require "bignum"
local fmt = require "fmt"

local function solve_pell(n)
    n = bigint.new(n)
    local x = bigint.sqrt(n)
    local y = x
    local z = bigint.one
    local r = x * 2
    local e1 = bigint.one
    local e2 = bigint.zero
    local f1 = bigint.zero
    local f2 = bigint.one
    while true do
        y = r * z - y
        z = (n - y * y) / z
        r = (x + y) / z
        e1, e2 = e2, r * e2 + e1
        f1, f2 = f2, r * f2 + f1
        local a = e2 + x * f2
        local b = f2
        if (a * a - n * b * b == bigint.one) then return {a, b} end
    end
end

for {61, 109, 181, 277} as n do
    local [a, b] = solve_pell(n)
    fmt.print("x² - %3dy² = 1 for x = %-21s and y = %s", n, a, b)
end