Check whether the given number is Wagstaff prime or not in Python

A Wagstaff prime is a prime number that can be expressed in the form (2^q + 1)/3, where q is an odd prime number. These numbers are named after mathematician Samuel S. Wagstaff Jr.

Example

For n = 683, we can verify it's a Wagstaff prime because 683 = (2¹¹ + 1)/3, where q = 11 is an odd prime.

Algorithm

To check if a number is a Wagstaff prime, we need to:

• Verify the number itself is prime
• Check if (num × 3 + 1) is a power of 2
• Verify that the exponent is an odd prime

Implementation

def isPrime(num):
    if num < 2:
        return False
    if num == 2:
        return True
    if num % 2 == 0:
        return False
    for i in range(3, int(num**0.5) + 1, 2):
        if num % i == 0:
            return False
    return True

def power_of_two(num):
    return num > 0 and (num & (num - 1)) == 0

def find_exponent(power_of_2):
    """Find the exponent q where 2^q = power_of_2"""
    if power_of_2 <= 0:
        return -1
    exponent = 0
    while power_of_2 > 1:
        power_of_2 //= 2
        exponent += 1
    return exponent

def is_wagstaff_prime(num):
    # Check if num is prime
    if not isPrime(num):
        return False
    
    # Check if (num * 3 + 1) is a power of 2
    candidate = num * 3 + 1
    if not power_of_two(candidate):
        return False
    
    # Find the exponent q
    q = find_exponent(candidate)
    
    # Check if q is an odd prime
    if q <= 1 or q % 2 == 0:
        return False
    
    return isPrime(q)

# Test with the given example
n = 683
result = is_wagstaff_prime(n)
print(f"Is {n} a Wagstaff prime? {result}")

# Verify the calculation
if result:
    q = find_exponent(n * 3 + 1)
    verification = (2**q + 1) // 3
    print(f"Verification: (2^{q} + 1) / 3 = {verification}")

Is 683 a Wagstaff prime? True
Verification: (2^11 + 1) / 3 = 683

Testing with More Examples

# Test with known Wagstaff primes
wagstaff_candidates = [3, 11, 43, 683, 2731, 43691]

for num in wagstaff_candidates:
    result = is_wagstaff_prime(num)
    if result:
        q = find_exponent(num * 3 + 1)
        print(f"{num} is a Wagstaff prime with q = {q}")
    else:
        print(f"{num} is not a Wagstaff prime")

3 is a Wagstaff prime with q = 3
11 is a Wagstaff prime with q = 5
43 is a Wagstaff prime with q = 7
683 is a Wagstaff prime with q = 11
2731 is a Wagstaff prime with q = 13
43691 is a Wagstaff prime with q = 17

Key Points

• Wagstaff primes are rare and follow the formula (2^q + 1)/3
• The exponent q must be an odd prime number
• We verify by checking if num × 3 + 1 equals 2^q
• Only a few Wagstaff primes are known for small values of q

Conclusion

A Wagstaff prime checker verifies if a number fits the form (2^q + 1)/3 where q is an odd prime. The algorithm checks primality, power-of-two properties, and validates the exponent.