def longest_period(n):
longest = [0, 0] # length, num
for x in range(3, n): # check all up to d for 1/d
pf = factorization(x)
if all(p == 2 or p == 5 for p in pf): # doesn't repeat
continue
elif len(pf) == 1: # run prime function
if longest[0] < period_if_prime(x):
longest = [period_if_prime(x), x]
else:
fact = pf
periods = []
for k in fact:
if k != 2 and k != 5:
if fact.count(k) == 1:
periods.append(period_if_prime(k))
else:
temp = k**(fact.count(k) - 1)
periods.append((period_if_prime(k)) * temp)
if lcm(periods) > longest[0]:
longest = [lcm(periods), x]
return longest[1]
def learn_mul_comparisons(H):
##############################
#
# IVar size info
#
##############################
def ge_one(i):
if i==0: return True
if (0, i) in H.term_comparisons.keys():
if [c for c in H.term_comparisons[0, i] if c.comp in [LT, LE] and
c.coeff > 0 and c.coeff <= 1]:
return True
return False
def le_one(i):
if i==0: return True
if (0, i) in H.term_comparisons.keys():
if [c for c in H.term_comparisons[0, i] if (c.comp in [GT, GE] and
c.coeff >= 1) or (c.comp in [LT, LE] and c.coeff < 0)]:
return True
return False
def le_neg_one(i):
if (0, i) in H.term_comparisons.keys():
if [c for c in H.term_comparisons[0, i] if c.comp in [LT, LE] and
c.coeff < 0 and c.coeff >= -1]:
return True
return False
def ge_neg_one(i):
if i==0: return True
if (0, i) in H.term_comparisons.keys():
if [c for c in H.term_comparisons[0, i] if (c.comp in [GT, GE] and
c.coeff <= -1) or (c.comp in [LE, LT] and c.coeff > 0)]:
return True
return False
def abs_ge_one(i):
return ge_one(i) or le_neg_one(i)
def abs_le_one(i):
return le_one(i) and ge_neg_one(i)
#############################
#
# Processing learned comparisons
#
#############################
# Takes comparison of the form |a_i|^{e0} comp coeff * |a_j|^{e1}
# Assumes coeff > 0
# Assumes that setting e1=e0 preserves the truth of the inequality.
# Takes e0th root of each side and learns the comparison
def take_roots_and_learn(i, j, e0, e1, comp, coeff):
if e0 < 0:
comp = comp_reverse(comp)
e0, e1, coeff = 1, 1, coeff ** Fraction(1, Fraction(e0))
coeff = round_coeff(coeff, comp)
comp, coeff = make_term_comparison_unabs(i, j, 1, 1, comp, coeff)
H.learn_term_comparison(i, j, comp, coeff, MUL)
# Takes a one_comparison of the form a_i^{e_i} * a_j^{e_j} * const >(=) 1,
# where a_i really represents |a_i|.
def learn_mul_comparison(i,j,comp,coeff,e0,e1):
if e0==e1==0:
return
elif e0==0:
i=0
elif e1==0:
j=0
if j<i:
i,j,e0,e1 = j,i,e1,e0
coeff = 1/Fraction(coeff)
if coeff<0:
comp = comp_reverse(comp)
e1 = -e1
if coeff < 0:
if comp in [LT, LE]: # pos < neg
H.raise_contradiction(MUL)
return # pos > neg. not useful.
if i==0: #a_i = 1, so we can set e0 to whatever we want.
e0 = e1
# Otherwise, both sides of the inequality are positive
# a_i and a_j are still abs values, coeff is positive
if (
(e0 == e1) # we have |a_i|^p comp coeff * |a_j|^p
or (e0 < e1 and comp in [LE, LT] and abs_le_one(j)) # making e1 smaller makes rhs bigger, which doesn't mess up comparison.
or (e0 > e1 and comp in [GE, GT] and abs_le_one(j)) # making e1 bigger makes rhs smaller
or (e0 < e1 and comp in [GE, GT] and abs_ge_one(j)) # making e1 smaller makes RHS smaller
or (e0 > e1 and comp in [LE, LT] and abs_ge_one(j)) # making e1 bigger makes RHS bigger
):
take_roots_and_learn(i, j, e0, e1, comp, coeff)
###################################
#
# Sign inference helper functions
#
###################################
def ivar_zero_comparison(i):
if i in H.zero_comparisons.keys():
return H.zero_comparisons[i].comp
return None
def mulpair_zero_comparison(m):
if m.term.index in H.zero_comparisons.keys():
comp = H.zero_comparisons[m.term.index].comp
if m.exp % 2 == 0 or (isinstance(m.exp, Fraction) and m.exp.numerator % 2 == 0):
if comp in [GT, LT]:
return GT
else:
return GE
else:
return comp
elif m.exp % 2 == 0:
return GE
else:
return None
def prod_zero_comparisons(comps):
if None in comps:
return None
else:
notstrict = LE in comps or GE in comps
if len([m for m in comps if m in [LT, LE]]) % 2 == 0:
if not notstrict:
return GT
else:
return GE
else:
if not notstrict:
return LT
else:
return LE
def sign_pow(s, n):
if s == 1 or (s == -1 and n % 2 == 0):
return 1
elif s == -1:
return -1
else:
raise Error("sign should be 1 or -1")
def multerm_zero_comparison(t):
comps = [mulpair_zero_comparison(m) for m in t.mulpairs]
return prod_zero_comparisons(comps)
def infer_signs_from_learned_equalities():
for t in (d for d in H.zero_equations if isinstance(d, Mul_term)):
# we have t=0.
comps = [mulpair_zero_comparison(m) for m in t.mulpairs]
# If there are more than two unknowns, we can't do anything.
if comps.count(None) > 1:
return
# If there is one unknown and every other is strict, that unknown must be equal to 0.
if comps.count(None) == 1 and comps.count(LE) + comps.count(GE) == 0:
index = comps.index(None)
i = t.mulpairs[index].term.index
H.learn_zero_equality(i, MUL)
elif comps.count(None) == 0:
ges, les = comps.count(GE), comps.count(LE)
# If there are no unknowns and only one is weak, that weak one must be equal to 0.
if ges == 1 and les == 0:
index = comps.index(GE)
H.learn_zero_equality(t.mulpairs[index].term.index, MUL)
elif les == 1 and ges == 0:
index = comps.index(LE)
H.learn_zero_equality(t.mulpairs[index].term.index, MUL)
# If there are no weak inequalities, we have a contradiction.
elif ges == 0 and les == 0:
H.raise_contradiction(MUL)
# This method looks at term definitions and tries to infer signs of subterms.
# For instance, if a_i = a_j * a_k, a_i > 0, a_j < 0, then it will learn a_k < 0.
# Provenance is HYP, so this info is available to mul routine
def infer_signs_from_definitions():
for i in (j for j in range(H.num_terms) if isinstance(H.name_defs[j], Mul_term)):
t = H.name_defs[i]
# have the equation a_i = t.
leftsign = ivar_zero_comparison(i)
comps = [mulpair_zero_comparison(m) for m in t.mulpairs]
rightsign = prod_zero_comparisons(comps)
if ((leftsign in [GE, GT] and rightsign == LT) or (leftsign in [LT, LE] and rightsign == GT)
or (leftsign == GT and rightsign in [LE, LT]) or (leftsign == LT and rightsign in [GE, GT])):
H.raise_contradiction(MUL)
if leftsign in [GT, LT]: # strict info on left implies strict info for all on right.
for j in range(len(comps)):
if comps[j] in [GE, LE]:
newcomp = (GT if comps[j] == GE else LT)
H.learn_zero_comparison(t.mulpairs[j].term.index, newcomp, HYP)
elif (rightsign in [GT, LT] and leftsign not in [GT, LT]) or (leftsign==None and rightsign!=None): # strict info on right implies strict info on left
H.learn_zero_comparison(i, rightsign, HYP)
elif (rightsign == GE and leftsign == LE) or (rightsign == LE and leftsign == GE): # we have zero equality.
H.learn_zero_equality(i, MUL)
if comps.count(GE) == 1 and comps.count(LE) == 0:
index = comps.index(GE)
H.learn_zero_equality(t.mulpairs[index].term.index, HYP)
elif comps.count(LE) == 1 and comps.count(GE) == 0:
index = comps.index(LE)
H.learn_zero_equality(t.mulpairs[index].term.index, HYP)
# Reset these with potentially new info
leftsign = ivar_zero_comparison(i)
comps = [mulpair_zero_comparison(m) for m in t.mulpairs]
rightsign = prod_zero_comparisons(comps)
# Two possibilities here.
# Case 1: lhs is strong, rhs is missing one. All others on rhs must be strong by above. Missing one must be strong too, and we can figure out what it is.
# Case 2: lhs is weak, rhs is missing one, all others on rhs are strong. Missing one must be weak.
if rightsign == None and comps.count(None) == 1 and (leftsign in [GT, LT] or (leftsign in [GE, LE] and comps.count(GE) + comps.count(LE) == 0)):
index = comps.index(None)
m = t.mulpairs[index]
comps.pop(index)
comps.append(leftsign)
new_comp = prod_zero_comparisons(comps)
index2 = m.term.index
H.learn_zero_comparison(index2, new_comp, HYP)
################################################
#
# Handle absolute values for elimination routine
#
#################################################
# this first routine takes i, j, comp, C representing
# ai comp C aj
# and returns a new pair comp', C', so that
# abs(ai) comp' C' abs(aj)
# is equivalent to the original comparison.
# assume signs are nonzero
def make_term_comparison_abs(i, j, comp, C):
C1 = C * H.sign(i) * H.sign(j)
if H.sign(i) == 1:
comp1 = comp
else:
comp1 = comp_reverse(comp)
return comp1, C1
# this routine takes i, j, ei, ej, comp1, C1 representing
# |ai|^{ei} comp1 C1 |aj|^{ej}
# and returns a new pair comp, C, so that
# ai^{ei} comp C aj^{aj}
# is equivalent to the original comparison.
# assume signs are nonzero
def make_term_comparison_unabs(i, j, ei, ej, comp1, C1):
correction = (H.sign(i) ** ei) * (H.sign(j) ** ej)
correction = 1 if correction > 0 else -1 # Make correction an int instead of a float
C = C1 * correction
if H.sign(i) ** ei == 1:
comp = comp1
else:
comp = comp_reverse(comp1)
return comp, C
# Assumes ai, aj > 0 and j is not 0
def one_comparison_from_term_comparison(i, j, comp, C):
if C <= 0:
if comp in [LE, LT]: # pos < neg. contradiction
H.raise_contradiction(MUL)
else: # pos > neg. useless
return None
if comp == GT or comp == GE:
new_comp = comp
if i == 0:
t = Mul_term([(IVar(j), -1)], Fraction(1, C))
else:
t = Mul_term([(IVar(i), 1), (IVar(j), -1)], Fraction(1, C))
else:
new_comp = comp_reverse(comp)
if i == 0:
t = Mul_term([(IVar(j), 1)], C)
else:
t = Mul_term([(IVar(i), -1), (IVar(j), 1)], C)
return One_comparison(t, new_comp)
########################
#
# MAIN ROUTINE
#
#######################
#H.info_dump()
if H.verbose:
print 'Learning multiplicative facts (polyhedron style...)'
print
infer_signs_from_definitions()
infer_signs_from_learned_equalities()
mul_eqs = [H.name_defs[i]*IVar(i)**(-1) for i in range(H.num_terms) if
(isinstance(H.name_defs[i],Mul_term) and
all(H.sign(p.term.index)!=0 for p in H.name_defs[i].mulpairs))]
if H.verbose:
print 'Multiplicative equations:'
for e in mul_eqs:
print e,'= 1'
print
mul_comps = []
for (i,j) in H.term_comparisons.keys():
if H.sign(i)!=0 and H.sign(j)!=0:
for c in (c for c in H.term_comparisons[i,j] if c.provenance!=MUL):
comp1, C1 = make_term_comparison_abs(i, j, c.comp, c.coeff)
onecomp = one_comparison_from_term_comparison(i, j, comp1, C1)
if onecomp:
mul_comps.append(onecomp)
if H.verbose:
print 'Multiplicative comparisons:'
print '(Note: here, a_i represents |a_i|)'
for c in mul_comps:
print c.term,comp_str[c.comp],'1'
print
#mul_eqs is a list of terms that are equal to 1
#mul_cops is a list of one_comparisons.
#turn these into zero_comps by taking logs,
#and store this info in a matrix to pass to lrs.
#Maps a prime number to the index of its ivar
index_of_prime = {}
class ind_store:
i = H.num_terms
#Inverse of above
prime_of_index = {}
#p is a prime
#returns the index of the ivar corresponding to p, or creates one
def get_index_of(p):
try:
return index_of_prime[p]
except KeyError:
index_of_prime[p] = ind_store.i
prime_of_index[ind_store.i] = p
ind_store.i+=1
return ind_store.i-1
add_eqs = []
for e in mul_eqs:
#take the log of each piece
p = e.mulpairs[0]
term = p.term*p.exp
for p in e.mulpairs[1:]:
term += p.term*p.exp
if e.const != 1:
if e.const < 0: #since we have taken absolute value, this should never happen
raise Exception
c_fract = Fraction(e.const)
fac = primes.factorization(c_fract.numerator)
for i in fac.keys():
term += fac[i]*IVar(get_index_of(i))
fac = primes.factorization(c_fract.denominator)
for i in fac.keys():
term -= fac[i]*IVar(get_index_of(i))
add_eqs.append(term)
add_comps = []
for c in mul_comps:
e = c.term
p = e.mulpairs[0]
term = p.term*p.exp
for p in e.mulpairs[1:]:
term += p.term*p.exp
if e.const != 1:
if e.const < 0:
raise Exception
c_fract = Fraction(e.const)
fac = primes.factorization(c_fract.numerator)
for i in fac.keys():
term += fac[i]*IVar(get_index_of(i))
fac = primes.factorization(c_fract.denominator)
for i in fac.keys():
term -= fac[i]*IVar(get_index_of(i))
#print c,'became:',term
add_comps.append(Zero_comparison(term,c.comp))
plist = sorted(index_of_prime.keys())
if H.verbose:
print 'We have',len(plist),'prime factors in play.'
for i in range(len(plist)-1):
add_comps.append(Zero_comparison(IVar(index_of_prime[plist[i+1]])-IVar(index_of_prime[plist[i]]),GT))
#add_comps.append(Zero_comparison(IVar(index_of_prime[plist[i]]) - under*IVar(index_of_prime[plist[i+1]]),GT))
#add_comps.append(Zero_comparison(IVar(index_of_prime[plist[i]]) - over*IVar(index_of_prime[plist[i+1]]),LT))
#if plist: add_comps.append(Zero_comparison(IVar(index_of_prime[plist[0]]),GT))
vertices,lin_set = lrs_util.get_generators(add_eqs,add_comps,H.num_terms+len(plist))
if H.verbose:
print 'Polyhedron vertices:',['Column 1 = delta.']+['Column '+str(j+1)+' = '+str(prime_of_index[j]) for j in sorted(prime_of_index.keys())]
for r in vertices:
print r[1:],('(ray)' if r[0]==0 else '(point)')
print
print 'Linear set:'
print lin_set
#H.info_dump()
for (i,j) in combinations(range(H.num_terms),2):
base_matrix = [[vertices[k][0],vertices[k][i+2],vertices[k][j+2]]+vertices[k][H.num_terms+2:] for k in range(len(vertices)) if k not in lin_set]
matrix = cdd.Matrix(base_matrix,number_type = 'fraction')
matrix.rep_type = cdd.RepType.GENERATOR
for k in lin_set:
matrix.extend([[vertices[k][0],vertices[k][i+2],vertices[k][j+2]]+vertices[k][H.num_terms+2:]],linear=True)
#print (i,j)
#print matrix
#matrix.canonicalize()
#print len(matrix),'vertices'
#print 'generating::'
#print lrs.redund_and_generate(matrix)
#exit()
#timecount.start()
#ineqs, lin_set2 = lrs.get_inequalities(matrix)
ineqs = cdd.Polyhedron(matrix).get_inequalities()
#timecount.stop()
#print len(ineqs), 'inequalities'
#ineqs,lin_set2 = lrs.redund_and_generate(matrix) #This will have the same effect, but right now it is slower. Need to import lrs to use
#print len([c for c in ineqs if c[2]!=0 or c[3]!=0]),'inequalities for',(i,j)
for c in ineqs:
if c[2]==c[1]==0: #no comp
continue
strong = not any(v[1]!=0 and v[i+2]*c[1]+v[j+2]*c[2]+sum(c[k]*v[H.num_terms+k-1] for k in range(3,len(c)))==0 for v in vertices)
const = 1
#Don't want constant to a non-int power
scale = int(reduce(operator.mul,(Fraction(c[k]).denominator for k in range(3,len(c))),1))
if scale!=1:
c = [c[0]]+[scale*v for v in c[1:]]
skip = False
for k in range(3,len(c)):
if c[k]!=0:
if c[k]>=1000000 or c[k]<=-1000000:
#Not going to get much here. Causes arithmetic errors.
skip = True
break
else:
const*=(prime_of_index[k+H.num_terms-3]**c[k])
if skip:
continue
if c[1]==0: #const*a_j^c[3] comp 1
learn_mul_comparison(0,j,(GT if strong else GE),const,1,c[2])
elif c[2]==0:
if i!=0:
learn_mul_comparison(0,i,(GT if strong else GE),const,1,c[1])
else:
learn_mul_comparison(i,j,(GT if strong else GE),const,c[1],c[2])
if H.verbose:
print
def jacoby_symbol(a, m):
fact = primes.factorization(m)
jac = 1
for p in fact:
jac = jac * legendre_symbol(a, p)
return jac
if candidates_indices[index] > i:
candidates[index] = number
except IndexError:
candidates[index] = number
yield number
limit = 12000
minNValues = [0] * (limit + 1)
setMinNValues = set()
NToKList = [set(), set(), set(), set()]
N = 4
count = 1
while count < limit:
kValues = set()
prime_factorization = factorization(N)
factors = pop_factor(prime_factorization)
factor = factors.next()
root = N**.5
while factor <= root:
k = find_k((factor, N / factor))
if k > N:
break
kValues.add(k)
kSet = NToKList[N / factor]
if len(kSet) == 0:
break
for i in kSet:
kValues.add(k + i - 1)
factor = factors.next()
NToKList.append(kValues)
import math from primes import factorization n = 600851475143 # primes = [] # while n > 1: # for i in range(2, int(math.sqrt(n))): # if n % i == 0: # primes.append(i) # n /= i # break # print primes print factorization(n)
candidates[index] = prime_factorization[index][0]*nums[candidates_indices[index]] if candidates_indices[index] > i: candidates[index] = number except IndexError: candidates[index] = number yield number limit = 12000 minNValues = [0]*(limit + 1) setMinNValues = set() NToKList = [set(), set(), set(), set()] N = 4 count = 1 while count < limit: kValues = set() prime_factorization = factorization(N) factors = pop_factor(prime_factorization) factor = factors.next() root = N**.5 while factor <= root: k = find_k((factor, N/factor)) if k > N: break kValues.add(k) kSet = NToKList[N/factor] if len(kSet) == 0: break for i in kSet: kValues.add(k + i - 1) factor = factors.next() NToKList.append(kValues)