P

```
-- Finds the simplist radical form of sqrt x
return function(
x -- number
)
if x == 0 then
return 1, 0
end -- 0 = 1 sqrt 0
assert(x > 0)
local coefficient = 1
local root_term = 1
for factor = 2, math.sqrt(x) do -- Prime-factorize x
local count = 0
while x % factor == 0 do
x = x / factor
count = count + 1
end
coefficient = coefficient * factor ^ math.floor(count / 2) -- extract sqrt(factor^(2y)) = factor^y
root_term = root_term * factor ^ (count % 2) -- possible leftover prime factor is multiplied with root term
end
-- x may itself be prime; the prime factorization only iterates up to sqrt x for efficiency, not handling that case
root_term = root_term * x -- multiply by leftover prime factor x (which is either the original x or 1)
-- Simplist radical form as coefficient * sqrt(root_term)
return coefficient, root_term
end
```