Project is now named Celeste

celeste/math/factor/pftrialdivision.py

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+'''
+The trial division algorithm is essentially the idea that
+all factors of an integer n are less than or equal to isqrt(n),
+where isqrt is floor(sqrt(n)).
+
+Moreover, if p divides n, then all other factors of n must
+be factors of n//p. Hence they must be <= isqrt(n//p).
+'''
+from math import isqrt # integer square root
+
+# Returns the "multiplicity" of a prime factor
+def pf_multiplicity(n: int, p: int) -> int:
+    mult = 0
+    while n % p == 0: 
+        n //= p
+        mult += 1
+    return n, mult
+
+'''
+Trial division prime factorisation algorithm.
+Returns a list of tuples (p, m) where p is
+a prime factor and m is its multiplicity.
+'''
+def trial_division(n: int) -> list[tuple[int, int]]:
+    factors = []
+
+    # determine multiplicity of the only even prime (2)
+    n, mult_2 = pf_multiplicity(n, 2)
+    if mult_2: factors.append((2, mult_2))
+    # determine odd factors and their multiplicities
+    p = 3
+    mult = 0
+    limit = isqrt(n)
+    while p <= limit:
+        n, mult = pf_multiplicity(n, p)
+        if mult:
+            factors.append((p, mult))
+            limit = isqrt(n) # recalculate limit
+            mult = 0 # reset
+        else:
+            p += 2
+    # if n is still greater than 1, then n is a prime factor
+    if n > 1:
+        factors.append((n, 1))
+    return factors
+