def try_all_beals(lim=1000):
populate_all()
#b is in the primes since composite moduli don't have primitve roots. Techincally, b could be in the prime powers or 2p^k, but the first is redundant for this application while the second isn't satisfied since b must be a power greater than two
for b in prob_primes:
print(b)
b_gp = get_gen_psq(b)
for a in range(3, lim):
if is_gen_n_k(a, b, all_fact[b]):
for c in range(3, lim):
if odd_num_evens(a, b, c) and is_coprime(a, c) and is_coprime(b, c):
for z in range(3, lim):
r = beals_solver(a, b, c, z, b_gp)
#print(a, b, c, z)
if r[2][0]:
if r[0] > 2 and r[1] > 2:
print("\t!!!!{!s}^{!s} {} {!s}^{!s} = {!s}^{!s}!!!!".format(a, r[0], r[2][1], b, r[1], c, z))
#wait = input("\tPossible solution found. Press enter to continue.")
elif r[0] > 1 and r[1] > 1 and max(r[0], r[1]) > 2:
print("\t{!s}^{!s} {} {!s}^{!s} = {!s}^{!s}".format(a, r[0], r[2][1], b, r[1], c, z))
def get_gen_psq(x):
gen_prime_sq = []
for i, pprime in enumerate(prob_primes):
if is_gen_n_k(x, pprime, all_fact[pprime]):
gen_prime_sq.append([pprime, pprime**2, prime_sq_fact[i][0], [pprime, 1]])
return gen_prime_sq
def test_beals_solver(n_tests=100):
populate_all()
num_pp = len(prob_primes)
num_af = len(all_fact)
print("Beginning {!s} tests of Beals Solver...".format(n_tests))
while (n_tests > 0):
a = randrange(num_af)
b = prob_primes[randrange(num_pp)]
if is_gen_n_k(a, b, all_fact[b]):
x = randrange(num_af)
#need to ensure that y is large enough that phi(b^y) > x
min_y = tot_cover(b, num_af)
y = randrange(min_y, num_af)
ax = a**x
by = b**y
if randrange(0, 2):
c = ax + by
s = '+'
else:
c = ax - by
if c < 0:
continue
else:
s = '-'
b_gp = get_gen_psq(b)
#print("\Solving {!s}^{!s} {!s} {!s}^{!s}...".format(a, x, s, b, y))
r = beals_solver(a, b, c, 1, b_gp)
assert r[0] == x and r[1] == y and r[2][0] and r[2][1] == s
n_tests -= 1
print("All tests passed!")
def get_y(a, x, c, z, b, lrab, max_yc):
y_res_mod = []
max_ya = x * lrab
max_y = max(max_ya, max_yc)
p = 1
mr = MontgomeryReduction()
for i, pprime in enumerate(prob_primes):
if is_gen_n_k(b, pprime, all_fact[pprime]):
m = pprime**2
#squaring ensures totient is divisible by prime
mr.set_modulus(m, prime_sq_fact[i])
ra, rc, rb = mr.convert(a), mr.convert(c), mr.convert(b)
rax, rcz = mr.exp(ra, x), mr.exp(rc, z)
rd = (rax - rcz) % m
if is_coprime(rd, m):
y_sp = mr.dlog_mod(rb, rd, [pprime, 1])
y_res_mod.append([y_sp, pprime])
p *= pprime
if p > max_y:
break
else:
ipdb.set_trace()
print("Not enough primes to get y")
#raise Error("Not enough primes to get y")
return crt(y_res_mod)