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add is_prime package level documentation #55

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Jan 26, 2025
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add is_prime package level documentation #55

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141 changes: 132 additions & 9 deletions docs/example.ipynb
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" - Additional options to calculate exponential growth and other non linear progressions (e.g., geometric, harmonic, etc.)."
]
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"source": [
"## **Is Prime**\n",
"\n",
"The `is_prime` package provides a simple, efficient, and reliable way to determine whether a given integer is prime. This lightweight utility is perfect for educational purposes, mathematical explorations, and real-world applications requiring prime number checks.\n",
"\n",
"---\n",
"### Parameters\n",
"\n",
"**n : int** \n",
" The number to check for primality. Must be a positive integer greater than 1. \n",
"\n",
"### Returns\n",
"\n",
"**bool** \n",
" `True` if the number is prime, otherwise `False`.\n",
"\n",
"### Raises\n",
"\n",
"**ValueError** \n",
" If the input is not a positive integer greater than 1.\n",
"\n",
"---\n",
"\n",
"### Example Usage\n",
"```python\n",
"from num_theory.is_prime import is_prime\n",
"\n",
"print(is_prime(7)) # True\n",
"print(is_prime(10)) # False\n",
"```\n",
"\n",
"---\n",
"### Application 1: Prime Pattern Finder\n",
"\n",
"By filtering prime numbers that satisfy the pattern $p = 6k \\pm 1$, this demonstrates how mathematical properties can be utilized to further refine prime numbers."
]
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"text": [
"Special primes (6k ± 1) between 2 and 200: [5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97, 101, 103, 107, 109, 113, 127, 131, 137, 139, 149, 151, 157, 163, 167, 173, 179, 181, 191, 193, 197, 199]\n"
]
}
],
"source": [
"from num_theory.is_prime import is_prime\n",
"\n",
"# Find primes matching the pattern\n",
"def find_special_primes(start, end):\n",
" primes = [n for n in range(start, end + 1) if is_prime(n)]\n",
" special_primes = [p for p in primes if (p - 1) % 6 == 0 or (p + 1) % 6 == 0]\n",
" return special_primes\n",
"\n",
"# Find special primes in the range\n",
"special_primes = find_special_primes(2, 200)\n",
"print(f\"Special primes (6k ± 1) between 2 and 200: {special_primes}\")"
]
},
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"source": [
"### Application 2: Generate RSA Keys\n",
"\n",
"In encryption applications, generating an RSA key pair (public and private keys) requires two large prime numbers and modulus calculations. This code uses `is_prime` to verify the generated prime numbers and constructs the RSA public and private keys, showcasing a practical application in a real-world encryption system."
]
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"name": "stdout",
"output_type": "stream",
"text": [
"RSA Keys:\n",
"Public Key: (52931, 126169)\n",
"Private Key: (44011, 126169)\n"
]
}
],
"source": [
"import random\n",
"from math import gcd\n",
"\n",
"# Generate large primes\n",
"def generate_large_prime(start, end):\n",
" while True:\n",
" n = random.randint(start, end)\n",
" if is_prime(n):\n",
" return n\n",
"\n",
"# Generate RSA keys\n",
"def generate_rsa_keys():\n",
" p = generate_large_prime(100, 999)\n",
" q = generate_large_prime(100, 999)\n",
" n = p * q\n",
" phi = (p - 1) * (q - 1)\n",
"\n",
" # Choose e such that 1 < e < phi and gcd(e, phi) = 1\n",
" e = random.choice([x for x in range(2, phi) if gcd(x, phi) == 1])\n",
"\n",
" # Calculate d such that (d * e) % phi = 1\n",
" d = pow(e, -1, phi)\n",
" return {\"public_key\": (e, n), \"private_key\": (d, n)}\n",
"\n",
"# Generate and display RSA keys\n",
"keys = generate_rsa_keys()\n",
"print(\"RSA Keys:\")\n",
"print(f\"Public Key: {keys['public_key']}\")\n",
"print(f\"Private Key: {keys['private_key']}\")"
]
},
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"cell_type": "markdown",
"metadata": {},
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