Answer by Mad Physicist for Checking if float is equivalent to an integer value in python

Python has unlimited integer precision, but only 53 bits of float precision. When you square a number, you double the number of bits it requires. This means that the ULP of the original number is (approximately) twice the ULP of the square root.

You start running into issues with numbers around 50 bits or so, because the difference between the fractional representation of an irrational root and the nearest integer can be smaller than the ULP. Even in this case, checking if you are within tolerance will do more harm than good (by increasing the number of false positives).

For example:

>>> x = (1 << 26) - 1>>> (math.sqrt(x**2)).is_integer()True>>> (math.sqrt(x**2 + 1)).is_integer()False>>> (math.sqrt(x**2 - 1)).is_integer()False>>> y = (1 << 27) - 1>>> (math.sqrt(y**2)).is_integer()True>>> (math.sqrt(y**2 + 1)).is_integer()True>>> (math.sqrt(y**2 - 1)).is_integer()True>>> (math.sqrt(y**2 + 2)).is_integer()False>>> (math.sqrt(y**2 - 2)).is_integer()True>>> (math.sqrt(y**2 - 3)).is_integer()False

You can therefore rework the formulation of your problem slightly. If an integer x is a triangular number, there exists an integer n such that x = n * (n + 1) // 2. The resulting quadratic is n**2 + n - 2 * x = 0. All you need to know is if the discriminant 1 + 8 * x is a perfect square. You can compute the integer square root of an integer using math.isqrt starting with python 3.8. Prior to that, you could use one of the algorithms from Wikipedia, implemented on SO here.

You can therefore stay entirely in python's infinite-precision integer domain with the following one-liner:

def is_triangular(x):    return math.isqrt(k := 8 * x + 1)**2 == k

Now you can do something like this:

>>> x = 58686775177009424410876674976531835606028390913650409380075>>> math.isqrt(k := 8 * x + 1)**2 == kTrue>>> math.isqrt(k := 8 * (x + 1) + 1)**2 == kFalse>>> math.sqrt(k := 8 * x + 1)**2 == kFalse

The first result is correct: x in this example is a triangular number computed with n = 342598234604352345342958762349.