>>136614
i think he was asking for the proper x and y so that x^2 - ny^2 = 1, not to simplify it

>>136604
you use the structure x^2-ny^2 = 1, so this is pell's equation

x^2 - ny^2 = 1 where n = 61

we have to find the continued fraction of n, 61
sqrt(61) = 7.8102, but we'll round down to represent it as 7. but in memory keep the decimal to about 4 points, we'll use it later. we will create a set that we'll use to compute the converging points, P.
P = {7}

from there bring back the 7.8102 and subtract the 7 for 0.8102, divide 1 by that and you'll get 1.2342, rounding down for 1 so add that to the set
P = {7, 1}
bring back 1.2342 and do the same thing as we did for 7

we're gonna repeat this process a bunch until we end up with
{7, 1, 4, 3, 1, 2, 2, 1, 3, 4, 1, 14...} we'll stop there

now lets get the convergents
we'll use Cq = xq/yq

the initial value of h is our first element and the denominator initializes to 1
x0 = 7
y0 = 1

so C0 = 7 / 1

before we move further let's bring back the original equation and try plugging shit in
(x0)^2 - 61(y0)^2 = (7)^2 - 61(1)^2 = 49 - 3721 = -3672 which =/= 1, which isn't what we're
looking for so this can't be it. lets continue the process

for the x1 and y1 values i'll use these equations
x1 = a0 * x1 + 1
y1 = a1

x1 = 1 * 7 + 1 = 8
y1 = 1 * 1 = 1

C1 = 8 / 1

lets try again

(x1)^2 - 61(y1)^2 = (8)^2 - 61(1)^2 = 64 - 3721 = -3657 =/= 1
nope

for anything greater than that i'll use this
xq = aqx(q-1) + x(q-2)
yq = aqy(q-1) + y(q-2)

so x2 = 1 * 8 + 7 = 15
y2 = 1* 1 + 1 = 2

C2 = 15 / 2

i dont want to write it down but this doesnt work either

and so on and so on. now we just keep repeating this process until we find a number that works, which isnt for a retardedly long time
it is so long we dont get to it until about the 19th convergent, of which the numbers are

C19 = 1766319049 / 226153980

lets try it to confirm

(1766319049)^2 - 61(226153980)^2 = 3119882982860264401 - 3119882982860264400 = 1

yay! x^2 - 61y^2 = 1! this intentionally obtuse equation is solved
alternatively you could plug the equation into desmos or look at the pell's equation wikipedia page
sry i did not use proper ascii notation i could not be fucked
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