def p027(N=10**3):
lim = 10 * N
table = set(numty.sieve(lim))
primes = numty.sieve(2 * N + 1)
best = 0
best_ab = (0, 0)
for b in primes:
if b > N:
break
for c in primes:
a = c - b - 1
if a > N:
break
count = 2
while True:
cand = count**2 + a * count + b
if cand < 0:
break
if cand < lim and cand not in table:
break
if cand >= lim and not numty.isprime(cand):
break
count += 1
if count > best:
best_ab = (a, b)
best = count
return best_ab[0] * best_ab[1]
def p027(N = 10**3):
lim = 10*N
table = set(numty.sieve(lim))
primes = numty.sieve(2*N+1)
best = 0
best_ab = (0,0)
for b in primes:
if b > N:
break
for c in primes:
a = c - b - 1
if a > N:
break
count = 2
while True:
cand = count**2+a*count+b
if cand < 0:
break
if cand < lim and cand not in table:
break
if cand >= lim and not numty.isprime(cand):
break
count += 1
if count > best:
best_ab = (a,b)
best = count
return best_ab[0]*best_ab[1]
def p050(N=10 ** 6, lim=4000):
primes = numty.sieve(lim)
primeset = set(primes)
bestprime = 0
bestlen = 0
bestbase = 0
i = 0
while True:
if i + bestlen > len(primes):
logger.error("Limit was too low!")
return
cand = sum(primes[i : i + bestlen])
if cand >= N:
break
j = i + bestlen
while cand < N:
if j >= len(primes):
logger.error("Limit was too low!")
return
cand += primes[j]
j += 1
while j > i + bestlen:
if numty.isprime(cand):
bestprime = cand
bestlen = j - i
bestbase = primes[i]
break
cand -= primes[j - 1]
j -= 1
i += 1
logger.debug((bestlen, primes[bestbase : bestbase + bestlen]))
return bestprime
def p050(N=10**6, lim=4000):
primes = numty.sieve(lim)
primeset = set(primes)
bestprime = 0
bestlen = 0
bestbase = 0
i = 0
while True:
if i + bestlen > len(primes):
logger.error('Limit was too low!')
return
cand = sum(primes[i:i + bestlen])
if cand >= N:
break
j = i + bestlen
while cand < N:
if j >= len(primes):
logger.error('Limit was too low!')
return
cand += primes[j]
j += 1
while j > i + bestlen:
if numty.isprime(cand):
bestprime = cand
bestlen = j - i
bestbase = primes[i]
break
cand -= primes[j - 1]
j -= 1
i += 1
logger.debug((bestlen, primes[bestbase:bestbase + bestlen]))
return bestprime
def p087(N=5*10**7):
primes = numty.sieve(int(math.sqrt(N)))
squares = [p**2 for p in primes if p**2 < N]
cubes = [p**3 for p in primes if p**3 < N]
fourthpows = [p**4 for p in primes if p**4 < N]
goodnums = set(a+b+c for a in squares for b in cubes for c in fourthpows if
a+b+c < N)
return len(goodnums)
def p087(N=5 * 10**7):
primes = numty.sieve(int(math.sqrt(N)))
squares = [p**2 for p in primes if p**2 < N]
cubes = [p**3 for p in primes if p**3 < N]
fourthpows = [p**4 for p in primes if p**4 < N]
goodnums = set(a + b + c for a in squares for b in cubes
for c in fourthpows if a + b + c < N)
return len(goodnums)
def p005(N=20):
primes = numty.sieve(N)
ret_val = 1
for p in primes:
tmp = p
while tmp <= N:
tmp *= p
ret_val *= p
return ret_val
def p005(N=20):
primes = numty.sieve(N)
ret_val = 1
for p in primes:
tmp = p
while tmp <= N:
tmp *= p
ret_val *= p
return ret_val
def p035(N=10 ** 6):
primes = numty.sieve(N)
primeset = set(primes)
count = 0
for p in primes:
str_p = str(p)
if all(int(str_p[n:] + str_p[:n]) in primeset for n in range(len(str_p))):
logger.debug(p)
count += 1
return count
def p046():
lim = 10**4
primes = numty.sieve(lim)
table = [1, 0] * (lim / 2)
table[1] = 1
for n in range(int(math.sqrt(lim / 2))):
i = 2 * n**2
for p in primes:
if p + i > lim:
break
table[p + i] = 1
return table.index(0)
def p046():
lim = 10**4
primes = numty.sieve(lim)
table = [1,0]*(lim/2)
table[1] = 1
for n in range(int(math.sqrt(lim/2))):
i = 2*n**2
for p in primes:
if p+i > lim:
break
table[p+i] = 1
return table.index(0)
def p035(N=10**6):
primes = numty.sieve(N)
primeset = set(primes)
count = 0
for p in primes:
str_p = str(p)
if all(
int(str_p[n:] + str_p[:n]) in primeset
for n in range(len(str_p))):
logger.debug(p)
count += 1
return count
def p051(N=6,target=8):
primes = numty.sieve(10**N)
primes = [str(p) for p in primes if len(str(p)) == N]
primeset = set(primes)
masks = [bin(x)[2:].zfill(N) for x in range(1,2**N-1)]
for prime in primes:
for mask in masks:
if len(set(''.join(b for a,b in zip(mask,prime) if a=='0'))) == 1:
family = [''.join((b if a == '1' else str(i)) for a,b in zip(mask, prime)) for i in range(10)]
family = [p for p in family if p in primeset]
if len(family) == target:
logger.debug(family)
return prime
def decomps(N):
decomps = [[] for n in range(N)]
primes = set(numty.sieve(N))
for n in range(2,N):
decomps[n].append([n])
if n in primes:
continue
for k in range(2,n):
if k > n/2:
break
if n % k == 0:
decomps[n] += [[k]+v for v in decomps[n/k] if
all(k<=f for f in v)]
return decomps
def gengraph(N):
primes = numty.sieve(N)
graph = defaultdict(list)
for i in xrange(len(primes)):
p = primes[i]
for j in xrange(i):
q = primes[j]
if (p+q)%3 == 0:
continue
pq = int(str(p)+str(q))
qp = int(str(q)+str(p))
if numty.isprime(pq) and utils.isprime(qp):
graph[p].append(q)
graph[q].append(p)
return graph
def p070():
primes = numty.sieve(10 ** 4)
best = (1000, 0)
for i, p in enumerate(primes):
if p < 1000:
continue
for q in primes[i - 1 :: -1]:
n = p * q
if n > 10 ** 7:
continue
phi = (p - 1) * (q - 1)
if sorted(str(phi)) == sorted(str(n)):
cand = (1.0 * n / phi, n)
if cand < best:
best = cand
return best[1]
def p070():
primes = numty.sieve(10**4)
best = (1000,0)
for i,p in enumerate(primes):
if p < 1000:
continue
for q in primes[i-1::-1]:
n = p*q
if n > 10**7:
continue
phi = (p-1)*(q-1)
if sorted(str(phi)) == sorted(str(n)):
cand = (1.0*n/phi, n)
if cand < best:
best = cand
return best[1]
def p051(N=6, target=8):
primes = numty.sieve(10**N)
primes = [str(p) for p in primes if len(str(p)) == N]
primeset = set(primes)
masks = [bin(x)[2:].zfill(N) for x in range(1, 2**N - 1)]
for prime in primes:
for mask in masks:
if len(set(''.join(b for a, b in zip(mask, prime)
if a == '0'))) == 1:
family = [
''.join((b if a == '1' else str(i))
for a, b in zip(mask, prime)) for i in range(10)
]
family = [p for p in family if p in primeset]
if len(family) == target:
logger.debug(family)
return prime
def p049():
found = 1487
primes = numty.sieve(10 ** 4)
table = defaultdict(list)
for p in primes:
str_p = str(p)
if len(str_p) < 4:
continue
table["".join(sorted(str_p))].append(p)
for key in table:
if len(table[key]) >= 3:
cands = sorted(table[key])
for b in cands:
for a in cands:
if a >= b:
break
if 2 * b - a in cands:
if a == found:
continue
return str(a) + str(b) + str(2 * b - a)
def p049():
found = 1487
primes = numty.sieve(10**4)
table = defaultdict(list)
for p in primes:
str_p = str(p)
if len(str_p) < 4:
continue
table[''.join(sorted(str_p))].append(p)
for key in table:
if len(table[key]) >= 3:
cands = sorted(table[key])
for b in cands:
for a in cands:
if a >= b:
break
if 2*b-a in cands:
if a == found:
continue
return str(a)+str(b)+str(2*b-a)
def p010(N=2*10**6):
return sum(numty.sieve(2*10**6))
def p010(N=2 * 10**6):
return sum(numty.sieve(2 * 10**6))
## What is the first number that can be written as the sum of primes in over
# 5000 different ways?
import lib.numty as numty
table = numty.sieve(5000)
cache = {}
def primep(n, i):
if n == 0:
return 1
if i < 0:
return 0
p = table[i]
if (n,p) not in cache:
cache[(n,p)] = sum(primep(n-p*k, i-1) for k in range(n/p+1))
return cache[(n,p)]
def primepartitions(n):
if n > table[-1]:
raise ValueError("n=%d unexpectedly larger than largest summand %d" % (n, table[-1]))
return primep(n,n)
def p077():
for n in range(5000):
if primepartitions(n) > 5000:
return n
