def prime_count_AP(n, a, d):
"""
Returns the count of prime numbers less than or equal to n in AP defined by first term as 'a' and common difference 'd'
Parameters
----------
n : int
upper bound for prime numbers
a : int
first term of AP
d : int
common difference of AP
return : int
returns the count of prime numbers
"""
if (n != int(n) or a != int(a) or d != int(d)):
raise ValueError("n , a and d are integers")
if (if_infinite_prime_AP(a, d) == False):
raise ValueError("a and d must be coprime")
x = a
count = 0
while (x <= n):
if (isPrime(x) == True):
count = count + 1
x = x + d
return count
def generate_euclid_perfect(k):
"""
Return the perfect number generated from euclid's theorm
If 2^k - 1 is prime, then (2^k)((2^k)-1) is perfect
Parameters
----------
k : int
denotes power of 2 in 2^k -1
return : int
returns perfect number
"""
if (k != int(k) or k < 1):
raise ValueError("k must be positive integer")
n = int(math.pow(2, k) - 1)
if (isPrime(n) == False):
raise ValueError("invalid k")
return int((math.pow(2, k - 1)) * (math.pow(2, k) - 1))
def quadratic_residues(p):
"""
Returns an array of quadratic residues mod p for a given prime p
Parameters
----------
p : int
denotes a prime number
return : array
returns an array of quadratic residues
"""
if (p != int(p) or p < 1):
raise ValueError("p must be positive integer")
if (isPrime(p) == False):
raise ValueError("p must be prime number")
quadratic_residues_array = []
for i in range(1, int(p / 2) + 1):
quadratic_residues_array.append((i * i) % p)
quadratic_residues_array = sorted(quadratic_residues_array)
return quadratic_residues_array
def jacobi_symbol(n, p):
"""
Returns jacobi symbol (1 or -1) for given integer n and given prime p
Parameters
----------
n : int
denotes positive integer
p : int
denotes a prime number
return : int
returns 1 or -1
"""
if (n != int(n) or n < 1 or p != int(p) or p < 1):
raise ValueError("n and p must be positive integers")
if (isPrime(p) == False):
raise ValueError("p must be prime number")
symbol = 1
for i in prime_factorization(p):
symbol = symbol * (math.pow(legendre_euler_symbol(n, i),
prime_factorization(p)[i]))
return int(symbol)
def legendre_euler_symbol(n, p):
"""
Returns 1 if n is quadratic residue mod p else return -1
Parameters
----------
n : int
denotes positive integer
p : int
denotes prime number
return : int
returns 1 if n is quadratic residue mod p else return -1
"""
if (n != int(n) or n < 1 or p != int(p) or p < 1):
raise ValueError("n and p must be positive integers")
if (isPrime(p) == False):
raise ValueError("p must be prime")
r = (math.pow(n, (p - 1) // 2)) % p
r = int(r)
if (r == 1):
return r
else:
return -1
def test_isPrime2():
assert isPrime(727) == True
def test_isPrime1():
assert isPrime(20) == False
