History of Hindu Mathematics: A Source Book v2.c17.s24.38

v2.c17.s24.38

From (2), by the method of the Square-nature, we get the values of (u, v) as (1, 2), (4, 7), (15, 26), Whence the values of x are 21, 3374/3, Alternatively¹ we assume u = 3y + 1; then from the equation (1) we get x = 3y(3y² + 3y + 1). Also from (2) we have 27y² + 18y + 4 &= v² / &= (my - 2)², say. Hence y = (18 + 4m)/(m² - 27). Therefore, the required value of x is x = 9((18 + 4m)/(m² - 27))³ + 9((18 + 4m)/(m² - 27))² + 3((18 + 4m)/(m² - 27)), where m is a rational number greater than 5. The first of the previous solutions is given by m = 9.