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
def perfect_power(n, candidates=None, big=True, factor=True):
"""
Return ``(a, b)`` such that ``n`` == ``a**b`` if ``n`` is a
perfect power; otherwise return ``None``.
By default, the base is recursively decomposed and the exponents
collected so the largest possible ``b`` is sought. If ``big=False``
then the smallest possible ``b`` (thus prime) will be chosen.
If ``candidates`` 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.
"""
n = int(n)
if n < 3:
return None
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 b in candidates:
if b < 3:
if b == 1 or b == 2 and not_square:
continue
if b > max_possible:
return
# see if there is a factor present
if factor:
if n % afactor == 0:
# find what the potential power is
if afactor == 2:
b = trailing(n)
else:
b = multiplicity(afactor, n)
# if it's a trivial power we are done
if b == 1:
return
# maybe the bth root of n is exact
r, exact = integer_nthroot(n, b)
if not exact:
# then remove this factor and check to see if
# any of b's factors are a common exponent; if
# not then it's not a perfect power
n //= afactor**b
m = perfect_power(n, candidates=primefactors(b), big=big)
if not m:
return
else:
r, m = m
# adjust the two exponents so the bases can
# be combined
g = igcd(m, b)
m //= g
b //= g
r, b = r**m*afactor**b, g
if not big:
b0 = primefactors(b)
if len(b0) > 1 or b0[0] != b:
b0 = b0[0]
r, b = r**(b//b0), b0
return r, b
else:
# get the next factor ready for the next pass through the loop
afactor = nextprime(afactor)
# Weed out downright impossible candidates
if logn/b < 40:
a = 2.0**(logn/b)
if abs(int(a + 0.5) - a) > 0.01:
continue
# now see if the plausible b makes a perfect power
r, exact = integer_nthroot(n, b)
if exact:
if big:
m = perfect_power(r, big=big, factor=factor)
if m:
r, b = m[0], b*m[1]
return int(r), b
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
def perfect_power(n, candidates=None, big=True, factor=True):
"""
Return ``(a, b)`` such that ``n`` == ``a**b`` if ``n`` is a
perfect power; otherwise return ``None``.
By default, the base is recursively decomposed and the exponents
collected so the largest possible ``b`` is sought. If ``big=False``
then the smallest possible ``b`` (thus prime) will be chosen.
If ``candidates`` 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.
"""
n = int(n)
if n < 3:
return None
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 b in candidates:
if b < 3:
if b == 1 or b == 2 and not_square:
continue
if b > max_possible:
return
# see if there is a factor present
if factor:
if n % afactor == 0:
# find what the potential power is
if afactor == 2:
b = trailing(n)
else:
b = multiplicity(afactor, n)
# if it's a trivial power we are done
if b == 1:
return
# maybe the bth root of n is exact
r, exact = integer_nthroot(n, b)
if not exact:
# then remove this factor and check to see if
# any of b's factors are a common exponent; if
# not then it's not a perfect power
n //= afactor**b
m = perfect_power(n, candidates=primefactors(b), big=big)
if not m:
return
else:
r, m = m
# adjust the two exponents so the bases can
# be combined
g = igcd(m, b)
m //= g
b //= g
r, b = r**m * afactor**b, g
if not big:
b0 = primefactors(b)
if len(b0) > 1 or b0[0] != b:
b0 = b0[0]
r, b = r**(b // b0), b0
return r, b
else:
# get the next factor ready for the next pass through the loop
afactor = nextprime(afactor)
# Weed out downright impossible candidates
if logn / b < 40:
a = 2.0**(logn / b)
if abs(int(a + 0.5) - a) > 0.01:
continue
# now see if the plausible b makes a perfect power
r, exact = integer_nthroot(n, b)
if exact:
if big:
m = perfect_power(r, big=big, factor=factor)
if m:
r, b = m[0], b * m[1]
return int(r), b