def prob_solution(a, x, b, y, c, z):
num_pp = len(safe_primes)
plus = True
minus = True
for i in range(num_ps_tests):
#using safe primes ensures that the order of a and b modulo m is high, thus ensuring the probability of a^x+/-b^y=c^z is low, on the order of 1/m
m = safe_primes[randrange(num_pp)]
ax, by, cz = exp_bs(a, x, m), exp_bs(b, y, m), exp_bs(c, z, m)
if plus and (ax + by) % m != cz:
plus = False
s = '-'
if not minus:
break
if minus and (ax - by) % m != cz:
minus = False
s = '+'
if not plus:
break
else:
return [True, s]
return [False, s]
def get_y(a, x, c, z, b, lrab, max_yc, b_gp):
y_res_mod = []
max_ya = x * lrab
max_y = max(max_ya, max_yc)
p = 1
for pprime, pprime_sq, prime_sq_tot, prime_div in b_gp:
#squared prime modulus ensures totient is divisible by prime
ax, cz = exp_bs(a, x, pprime_sq), exp_bs(c, z, pprime_sq)
d = (ax - cz) % pprime_sq
if is_coprime(d, pprime_sq):
y_sp = dlog_mod(b, d, pprime_sq, prime_sq_tot, prime_div)
y_res_mod.append([y_sp, pprime])
p *= pprime
if p > max_y:
break
else:
print("Not enough primes to get y")
#raise Error("Not enough primes to get y")
return crt(y_res_mod)
def get_x(a, x, c, z, m, mf, t, lt):
cz = exp_bs(c, z, m)
f = exp_bs(a, t - x, m)
g = exp_bs(a, lt, m)
h = f * cz % m
p = dlog(g, h, m, mf)
x += p * lt
return x
def beals_solver(b, y, c, z, max_a, max_x):
"""finds a and x s.t. a^x + b^y = c^z
b, c mutually coprime"""
crt_mod = 1
rep_crt = []
for pprime in prob_primes:
pp_sq = pprime**2
t = pprime * (pprime - 1)
by = exp_bs(b, y, pp_sq)
cz = exp_bs(c, z, pp_sq)
d = (cz - by) % pp_sq
res_exp_pairs = []
for a in range(pp_sq):
l = 0 if a == 0 else 1
for x in range(t):
if l == d:
re_pair = [a % pprime, x % pprime]
if re_pair not in res_exp_pairs:
res_exp_pairs.append(re_pair)
else:
l = l * a % pp_sq
rep_crt.append([pprime, res_exp_pairs])
crt_mod *= pprime
if crt_mod > max_a and crt_mod > max_x:
break
np = reduce(lambda a, b: a * len(b[1]), rep_crt, 1)
#np = reduce(lambda a, b: a * b[0], rep_crt, 1)
print("{!s} chinese remainder theorem permutations".format(np))
ax_pairs = crt_permutation(rep_crt, b, y, c, z)
for axp in ax_pairs:
a, x = axp
print("\t{!s}^{!s} + {!s}^{!s} = {!s}^{!s}".format(a, x, b, y, c, z))
def prob_beals(min_xyz=3, min_abc=2):
populate_pprimes_list()
#minimum sum is x+y+z=3+3+3=9 and a+b+c=2+3+5=10
s = 19
while True:
#print("Shell with sum {!s} completed!".format(s))
for a in range(min_abc, s + 1):
s_1 = s - a
for b in range(min_abc, s_1 + 1):
s_2 = s_1 - b
if b > a and is_coprime(a, b):
for c in range(min_abc, s_2 + 1):
s_3 = s_2 - c
if is_coprime(a, c) and is_coprime(b, c):
for x in range(min_xyz, s_3 + 1):
s_4 = s_3 - x
for y in range(min_xyz, s_4 + 1):
z = s_4 - y
if z >= min_xyz:
for p in prob_primes:
p_tot = p - 1
a_m, b_m, c_m = a % p, b % p, c % p
x_m, y_m, z_m = x % p_tot, y % p_tot, z % p_tot
a_x, b_y, c_z = exp_bs(a, x, p), exp_bs(b, y, p), exp_bs(c, z, p)
if (a_x + b_y) % p != c_z:
break
else:
print("{!s}^{!s} + {!s}^{!s} = {!s}^{!s}".format(a, x, b, y, c, z))
s += 1
def miller_rabin_test(n, pci=prob_comp_ind):
nm = n - 1
d = nm
s = 0
#number of tests required to establish probable primeness
k = math.ceil(-math.log(pci, 4))
while d % 2 == 0:
s += 1
d //= 2
sm = s - 1
for i in range(k):
a = randrange(2, nm)
x = exp_bs(a, d, n)
if x == 1 or x == nm:
continue
else:
for j in range(sm):
x = x**2 % n
#early stopping x=1 for all powers of s > j
if x == 1:
return False
#early stopping, the condition we're trying to prevent
if x == nm:
break
else:
return False
return True
