def check(self, f, force_prime=None):
# try for prime divisors of constant term (might add leading term later too)
const_coeff = abs(f.TC())
if const_coeff == 0:
return REDUCIBLE, None
if force_prime is not None:
if const_coeff % force_prime != 0:
return UNKNOWN, None
primes = [force_prime]
else:
primes = sympy.ntheory.factorint(const_coeff)
satisfies = []
for p in primes:
# calculate points for given prime p for all coeffs
points = []
for exp, coeff in poly_non_zero_exps(f):
if coeff != 0:
coeff_factors = sympy.ntheory.factorint(coeff)
padic = coeff_factors[p] if p in coeff_factors else 0
points.append((exp, padic))
# if this can be of any use, length of line between end points must not cross any integers
first = points[0]
last = points[-1]
x_diff = abs(first[0] - last[0])
y_diff = abs(first[1] - last[1])
if sympy.gcd(x_diff, y_diff) != 1:
continue
# now just check if all the other points are above this line
x1 = first[0]
y1 = first[1]
x2 = last[0]
y2 = last[1]
above = True
for i in range(1, len(points) - 1):
x, y = points[i]
if (y - y1) * (x2 - x1) <= (x - x1) * (y2 - y1):
above = False
break
if not above:
continue
satisfies.append(p)
if satisfies:
return IRREDUCIBLE, {"p": satisfies}
return UNKNOWN, None
def check(self, f):
# http://cms.dm.uba.ar/academico/materias/2docuat2011/teoria_de_numeros/Irreducible.pdf
# Theorem 1
h = 0
for exp, coeff in poly_non_zero_exps(f):
if exp != f.degree():
h = max(h, abs(coeff / f.LC()))
nmin = int(math.ceil(h + 2))
for n in range(nmin, nmin + 5):
val = f.eval(n)
if sympy.isprime(val) and (not self.max_p or val < self.max_p):
return IRREDUCIBLE, {'n': n, 'p': val}
return UNKNOWN, None
def check(self, f):
const_coeff = f.TC()
if const_coeff == 0:
return REDUCIBLE, None
lead_coeff = f.LC()
if lead_coeff != 1:
return UNKNOWN, None
a_nm1 = abs(get_coeff(f, f.degree() - 1))
s = 1
for exp, coeff in poly_non_zero_exps(f):
if exp < f.degree() - 1:
s += abs(coeff)
if a_nm1 > s:
return IRREDUCIBLE, None
return UNKNOWN, None
def check(self, f):
# first we need to ensure coefficients are non-negative
max_coeff = 0
for exp, coeff in poly_non_zero_exps(f):
if coeff < 0:
return UNKNOWN, None
max_coeff = max(max_coeff, coeff)
check_range = 30
for base in range(max_coeff + 1, max_coeff + 1 + check_range):
val = f.subs(VAR_X, base)
if sympy.isprime(val) and (not self.max_p or val < self.max_p):
return IRREDUCIBLE, {"base": base, "p": val}
return UNKNOWN, None
def check(self, f):
# Polynomials - Prasolov - Theorem 2.2.7 ([Os1]) part b)
lead_coeff = f.LC()
if lead_coeff != 1:
return UNKNOWN, None
const_coeff = abs(f.TC())
if not sympy.isprime(const_coeff):
return UNKNOWN, None
if self.max_p and const_coeff >= self.max_p:
return UNKNOWN, None
s = 0
for exp, coeff in poly_non_zero_exps(f):
if exp != f.degree() and exp != 0:
s += abs(coeff)
if const_coeff < 1 + s:
return UNKNOWN, None
import mpmath
# check if there are roots inside as well as outside of unit circle
try:
all = mpmath.polyroots(f.all_coeffs(), maxsteps=100)
inside_unit_circle = 0
outside_unit_circle = 0
on_unit_circle = 0
for root in all:
root_size = abs(root)
if root_size == 1:
on_unit_circle += 1
elif root_size < 1:
inside_unit_circle += 1
else:
outside_unit_circle += 1
if on_unit_circle == 0:
return IRREDUCIBLE, {'s': const_coeff,
"on": on_unit_circle}
except mpmath.libmp.libhyper.NoConvergence:
# could not get complex roots, too bad
pass
return UNKNOWN, None
def check(self, f):
# Polynomials - Prasolov - Theorem 2.2.7 ([Os1]) part a)
lead_coeff = f.LC()
if lead_coeff != 1:
return UNKNOWN, None
const_coeff = abs(f.TC())
if not sympy.isprime(const_coeff):
return UNKNOWN, None
if self.max_p and const_coeff >= self.max_p:
return UNKNOWN, None
s = 0
for exp, coeff in poly_non_zero_exps(f):
if exp != f.degree() and exp != 0:
s += abs(coeff)
if const_coeff > 1 + s:
return IRREDUCIBLE, {'s': const_coeff}
return UNKNOWN, None
def check(self, f):
# see https://www.lehigh.edu/~shw2/preprints/eisenstein.pdf,
# and https://math.stackexchange.com/questions/3589657/prove-polynomial-is-irreducible/3590300#3590300
const_coeff = f.TC()
if const_coeff == 0:
return REDUCIBLE, None
linear_coeff = get_coeff(f, 1)
if linear_coeff == 0:
return UNKNOWN, None
primes = sympy.ntheory.factorint(linear_coeff)
for p in primes:
power = primes[p]
if power == 1:
# good candidate, it must not divide leading term
lead_coeff = f.LC()
if lead_coeff % p == 0:
# bad luck
continue
# it must divide or other terms
for exp, coeff in poly_non_zero_exps(f):
if exp != f.degree():
if coeff % p != 0:
# bad luck...
return UNKNOWN, None
# finally the polynomial must not have rational roots
numerators = sympy.ntheory.factorint(const_coeff)
denominators = sympy.ntheory.factorint(lead_coeff)
for num in numerators:
for denom in denominators:
for sign in (-1, 1):
if f.eval(sympy.Rational(sign * num, denom)) == 0:
# rational root...
return UNKNOWN, None
return IRREDUCIBLE, {"p": p}
return UNKNOWN, None