Page:Popular Science Monthly Volume 67.djvu/670

664 It would be thought that, after this investigation by Amthor, the question of solving the cattle problem would have been finally dropped, but such was not the case. The certainty that numbers could be found satisfying all nine conditions existed, and until they had been actually computed the challenge of the author of the problem still remained open. The way to solve it was well understood from the theory of indeterminate analysis. Let the preceding equation be multiplied by 8 and unity added to each member, and let 2 n $$+$$ 1 be called y; then it reduces to

which is of the form $$y^{2} - Ax^{2} = 1$$, and it is known that when A is an integer there can always be found integral values of y and x which satisfy the equation. The method of obtaining such values of x and y can not well be explained here, but such a method was devised many years ago by Pell and by Fermat, and it is well known to those skilled in higher arithmetic. For example, take the simple case where $$A = 19$$, or $$y^{2} - 19 x^{2}- = 1$$, then the smallest integral values of y and x which satisfy this equation may be found to be 170 and 39, so that the square of 170 minus 19 times the square of 39 equals unity.

In 1889 A. H. Bell, a surveyor and civil engineer of Hillsboro, Illinois, began the work of solution. He formed the Hillsboro Mathematical Club, consisting of Edmund Fish, George H. Richards and himself, and nearly four years were spent on the work. They computed thirty of the left-hand figures and twelve of the right-hand figures of the value of $$x^{2}$$ without finding the intermediate ones. This value is

in which the dots indicate fifteen computed figures which it is here unnecessary to give and 206,487 uncomputed ones; the total number of figures in this number is 206,531. The final step is to multiply each of the numbers of the first solution by 4,456,749 and by this value of $$x^{2}$$, and thus are obtained

in which the dots represent 206,532 figures, the total number of figures in each line being either 206,545 or 206,544. In each of these lines there are omitted twenty-four figures at the left-end and six at the