Check if a number is a Trojan Numbers in Python

A Trojan Number is a special type of number that meets two criteria: it must be a strong number but not a perfect power. A strong number is one where every prime factor appears at least twice (squared or higher). For example, 72 = 2³ × 3² is strong because both prime factors (2 and 3) appear at least twice, and it's not a perfect power like 8 = 2³.

Understanding Strong Numbers

A number is strong when every prime factor appears with an exponent of 2 or more ?

from math import sqrt

def check_strong_num(n):
    # Count frequency of each prime factor
    count = {}
    original_n = n
    
    # Handle factor 2
    while n % 2 == 0:
        n = n // 2
        count[2] = count.get(2, 0) + 1
    
    # Handle odd factors
    for i in range(3, int(sqrt(n)) + 1, 2):
        while n % i == 0:
            n = n // i
            count[i] = count.get(i, 0) + 1
    
    # Handle remaining prime factor
    if n > 2:
        count[n] = count.get(n, 0) + 1
    
    # Check if all prime factors appear at least twice
    for value in count.values():
        if value == 1:
            return False
    return True

# Test examples
print(f"72 is strong: {check_strong_num(72)}")  # 72 = 2³ × 3²
print(f"12 is strong: {check_strong_num(12)}")  # 12 = 2² × 3¹ (not strong)

72 is strong: True
12 is strong: False

Checking Perfect Powers

A perfect power is a number that can be expressed as a^b where a and b are positive integers and b ? 2 ?

from math import sqrt

def check_perfect_pow(n):
    if n == 1:
        return True
    
    for base in range(2, int(sqrt(n)) + 1):
        exponent = 2
        power = base ** exponent
        
        while power <= n and power > 0:
            if power == n:
                return True
            exponent += 1
            power = base ** exponent
    
    return False

# Test examples
print(f"8 is perfect power: {check_perfect_pow(8)}")   # 8 = 2³
print(f"72 is perfect power: {check_perfect_pow(72)}") # 72 is not a^b

8 is perfect power: True
72 is perfect power: False

Complete Trojan Number Check

A Trojan number must be strong but not a perfect power ?

from math import sqrt

def check_perfect_pow(n):
    if n == 1:
        return True
    
    for base in range(2, int(sqrt(n)) + 1):
        exponent = 2
        power = base ** exponent
        
        while power <= n and power > 0:
            if power == n:
                return True
            exponent += 1
            power = base ** exponent
    
    return False

def check_strong_num(n):
    count = {}
    
    # Handle factor 2
    while n % 2 == 0:
        n = n // 2
        count[2] = count.get(2, 0) + 1
    
    # Handle odd factors
    for i in range(3, int(sqrt(n)) + 1, 2):
        while n % i == 0:
            n = n // i
            count[i] = count.get(i, 0) + 1
    
    # Handle remaining prime factor
    if n > 2:
        count[n] = count.get(n, 0) + 1
    
    # Check if all prime factors appear at least twice
    for value in count.values():
        if value == 1:
            return False
    return True

def isTrojan(n):
    return not check_perfect_pow(n) and check_strong_num(n)

# Test cases
test_numbers = [72, 108, 200, 8, 12]
for num in test_numbers:
    result = isTrojan(num)
    print(f"{num} is Trojan: {result}")

72 is Trojan: True
108 is Trojan: True
200 is Trojan: True
8 is Trojan: False
12 is Trojan: False

Examples Breakdown

Conclusion

A Trojan number must satisfy two conditions: it must be a strong number (all prime factors appear at least twice) and not be expressible as a perfect power. Use the combined check to identify Trojan numbers correctly.