def search_down():
global prod
global prod_row
global prod_col
global prod_direction
global prod_lst
for j in range(20):
for i in range(16):
lst = [mtrx[i + x][j] for x in range(4)]
m = mul(lst)
if m > prod:
prod = m
prod_row = i
prod_col = j
prod_direction = "down"
prod_lst = lst
def search_up_right():
global prod
global prod_row
global prod_col
global prod_direction
global prod_lst
for i in range(16):
for j in range(3, 20):
lst = [mtrx[i + x][j - x] for x in range(4)]
m = mul(lst)
if m > prod:
prod = m
prod_row = i
prod_col = j
prod_direction = "up-right"
prod_lst = lst
return d_totient[n]
except KeyError:
ret = n
if is_prime(n):
ret = n - 1
else:
fact = prime_factors(n)
for p in fact:
ret = ret*(p - 1)/p
k = 1
while True:
d_totient[n**k] = (n**(k - 1))*ret
k += 1
if n**k >= 10**6:
break
return d_totient[n]
if __name__ == '__main__':
# read the wiki: http://en.wikipedia.org/wiki/Euler%27s_totient_function
# in particular the chapter 'Growth of the function'
lst_primes = [x for x in range(2, 50) if is_prime(x)]
i = 0
while True:
n = mul(lst_primes[:i])
if n > 10**6:
n = mul(lst_primes[:(i - 1)])
break
i += 1
print n
d_gcd[(n, r)] = gcd(r, n - r*(n/r))
return d_gcd[(n, r)]
def simplify((a, b)):
d = gcd(a, b)
return (a/d, b/d)
def is_curious((a, b)):
possibilities = [(hdc(a), hdc(b)), (hdc(a), ldc(b)), (ldc(a), hdc(b)), (ldc(a), ldc(b))]
equalities = [(ldc(a), ldc(b)), (ldc(a), hdc(b)), (hdc(a), ldc(b)), (hdc(a), hdc(b))]
for i, frac in enumerate(possibilities):
if equalities[i][0] == equalities[i][1] and fractions_equal((a, b), frac):
print "%d/%d = %d/%d" % (a, b, frac[0], frac[1])
return True
return False
def find_fractions():
d = {}
for a in range(10, 100):
for b in range(a + 1, 100):
if is_curious((a, b)):
try:
tmp = d[simplify((a, b))]
except KeyError:
d[simplify((a, b))] = 1
return d
if __name__ == '__main__':
f = find_fractions()
print simplify((mul([x for x, y in f.keys()]), mul([y for x, y in f.keys()])))