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_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 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 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)