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Return ``(b, e)`` such that ``n`` == ``b**e`` if ``n`` is a perfect power; otherwise return ``False``. By default, the base is recursively decomposed and the exponents collected so the largest possible ``e`` is sought. If ``big=False`` then the smallest possible ``e`` (thus prime) will be chosen. If ``candidates`` for exponents are given, they are assumed to be sorted and the first one that is larger than the computed maximum will signal failure for the routine.(more...)

def perfect_power(n, candidates=None, big=True, factor=True): """ Return ``(b, e)`` such that ``n`` == ``b**e`` if ``n`` is a perfect power; otherwise return ``False``. By default, the base is recursively decomposed and the exponents collected so the largest possible ``e`` is sought. If ``big=False`` then the smallest possible ``e`` (thus prime) will be chosen. If ``candidates`` for exponents are given, they are assumed to be sorted and the first one that is larger than the computed maximum will signal failure for the routine. If ``factor=True`` then simultaneous factorization of n is attempted since finding a factor indicates the only possible root for n. This is True by default since only a few small factors will be tested in the course of searching for the perfect power. Examples ======== >>> from sympy import perfect_power >>> perfect_power(16) (2, 4) >>> perfect_power(16, big = False) (4, 2) """ n = int(n) if n < 3: return False logn = math.log(n, 2) max_possible = int(logn) + 2 # only check values less than this not_square = n % 10 in [2, 3, 7, 8] # squares cannot end in 2, 3, 7, 8 if not candidates: candidates = primerange(2 + not_square, max_possible) afactor = 2 + n % 2 for e in candidates: if e < 3: if e == 1 or e == 2 and not_square: continue if e > max_possible: return False # see if there is a factor present if factor: if n % afactor == 0: # find what the potential power is if afactor == 2: e = trailing(n) else: e = multiplicity(afactor, n) # if it's a trivial power we are done if e == 1: return False # maybe the bth root of n is exact r, exact = integer_nthroot(n, e) if not exact: # then remove this factor and check to see if # any of e's factors are a common exponent; if # not then it's not a perfect power n //= afactor**e m = perfect_power(n, candidates=primefactors(e), big=big) if m is False: return False else: r, m = m # adjust the two exponents so the bases can # be combined g = igcd(m, e) if g == 1: return False m //= g e //= g r, e = r**m*afactor**e, g if not big: e0 = primefactors(e) if len(e0) > 1 or e0[0] != e: e0 = e0[0] r, e = r**(e//e0), e0 return r, e else: # get the next factor ready for the next pass through the loop afactor = nextprime(afactor) # Weed out downright impossible candidates if logn/e < 40: b = 2.0**(logn/e) if abs(int(b + 0.5) - b) > 0.01: continue # now see if the plausible e makes a perfect power r, exact = integer_nthroot(n, e) if exact: if big: m = perfect_power(r, big=big, factor=factor) if m is not False: r, e = m[0], e*m[1] return int(r), e else: return False