Find the least positive nontrivial solution of 22 18y21 Sol

Find the least positive nontrivial solution of 22 - 18y21.

Given Pell equation is x² - 18y² = 1

The trivial solution of the equation is x = 1 & y = 0.

Nontrivial solution of equation can be find by following:

x² - 18y² = 1

or, x² = 18y² + 1

the above form implies that (18y² + 1) is a perfect square such that x exist.

So, we will try to find the minimum value of y for which (18y² + 1) is a perfect square,

for y =1, (18y² + 1) = 18+1 = 19; not a perfect square

for y = 2, (18y² + 1) = 18*4 + 1 = 73; not a perfect square

for y = 3, (18y² + 1) = 18*9+1 = 163; not a perfect square

for y = 4, (18y² + 1) = 18*16+1 = 289; perfect square of 17.

As y = 4 gives a perfect square of 17, so least positive nontrivial solution is x = 17 & y = 4.