In order to compare our two methods of solving a Diophantine equation of the form, Px + Qy = T, when the coefficients are such as to determine a relatively large number of potential solutions: Find all positive integral solutions for x and y such that x, y < 1,000 and . Standard Solution: The derived equations are , Let , , Let , , , . If , then , so . If , then , so . Therefore, . Substituting 0 through 15 for B in equations (7) and (8), we get , through , . Actually, once we have the first solution, all the others may be obtained from the fact that is a linear equation that may be written as , so for every increase of x by 7, there is a decrease of y by 9 and that yields all the other solutions.
New Solution: Using the original equation: If , then , so . If , then , so . there are fewer possibilities for x, namely, through . The smallest value for x that yields an integral value for y is 3, which yields a value of 139 for y. The rest of the solutions can be found using the slope of the given equation, as they were above.
We will deal with one more objection to this new method in our next posting.