def decompose_prime_square(n):
"""
Attempt to write n as the sum of a prime and twice a square.
>>> decompose_prime_square(9)
(7, 1)
>>> decompose_prime_square(15)
(7, 2)
>>> decompose_prime_square(21)
(3, 3)
>>> decompose_prime_square(25)
(7, 3)
>>> decompose_prime_square(27)
(19, 2)
>>> decompose_prime_square(33)
(31, 1)
"""
for p in up_to(n, primes()):
if p == 2:
continue
residue = n - p
assert residue % 2 == 0, residue
square = residue // 2
if is_perfect_square(square):
return (p, isqrt(square))
from utility import up_to
def fibonacci():
a = 0
b = 1
yield a
while True:
yield b
a, b = b, a + b
print(sum(f for f in up_to(4000000, fibonacci()) if f % 2 == 0))
for c in coefficients:
value = value * x + c
return value
def prime_sequence_length(coefficients):
"""
Find the length of the prime sequence generated by a polynomial.
>>> prime_sequence_length((1, 1, 41))
40
>>> prime_sequence_length((1, -79, 1601))
80
"""
for n in itertools.count():
if not is_prime(eval_polynomial(coefficients, n)):
return n
# We only need to check prime b, because we evaluate an n = 0, where the value == b.
MAX_NUM = 1000 - 1
max_poly = None
max_length = 0
for a in range(-MAX_NUM, MAX_NUM + 1):
for b in up_to(MAX_NUM, primes()):
poly = (1, a, b)
l = prime_sequence_length(poly)
if l > max_length:
max_poly = poly
max_length = l
_, max_a, max_b = max_poly
print(max_a * max_b)
from utility import primes, digits_of, up_to
# In order to maximize the totient, we need to look for "sharp" numbers, having
# few prime factors, with those factors being large.
limit = 10**7
solutions = []
seen_primes = []
for a in up_to(limit // 2, primes()):
for b in seen_primes:
n = a * b
if n > limit:
break
t = n
t -= t // a
if a != b:
t -= t // b
if sorted(digits_of(n)) == sorted(digits_of(t)):
solutions.append((n, n / t))
seen_primes.append(a)
solutions.sort(key=lambda x: x[1])
print(min(solutions, key=lambda x: x[1])[0])
from utility import up_to, figurate_numbers
def letter_value(c):
"""
>>> [letter_value(c) for c in 'abcdefghijklmnopqrstuvwxyz']
[1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26]
"""
c = c.lower()
return ord(c) - ord('a') + 1
def word_value(word):
"""
>>> word_value('sky')
55
"""
return sum(letter_value(c) for c in word)
with open('data/words.txt') as f:
word_data = f.read()
words = [w.strip('"').lower() for w in word_data.split(',')]
t_nums = set(up_to(30 * 26, figurate_numbers(3)))
print(sum(1 for w in words if word_value(w) in t_nums))
def circular_primes(limit):
prime_set = set(up_to(limit, primes()))
for p in prime_set:
if all((r in prime_set) for r in rotations(p)):
yield p