Permalink
Find file Copy path
Fetching contributors…
Cannot retrieve contributors at this time
160 lines (135 sloc) 6.03 KB
"""Search for counterexamples to Beal's conjecture
See http://norvig.com/beal.html and http://www.bealconjecture.com"""
from __future__ import division, print_function
from math import log
from itertools import combinations, product
from collections import defaultdict
try:
from math import gcd # For Python 3.6 and up
except ImportError:
from fractions import gcd # For older versions (works in 2.7 as well)
def beal(max_A, max_x):
"""See if any A ** x + B ** y equals some C ** z, with gcd(A, B) == 1.
Consider any 1 <= A,B <= max_A and x,y <= max_x, with x,y prime or 4."""
Apowers = make_Apowers(max_A, max_x)
Czroots = make_Czroots(Apowers)
for (A, B) in combinations(Apowers, 2):
if gcd(A, B) == 1:
for (Ax, By) in product(Apowers[A], Apowers[B]):
Cz = Ax + By
if Cz in Czroots:
C = Czroots[Cz]
x, y, z = exponent(Ax, A), exponent(By, B), exponent(Cz, C)
print('{} ** {} + {} ** {} == {} ** {} == {}'
.format(A, x, B, y, C, z, C ** z))
def make_Apowers(max_A, max_x):
"A dict of {A: [A**3, A**4, ...], ...}."
exponents = exponents_upto(max_x)
return {A: [A ** x for x in (exponents if (A != 1) else [3])]
for A in range(1, max_A+1)}
def make_Czroots(Apowers): return {Cz: C for C in Apowers for Cz in Apowers[C]}
def exponents_upto(max_x):
"Return all odd primes up to max_x, as well as 4."
exponents = [3, 4] if max_x >= 4 else [3] if max_x == 3 else []
for x in range(5, max_x, 2):
if not any(x % p == 0 for p in exponents):
exponents.append(x)
return exponents
def exponent(Cz, C):
"""Recover z such that C ** z == Cz (or equivalently z = log Cz base C).
For exponent(1, 1), arbitrarily choose to return 3."""
return 3 if (Cz == C == 1) else int(round(log(Cz, C)))
##############################################################################
def tests():
assert make_Apowers(6, 10) == {
1: [1],
2: [8, 16, 32, 128],
3: [27, 81, 243, 2187],
4: [64, 256, 1024, 16384],
5: [125, 625, 3125, 78125],
6: [216, 1296, 7776, 279936]}
assert make_Czroots(make_Apowers(5, 8)) == {
1: 1, 8: 2, 16: 2, 27: 3, 32: 2, 64: 4, 81: 3,
125: 5, 128: 2, 243: 3, 256: 4, 625: 5, 1024: 4,
2187: 3, 3125: 5, 16384: 4, 78125: 5}
Czroots = make_Czroots(make_Apowers(100, 100))
assert 3 ** 3 + 6 ** 3 in Czroots
assert 99 ** 97 in Czroots
assert 101 ** 100 not in Czroots
assert Czroots[99 ** 97] == 99
assert exponent(10 ** 5, 10) == 5
assert exponent(7 ** 3, 7) == 3
assert exponent(1234 ** 999, 1234) == 999
assert exponent(12345 ** 6789, 12345) == 6789
assert exponent(3 ** 10000, 3) == 10000
assert exponent(1, 1) == 3
assert exponents_upto(2) == []
assert exponents_upto(3) == [3]
assert exponents_upto(4) == [3, 4]
assert exponents_upto(40) == [3, 4, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37]
assert exponents_upto(100) == [
3, 4, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61,
67, 71, 73, 79, 83, 89, 97]
assert gcd(3, 6) == 3
assert gcd(3, 7) == 1
assert gcd(861591083269373931, 94815872265407) == 97
assert gcd(2*3*5*(7**10)*(11**12), 3*(7**5)*(11**13)*17) == 3*(7**5)*(11**12)
return 'tests pass'
##############################################################################
def beal_modp(max_A, max_x, p=2**31-1):
"""See if any A ** x + B ** y equals some C ** z (mod p), with gcd(A, B) == 1.
If so, verify that the equation works without the (mod p).
Consider any 1 <= A,B <= max_A and x,y <= max_x, with x,y prime or 4."""
assert p >= max_A
Apowers = make_Apowers_modp(max_A, max_x, p)
Czroots = make_Czroots_modp(Apowers)
for (A, B) in combinations(Apowers, 2):
if gcd(A, B) == 1:
for (Axp, x), (Byp, y) in product(Apowers[A], Apowers[B]):
Czp = (Axp + Byp) % p
if Czp in Czroots:
lhs = A ** x + B ** y
for (C, z) in Czroots[Czp]:
if lhs == C ** z:
print('{} ** {} + {} ** {} == {} ** {} == {}'
.format(A, x, B, y, C, z, C ** z))
def make_Apowers_modp(max_A, max_x, p):
"A dict of {A: [(A**3 (mod p), 3), (A**4 (mod p), 4), ...]}."
exponents = exponents_upto(max_x)
return {A: [(pow(A, x, p), x) for x in (exponents if (A != 1) else [3])]
for A in range(1, max_A+1)}
def make_Czroots_modp(Apowers):
"A dict of {C**z (mod p): [(C, z),...]}"
Czroots = defaultdict(list)
for A in Apowers:
for (Axp, x) in Apowers[A]:
Czroots[Axp].append((A, x))
return Czroots
##############################################################################
def simpsons(bases, powers):
"""Find the integers (A, B, C, n) that come closest to solving
Fermat's equation, A ** n + B ** n == C ** n.
Let A, B range over all pairs of bases and n over all powers."""
equations = ((A, B, iroot(A ** n + B ** n, n), n)
for A, B in combinations(bases, 2)
for n in powers)
return min(equations, key=relative_error)
def iroot(i, n):
"The integer closest to the nth root of i."
return int(round(i ** (1./n)))
def relative_error(equation):
"Error between LHS and RHS of equation, relative to RHS."
(A, B, C, n) = equation
LHS = A ** n + B ** n
RHS = C ** n
return abs(LHS - RHS) / RHS
if __name__ == '__main__':
print(tests())
print("Searching beal(500, 100)")
print(beal(500, 100))
print("Finding Simpson-esque near-solutions to Fermat's Equation")
def s(b, p): print('{0}^{3} + {1}^{3} = {2}^{3}'.format(*simpsons(b, p)))
s(range(1000, 2000), [11, 12, 13])
s(range(3000, 5000), [12])
print("Searching beal_modp(500, 100)")
print(beal_modp(500, 100))