member is an odd number divided by 2, which makes a fraction; and, we should have a whole number equal to a fraction, which is impossible. It is evident then, that the root of 2 cannot be expressed in numbers: although, we can approximate the root, until the difference between the approximate root, and the true one, is less than any number which can be assigned. By a method somewhat similar, it can be shown that the root of 3, 5, 7, &c. cannot be expressed in numbers. Any number, whether whole or fractional, the root of which can be expressed in numbers, is called a square number: thus, 1, 4, 9, 25, 1, †, &c., are called square numbers. The Diophantine Analysis teaches how to find square numbers, under given conditions, or square numbers, which shall have given relations to each other, or to other numbers. 1. Six times a certain number added to 5, is equal to a square number. What is the number? Now, we may assume 6x+5= any square number, as 4, 16, 25, 36, &c. |