def problem58():
length = 7
primes = 8
total = 13
while primes/total > 0.1:
length += 2
primes += quantify(corners(length**2), pred=is_prime)
total += 4
return length
def problem74():
# Known chain loop lengths given in problem description. We will use this
# dictionary to cache all further results as we calculate them.
known_loops = {145: 1, 169: 3, 1454: 3, 871: 2, 872: 2, 69: 5, 78: 4, 540: 2}
lengths = (chain_length(n) for n in range(1, 1000000+1))
def chain_length(n):
chain = [n]
next = sum_factorial_digits(n)
while True:
if next in chain: # We have found a new loop, add to the cache.
result = known_loops[n] = len(chain)
return result
if next in known_loops: # We have found a known loop, add its length to current chain.
result = known_loops[n] = len(chain) + known_loops[next]
return result
# We haven't found a loop, continue to investigate the chain.
chain.append(next)
next = sum_factorial_digits(next)
return quantify(lengths, pred=lambda x: x == 60)
def is_smallest_member(num):
"""Does the number satisfy the problem specification?"""
return any(quantify(family(num, indices), pred=is_prime) == 8
for indices in find_indices(num))
def problem55():
return quantify(range(1, 10000), pred=is_lychrel)
def problem57():
generate_tails = iterate(tail, 2) # yields 2, tail(2), tail(tail(2)), ...
expansions = (1 + Fraction(1, t) for t in generate_tails)
return quantify(take(1000, expansions), pred=check_numerator)
def problem64():
continued_fractions = (continued_fraction_sqrt(i) for i in range(2, 10000+1))
odd_period = lambda x: len(x) % 2 == 0 # The first element is not part of the period.
return quantify(continued_fractions, pred=odd_period)