def problem111a():
# 9 digits the same
# 1 digit is the same
total = 0
for digits in [str(i) for i in range(0,10)]:
count = 0
for index in range(0,10):
for a in range(0,10):
number = [digits]*9
number.insert(index,str(a))
a = int(''.join(number))
if len(str(a)) == 10 and isPrime(a):
total += a
#print(a)
for digits in ['2','8','0']:
count = 0
for index1,index2 in combinations([i for i in range(0,10)],2):
for a,b in itertools.product(range(10),range(10)):
number = [digits]*10
number[index1] = str(a)
number[index2] = str(b)
a = int(''.join(number))
if len(str(a)) == 10 and isPrime(a):
total += a
return total
def problem146():
total = 0
for i in range(10, 150000000 + 1, 10):
n = i ** 2
if n % 3 != 1:
continue
if n % 7 != 2 and n % 7 != 3:
continue
if n % 9 == 0 or n % 13 == 0 or n % 27 == 0:
continue
if all(isPrime(n + j) for j in [1, 3, 7, 9, 13, 27]) and not any(isPrime(n + j) for j in [19, 21]):
print(i)
total += i
return total
def problem41(): from itertools import permutations from PE_primes import isPrime # Let's just go over all the 7 digits pandigital numbers in reverse and find the first one that is prime for c in permutations(range(7,0,-1),7): if isPrime(numberFromList(c)): return numberFromList(c)
def isPrimeProofsqubeEven(p, q):
n = p ** 2 * q ** 3
if "200" not in str(n):
return False
for i in range(1, 10, 2): # the last digit is odd
if isPrime(int(str(n)[:-1] + str(i))):
return False
return True
def problem249():
primes = primesUpTo(5000)
sums = defaultdict(int)
sums[0] = 1
for p,total in runningTotal(primes):
for j in range(total,p-1,-1):
sums[j] = sums[j] + sums[j-p]
return sum([sums[k] for k in range(total+1) if isPrime(k)]) % 10**16
def primesInNbd(n, m):
# Check the top and bottom row as the elements left and right cannot be primes
# n,m is a prime number
locations = []
for j in [-1, 1]:
for i in [-1, 0, 1]:
if isPrime(a(n + j, m + i)):
locations.append((n + j, m + i))
return locations
def isPrimeProofsqubeOdd(p, q):
n = p ** 2 * q ** 3
if "200" not in str(n):
return False
for digit in range(len(str(n))):
for i in range(0, 10):
m = str(n)[:digit] + str(i) + str(n)[digit + 1:]
if isPrime(int(m)):
return False
return True
def problem131a(): count = 0 for a in range(1,10**5): a3 = a**3 factorsOfa = factorize(a) div = divisorsFromFactors( factorsOfa + factorsOfa + factorsOfa ) for k in div: n = isSquare(a3 // k) if n and k-n >0 and isPrime(k - n): print(k-n)
def problem131(): count = 0 primes = set() for a in range(1,10**10): for n in range(1,a+1): if 0 < (a**3 - n**3) / n**2 < 10**6 and (a**3 - n**3) % n**2 == 0 and isPrime((a**3 - n**3) // n**2): count += 1 print((a**3 - n**3) // n**2, len(primes), a,n) primes.add((a**3 - n**3) // n**2) break
def allPrimes(digits,soFar=set()):
if len(digits) == 0:
seen.add(tuple(sorted(tuple(soFar))))
print(len(seen))
else:
for combi in powerset(digits):
if combi == (): continue
for perm in permutations(combi, len(combi)):
number = int(''.join([str(s) for s in perm]))
if isPrime(number):
allPrimes(digits.difference(perm), soFar.union({number}))
def S(n):
total = 0
previousIsPrime = False # A small optimization that doesn't do much
for m in range(1, n + 1):
if previousIsPrime:
previousIsPrime = False
continue
if isPrime(a(n, m)):
previousIsPrime = True
if isPrimeTriplet(n, m):
total += a(n, m)
return total
def problem146():
total = 0
# n % 210 = 10
for n in range(10, 15 * 10 ** 7, 210):
if all([isPrime(n ** 2 + i) for i in [1, 3, 7, 9, 13, 27]]):
print(n)
total += n
# n % 210 = 80
for n in range(80, 15 * 10 ** 7, 210):
if all([isPrime(n ** 2 + i) for i in [1, 3, 7, 9, 13, 27]]):
print(n)
total += n
# n % 210 = 130
for n in range(130, 15 * 10 ** 7, 210):
if all([isPrime(n ** 2 + i) for i in [1, 3, 7, 9, 13, 27]]):
print(n)
total += n
# n % 210 = 200
for n in range(200, 15 * 10 ** 7, 210):
if all([isPrime(n ** 2 + i) for i in [1, 3, 7, 9, 13, 27]]):
print(n)
total += n
return total
def problem131(): seen = 0 for m in count(2): factors = set(factorize(m)) n = m**3 for k in range(0,n,product(factors)**2): if any( k % p != 0 for p in factors): continue if k % product(factors) != 0: print(k) if (n**3 - k**3) % n**2 == 0 and isPrime((n**3 - k**3) // n**2): p = (n**3 - k**3) // n**2 if p > 10**6: return seen print(n, p, k) seen += 1
def problem216():
count = 0
for p in iPrime():
print(jacobi(( p + 1)//2, p))
if p > 100: break
return
for index, n in enumerate(t(100)):
index += 2
#print( 2*(index)**2 -1 , n, isPrime(n), jacobi(( index + 1)//2, index))
if isPrime(n):
count += 1
print("gave a prime ", jacobi(( n + 1)//2, n))
else:
for p in factorize(n):
print("Didn't give a prime", p,jacobi(( p + 1)//2, p))
return count
def problem293(): GOAL = 10**9 primes = [] maximum = 1 for p in iPrime(): maximum *= p primes.append(p) # need one more beyond what is required if maximum > GOAL: break admissible = [] total = 0 num = 0 for c in generateAdmissible(GOAL,primes): #print(c, sorted(factorize(c))) if c > GOAL: continue num += 1 for d in count(3,2): if isPrime(c+d): total += d break return num, total
def problem128():
count = 1
limit = 2000
n = 0
number = 0
while count < limit:
n += 1
if (isPrime(6 * n - 1) and isPrime(6 * n + 1) and isPrime(12 * n + 5)):
count += 1;
number = (3 * n * n - 3*n + 2)
if (count >= limit):
break
if (isPrime(6 * n + 5) and isPrime(6 * n - 1) and isPrime(12 * n - 7) and n != 1):
count += 1
number = (3 * n * n + 3*n + 1)
return number
