History of Hindu Mathematics: A Source Book v2.c17.s9.11

v2.c17.s9.11

Thus $3 (x⁶ - 1)/(x - 1) = 4095,$ or $3(x⁵ + x⁴ + x³ + x² + x + 1) = 4095,$ a quintic equation. Here dividing 4095 by 3 we get 1365. Now let us try with the divisor 4; we have (1365 - 1)/4 = 341; (341 - 1)/4 = 85; (85 - 1)/4 = 21; (21 - 1)/4 = 5; (5 - 1)/4 = 1; (1 - 1)/4 = 0. So the number 1365 is exhausted on 6 successive divisions by 4, in the way indicated in the rule. Hence x = 4. What suggested the method is clearly this : $a (xn - 1)/(x - 1) ÷ a = (xn - 1)/(x - 1); (xn - 1)/(x - 1) - 1 = x((xn-1 - 1)/(x - 1));$ which is divisible by x. However, the solution is obtained in every case by trial only.