def factor(n):
if m_r(n):
return [n]
factorList = []
while n != 1:
if m_r(n):
return (factorList + [n])
d = pollard_rho(n)
while n % d == 0:
factorList.append(d)
n /= d
return factorList
def factor(n):
if m_r(n):
return [n]
factorList = []
while n != 1:
if m_r(n):
return (factorList + [n])
d = pollard_rho(n)
while n % d == 0:
factorList.append(d)
n /= d
return factorList
def main():
sumz = 0
for i in xrange(LOWER_LIM, UPPER_LIM):
if m_r(i):
result = f(i, math.sqrt(i))
sumz += result
print i, result, '\t', time.time() - START
print sumz % MOD
def primeProof(n):
for digit in xrange(1, int(log(n, 10)) + 2):
relevantDig = n % 10**digit - (n % 10**(digit - 1))
m = int(n - relevantDig)
digitChange = 10**(digit - 1)
for i in xrange(0, 10):
if m_r(m + i * digitChange):
return False
return True
def primeProof(n):
for digit in xrange(1,int(log(n,10))+2):
relevantDig = n % 10**digit - (n % 10** (digit-1))
m = int(n - relevantDig)
digitChange = 10**(digit-1)
for i in xrange(0, 10):
if m_r(m + i* digitChange):
return False
return True
def D(a,b,k):
sumz = 0
for p in xrange(a,a+b):
if m_r(p):
inv = [0] + [ multInverse(x,p) for x in xrange(1, p)]
tempSum = 0
rollingNcr = 1
for i in xrange(p-1,k-2,-1):
index = p - i
tempSum += inv[i] * rollingNcr
rollingNcr = (rollingNcr * (k + index - 1) * inv[index]) % p
sumz += tempSum % p
print p, tempSum % p
return sumz
def D(a, b, k):
sumz = 0
for p in xrange(a, a + b):
if m_r(p):
inv = [0] + [multInverse(x, p) for x in xrange(1, p)]
tempSum = 0
rollingNcr = 1
for i in xrange(p - 1, k - 2, -1):
index = p - i
tempSum += inv[i] * rollingNcr
rollingNcr = (rollingNcr * (k + index - 1) * inv[index]) % p
sumz += tempSum % p
print p, tempSum % p
return sumz
import time
from primes import m_r
START = time.time()
num = 10**12
start_val = 2 * 10**9
end_val = 2 * 10**9 + 2000
power = 10**4
running_sum = 0
for p in xrange(start_val, end_val):
# ignore non primes
if m_r(p):
nModp = num % p
running_sum += (-1 * sum( [ (nModp + 1 - x) * pow(x, power, p) \
for x in xrange((num+1)%p, p+1) ])) % p
print p, running_sum
print running_sum
print "Time Taken:", time.time() - START
"""
Congratulations, the answer you gave to problem 487 is correct.
You are the 358th person to have solved this problem.
Answer : 106650212746
Time Taken : 16.3226361275
Quick summary on this problem.
The problem asks us to find:
sum_{2*10^9 <= p <= 2*10^9, p prime} (
START = time.time()
LIM = 10**12
validHammingNumbers = set()
validPrimes = set()
for i in xrange(0,int(math.log(LIM, 2)) +1):
for j in xrange(0,int(math.log(LIM / 2**i, 3)) + 1):
for k in xrange(0,18):
if 2**i * 3**j * 5**k > LIM:
break
validHammingNumbers.add(2**i * 3**j * 5**k)
# Figure out which primes p have the property totient(p) = hammingNumber
for hammingNumber in validHammingNumbers:
if m_r(hammingNumber +1):
validPrimes.add(hammingNumber +1)
validPrimes = validPrimes.difference(set([2,3,5])) # these are included in hamming numbers already
validPrimes = sorted(validPrimes)[::-1]
validHammingNumbers.remove(1) # unnecessary, and causes slowdown
validHammingNumbers = sorted(validHammingNumbers)[::-1]
# compute every single product of primes ignoring 2,3,5 which results in totient(product) = hammingNumber
newSet = set([1])
for prime in validPrimes:
oldSet = newSet.union(set())
for oldNumber in oldSet:
if oldNumber * prime <= LIM:
newSet.add(prime * oldNumber)
s, t = extended_gcd(b, r)
return (t, s - q * t)
ext_gcd = extended_gcd
#############################################################
##################### Getting 100,000 primes ################
#############################################################
i = 10**14
count = 0
prime_lst = []
while count < 10**5:
i += 1
if m_r(i):
count += 1
prime_lst.append(i)
print "Time Taken:", time.time() - start
#############################################################
################## Finding the nth fibo num ################
#############################################################
pToTest = [3, 7, 13, 67, 107, 630803] #Factorization of 1234567891011
fibo_lsts = []
for prime in pToTest:
fib = [0, 1, 1, 2]
while fib[-1] != 1 or fib[-2] != 0:
def get_primes(start, end):
primes = []
for i in xrange(start, end):
if m_r(i):
primes.append(i)
return primes
import time
from primes import m_r
START = time.time()
num = 10**12
start_val = 2*10**9
end_val = 2*10**9 + 2000
power = 10**4
running_sum = 0
for p in xrange(start_val, end_val):
# ignore non primes
if m_r(p):
nModp = num % p
running_sum += (-1 * sum( [ (nModp + 1 - x) * pow(x, power, p) \
for x in xrange((num+1)%p, p+1) ])) % p
print p, running_sum
print running_sum
print "Time Taken:", time.time() - START
"""
Congratulations, the answer you gave to problem 487 is correct.
You are the 358th person to have solved this problem.
Answer : 106650212746
Time Taken : 16.3226361275
Quick summary on this problem.
q, r = a/b, a%b
s, t = extended_gcd(b, r)
return (t, s - q * t)
ext_gcd = extended_gcd
#############################################################
##################### Getting 100,000 primes ################
#############################################################
i = 10**14
count = 0
prime_lst = []
while count < 10**5:
i +=1
if m_r(i):
count +=1
prime_lst.append(i)
print "Time Taken:", time.time() - start
#############################################################
################## Finding the nth fibo num ################
#############################################################
pToTest = [3,7,13,67,107,630803] #Factorization of 1234567891011
fibo_lsts = []
for prime in pToTest:
fib = [0,1,1,2]
while fib[-1] != 1 or fib[-2] != 0:
pos = n - row_start
if pos != 0:
if row % 2 == 0:
neighbors = [n - row + 2, n - row, n + row]
else:
neighbors = [n - row + 1, n + row - 1, n + row + 1]
else:
neighbors = [n + row, n + row + 1, n - row + 1, n - row + 2]
return neighbors
sumz = 0
for p in pos:
i = p * (p - 1) / 2 + 1
while i < p * (p + 1) / 2 - 1:
if m_r(i):
neigh = neighbors(i, p)
neigh_sum = sum([m_r(x) for x in neigh])
if neigh_sum >= 2:
sumz += i
else:
for num in neigh:
row = p - 1 if num < i else p + 1
if m_r(num):
neigh_sum = sum([m_r(x) for x in neighbors(num, row)])
if neigh_sum >= 2:
sumz += i
break
i += 1
print sumz
print sumz
import time, math
from primes import m_r, findNumLessThan
START = time.time()
SIZE = 10**11
RELEVANT_SIZE = SIZE / 21125 # 21125 = 5^3 x 13^2
oneMod4Primes = filter(lambda x: m_r(x), xrange(5, RELEVANT_SIZE + 1, 4))
threeMod4Primes = [2] + filter(lambda x: m_r(x), xrange(
3, RELEVANT_SIZE + 1, 4))
large3Mod4PrimesIndex = findNumLessThan(threeMod4Primes, RELEVANT_SIZE**.5)
threeMod4Products = set([1] + threeMod4Primes[large3Mod4PrimesIndex + 1:])
for p in threeMod4Primes[large3Mod4PrimesIndex:0:-1] + [2]:
for power in xrange(1, int(math.log(RELEVANT_SIZE, p) + 1)):
threeMod4Products = threeMod4Products.union( \
filter((lambda x: x < RELEVANT_SIZE), \
[ x*p**power for x in threeMod4Products]))
threeMod4Products = sorted(threeMod4Products)
def compute(power1, power2, power3=0):
sumz = 0
for prime1 in oneMod4Primes:
if prime1**power1 * 5**power2 > SIZE:
break
for prime2 in oneMod4Primes:
#print prime1, prime2, prime1**power1*prime2**power2
if prime1 == prime2:
->
(x^2 - xy + y^2) | y^3
For now, assume:
x^2 - xy + y^2 = y^3
Then:
x = (y + y * sqrt(4y - 3)) / 2
"""
count = 0
for i in xrange(1, int(math.sqrt(LIM)), 2):
for v in xrange(1, 100):
y = v * (i**2 + 3) / 4
x = y * (1 + i) / (2 * v)
if x < y:
continue
if (x**4 - y**4) % (x**3 + y**3) != 0:
# print x, y, (x**4 - y**4) % (x**3 + y**3), "ERROR"
continue
number = (x**4 - y**4) / (x**3 + y**3)
if number > LIM:
break
if m_r(number):
print number, x, y, v, x**4 - y**4, x**3 + y**3
count += number
print "answer:", count
print "Time taken:", time.time() - START
import time
from primes import get_primes, divisors, m_r
START = time.time()
SIZE = 10**6 * 5
LIM = 10**5
primes = get_primes(SIZE)
div308 = divisors(308)
results = []
for p in primes:
if len(results) > LIM: break
if not any([m_r(div * p + 1) for div in div308]):
results.append(p)
print results[LIM - 2] * 308, len(results)
for repetition in xrange(10):
resultSet = set(results)
for index, a in enumerate(results):
if a * results[0] > results[LIM]: break
for bIndex in xrange(index + 1):
b = results[bIndex]
p = a * b
if a * b > results[LIM]: break
if not any([m_r(div * p + 1) for div in div308]):
if len(filter(lambda x: x not in resultSet,
divisors(p)[1:-1])) == 0:
from primes import m_r, primes
import time
START = time.time()
SIZE = 10**6
MAX_PRIMES_LIM = 600 # sum of [2,3,5,7, ...] first 546 primes = 997,661
maxNumPrimes = 0
for startingPrime in xrange(0, MAX_PRIMES_LIM):
if sum(primes[startingPrime:startingPrime + maxNumPrimes]) > SIZE:
break
lst = primes[:MAX_PRIMES_LIM]
k = startingPrime + 1
while lst[k-1] < SIZE:
lst[k] += lst[k-1]
k += 1
for i in xrange(k-1,0,-1):
if m_r(lst[i]):
if maxNumPrimes < i:
print lst[i], i
maxNumPrimes = i
break
print "Time taken:",time.time() -START
primeSets = [set() for i in xrange(6)]
baseDict = {1 : 0, 2 : 0, 3 : 0, 5 : 0, 7 : 0, 11 : 0, 13 : 0}
TOTIENT_VAL = math.factorial(13)
INDEX = 150000 -1
possiblePrimes = set()
possibleTotients = set([1])
for prime in primeFactors.keys():
newSet = set()
for power in xrange(1, primeFactors[prime]+1):
newSet.update(set([ prime**power * num for num in possibleTotients]))
possibleTotients.update(newSet)
for num in possibleTotients:
if m_r(num+1):
possiblePrimes.add(num+1)
# Arrange products into sets based on their largest prime factor
for num in list(possiblePrimes):
for pIndex in xrange(len(primes)):
if (num - 1) % primes[pIndex] == 0:
primeSets[pIndex].add(num)
break
def factorAndAddToDict(num, valueDict):
factors = factor(num)
for num in factors:
valueDict[num] += 1
def checkPrimePowers(valDict, func):
row_start = row*(row-1)/2 + 1
pos = n - row_start
if pos != 0:
if row % 2 == 0:
neighbors = [ n-row+2,n-row,n+row]
else:
neighbors = [n-row+1,n+row-1,n+row+1]
else:
neighbors = [n+row,n+row+1,n-row+1,n-row+2]
return neighbors
sumz = 0
for p in pos:
i = p*(p-1)/2+1
while i < p*(p+1)/2 - 1:
if m_r(i):
neigh = neighbors(i,p)
neigh_sum = sum([m_r(x) for x in neigh])
if neigh_sum >= 2:
sumz += i
else:
for num in neigh:
row = p-1 if num < i else p+1
if m_r(num):
neigh_sum = sum([m_r(x) for x in neighbors(num,row)])
if neigh_sum >= 2:
sumz +=i
break
i+=1
print sumz
print sumz
primeSets = [set() for i in xrange(6)]
baseDict = {1 : 0, 2 : 0, 3 : 0, 5 : 0, 7 : 0, 11 : 0, 13 : 0}
TOTIENT_VAL = math.factorial(13)
INDEX = 150000 -1
possiblePrimes = set()
possibleTotients = set([1])
for prime in primeFactors.keys():
newSet = set()
for power in xrange(1, primeFactors[prime]+1):
newSet.update(set([ prime**power * num for num in possibleTotients]))
possibleTotients.update(newSet)
for num in possibleTotients:
if m_r(num+1):
possiblePrimes.add(num+1)
# Arrange products into sets based on their largest prime factor
for num in list(possiblePrimes):
for pIndex in xrange(len(primes)):
if (num - 1) % primes[pIndex] == 0:
primeSets[pIndex].add(num)
break
def factorAndAddToDict(num, valueDict):
factors = factor(num)
for num in factors:
valueDict[num] += 1
def checkPrimePowers(valDict, func):
import time, math
from primes import m_r, findNumLessThan
START = time.time()
SIZE = 10**11
RELEVANT_SIZE = SIZE / 21125 # 21125 = 5^3 x 13^2
oneMod4Primes = filter(lambda x: m_r(x), xrange(5,RELEVANT_SIZE+1,4))
threeMod4Primes = [2] + filter(lambda x: m_r(x), xrange(3,RELEVANT_SIZE+1,4))
large3Mod4PrimesIndex = findNumLessThan(threeMod4Primes, RELEVANT_SIZE**.5)
threeMod4Products = set([1] + threeMod4Primes[large3Mod4PrimesIndex+1:])
for p in threeMod4Primes[large3Mod4PrimesIndex:0:-1] + [2]:
for power in xrange(1,int(math.log(RELEVANT_SIZE,p)+1)):
threeMod4Products = threeMod4Products.union( \
filter((lambda x: x < RELEVANT_SIZE), \
[ x*p**power for x in threeMod4Products]))
threeMod4Products = sorted(threeMod4Products)
def compute(power1,power2, power3=0):
sumz = 0
for prime1 in oneMod4Primes:
if prime1**power1 * 5**power2 > SIZE:
break
for prime2 in oneMod4Primes:
#print prime1, prime2, prime1**power1*prime2**power2
if prime1 == prime2:
continue
if prime1**power1 * prime2**power2 > SIZE:
import time
START = time.time()
from primes import m_r
rings = [1] + [3 * x * (x - 1) + 2 for x in xrange(1, 100000)]
sol = [0, 1]
for i in xrange(2, len(rings) - 2):
val = m_r(rings[i+1] - rings[i] +1) + \
m_r(rings[i+2] -rings[i] -1) + \
m_r(rings[i+1] - rings[i] -1)
if val == 3:
sol += [rings[i]]
for i in xrange(2, len(rings) - 2):
a = rings[i] - 1
val = m_r(a - rings[i-1]) + \
m_r(a -rings[i-2]) + \
m_r(rings[i+1] - 2 - a)
if val == 3:
sol += [a]
sol.sort()
print len(sol)
if len(sol) > 2000:
print sol[1999]
print sol[:10]
print "Time Taken:", time.time() - START
"""
13:57 ~/Desktop/python_projects/proj_euler $ python prob128.py
2635
->
(x^2 - xy + y^2) | y^3
For now, assume:
x^2 - xy + y^2 = y^3
Then:
x = (y + y * sqrt(4y - 3)) / 2
"""
count = 0
for i in xrange(1, int(math.sqrt(LIM)), 2):
for v in xrange(1,100):
y = v * (i**2 + 3 ) / 4
x = y * (1 + i) / (2*v)
if x < y:
continue
if (x**4 - y**4) % (x**3 + y**3) != 0:
# print x, y, (x**4 - y**4) % (x**3 + y**3), "ERROR"
continue
number = (x**4 - y**4) / (x**3 + y**3)
if number > LIM:
break
if m_r(number):
print number, x,y, v, x**4 - y**4, x**3 + y**3
count += number
print "answer:", count
print "Time taken:", time.time() - START
old_set = new_set
new_set = set()
for item in old_set:
admiss_set.add(item)
admiss_set.remove(1)
print len(admiss_set), "is how many admissible numbers we need to deal with,"
print "Time Taken:", time.time() - start
######################################### Above this is finding admissible numbers
fort_num = set() # the pseudo-fortunate numbers
for num in admiss_set:
n = num+2
while not m_r(n):
n += 1
fort_num.add(n-num)
print sum(fort_num), "is our final answer"
print "Time Taken:", time.time() - start
"""
Congratulations, the answer you gave to problem 293 is correct.
You are the 1292nd person to have solved this problem.
22:17 ~/Desktop/python_projects/proj_euler $ python prob293.py
6656 is how many admissible numbers we need to deal with,
Time Taken: 0.0592079162598
2209 is our final answer
old_set = new_set
new_set = set()
for item in old_set:
admiss_set.add(item)
admiss_set.remove(1)
print len(admiss_set), "is how many admissible numbers we need to deal with,"
print "Time Taken:", time.time() - start
######################################### Above this is finding admissible numbers
fort_num = set() # the pseudo-fortunate numbers
for num in admiss_set:
n = num + 2
while not m_r(n):
n += 1
fort_num.add(n - num)
print sum(fort_num), "is our final answer"
print "Time Taken:", time.time() - start
"""
Congratulations, the answer you gave to problem 293 is correct.
You are the 1292nd person to have solved this problem.
22:17 ~/Desktop/python_projects/proj_euler $ python prob293.py
6656 is how many admissible numbers we need to deal with,
Time Taken: 0.0592079162598
2209 is our final answer
Time Taken: 0.359772920609
import time, math
from primes import factor, m_r
START = time.time()
SIZE = 10**11
LIM = SIZE / 12101
MOD = 2017
vals = dict()
for n in sorted(range(1722, int(SIZE**(1/2.)), MOD) + range(294, int(SIZE**(1/2.)), MOD)):
if n**2 > SIZE:
break
if m_r(n):
vals[n] = 2
for n in sorted(range(1788, int(SIZE**(1/3.)), MOD) + range(229, int(SIZE**(1/3.)), MOD)):
if n**3 > SIZE:
break
if m_r(n):
vals[n] = 3
ansNum = 0
for n in xrange(MOD-1, SIZE, MOD):
if m_r(n):
if n > LIM:
# don't bother storing large numbers, they won't run into the duplicate problem
for j in xrange(1,SIZE/n+1):
ansNum += j * n
else:
vals[n] = 1
print "Time Taken:", time.time() - START
def get_primes(start,end):
primes = []
for i in xrange(start,end):
if m_r(i):
primes.append(i)
return primes