Udayadivākara

Udayadivākara used his method for solving the equation ${\displaystyle Nx^{2}+C=y^{2}}$ to obtain some particular solutions to a certain algebraic problem. The problem and Udayadivākara's solution to the problem are presented below only to illustrate the techniques used by Indian astronomers for solving algebraic equations.^[2]

Find positive integers ${\displaystyle x}$ and ${\displaystyle y}$ satisfying the following conditions:

${\displaystyle {\begin{aligned}x+y&={\text{a prefect square}},\\x-y&={\text{a prefect square}},\\xy+1&={\text{a prefect square}}.\end{aligned}}}$

To solve the problem, Udayadivākara makes a series of apparently arbitrary assumptions all aimed at reducing the problem to one of solving an indeterminate equation of the form ${\displaystyle Nx^{2}+C=y^{2}}$.

Udayadivākara begins by assuming that ${\displaystyle xy+1=(2y+1)^{2}}$ which can be written in the form ${\displaystyle x-y=3y+4}$. He next assumes that ${\displaystyle 3y+4=(3z+2)^{2}}$ which, together with the earlier equation, yields

${\displaystyle {\begin{aligned}x&=12z^{2}+16z+4,\\y&=3z^{2}+4z,\\x+y&=15z^{2}+20z+4.\end{aligned}}}$

Now, Udayadivākara puts

${\displaystyle 15z^{2}+20z+4=u^{2}}$

which is then transformed to the equation

${\displaystyle (30z+20)^{2}=60u^{2}+160.}$

This equation is of the form ${\displaystyle Nx^{2}+C=\lambda ^{2}}$ with ${\displaystyle N=60}$, ${\displaystyle C=160}$ and ${\displaystyle \lambda =30z+20}$. Using the method for solving the equation ${\displaystyle Nx^{2}+C=y^{2}}$, Udayadivākara finds the following solutions ${\displaystyle (u=2,\lambda =20)}$, ${\displaystyle (u=18,\lambda =140)}$ and ${\displaystyle (u=8802,\lambda =68180)}$ from which the values of ${\displaystyle x}$ and ${\displaystyle y}$ are obtained by back substitution.

• List of astronomers and mathematicians of the Kerala school