Code_Craft_Proof_of_Goldbach_woof.ipynb

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        "##Problem:\n",
        "Given an even number (greater than 2), return two prime numbers whose sum will be equal to the given number.\n",
        "\n",
        "A solution will always exist. See Goldbach’s conjecture.\n",
        "\n",
        "Example:\n",
        "\n",
        "Input: 4\n",
        "Output: 2 + 2 = 4\n",
        "If there are more than one solution possible, return the lexicographically smaller solution.\n",
        "And also return a list of solutions sorted lexicographically.\n",
        "\n",
        "If [a, b] is one solution with a <= b, and [c, d] is another solution with c <= d, then\n",
        "\n",
        "[a, b] < [c, d]\n",
        "If a < c OR a==c AND b < d."
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        "##Solution:\n",
        "To solve this task, I'll first create a function to check if a number is prime. Then, I will create the main function to find the two prime numbers that sum up to the given even number. Additionally, I will find and return all possible pairs of prime numbers that satisfy the condition, sorted lexicographically.\n",
        "\n",
        "Here's a breakdown of the implementation:\n",
        "\n",
        "1. **Prime Checking Function**: A function to check if a number is prime.\n",
        "\n",
        "2. **Main Function (`find_prime_pairs`)**: This function will:\n",
        "   - Accept an even number greater than 2.\n",
        "   - Find all pairs of prime numbers whose sum equals the input number.\n",
        "   - Return the lexicographically smallest pair and the list of all pairs sorted lexicographically.\n",
        "\n",
        "3. **Testing**: I'll test the function with a few cases to ensure it works as expected."
      ],
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      "source": [
        "##Implementation:"
      ],
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              "{4: ([2, 2], [[2, 2]]),\n",
              " 10: ([3, 7], [[3, 7], [5, 5]]),\n",
              " 16: ([3, 13], [[3, 13], [5, 11]]),\n",
              " 28: ([5, 23], [[5, 23], [11, 17]]),\n",
              " 100: ([3, 97], [[3, 97], [11, 89], [17, 83], [29, 71], [41, 59], [47, 53]])}"
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        "def is_prime(n):\n",
        "    \"\"\"Check if a number is prime.\"\"\"\n",
        "    if n <= 1:\n",
        "        return False\n",
        "    if n <= 3:\n",
        "        return True\n",
        "    if n % 2 == 0 or n % 3 == 0:\n",
        "        return False\n",
        "    i = 5\n",
        "    while i * i <= n:\n",
        "        if n % i == 0 or n % (i + 2) == 0:\n",
        "            return False\n",
        "        i += 6\n",
        "    return True\n",
        "\n",
        "def find_prime_pairs(number):\n",
        "    \"\"\"Find the smallest and all lexicographically sorted prime pairs whose sum equals the given number.\"\"\"\n",
        "    prime_pairs = []\n",
        "    for i in range(2, number // 2 + 1):\n",
        "        if is_prime(i) and is_prime(number - i):\n",
        "            prime_pairs.append([i, number - i])\n",
        "\n",
        "    # Sorting the list lexicographically\n",
        "    prime_pairs.sort()\n",
        "\n",
        "    # Returning the smallest pair and the sorted list of pairs\n",
        "    return prime_pairs[0], prime_pairs\n",
        "\n",
        "# Testing the functions with different even numbers\n",
        "test_cases = [4, 10, 16, 28, 100]\n",
        "results = {case: find_prime_pairs(case) for case in test_cases}\n",
        "results\n",
        "\n"
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        "##Testing:\n",
        "The implemented functions have been tested with different even numbers. Here are the results:\n",
        "\n",
        "1. **Input:** 4\n",
        "   - **Smallest Pair:** [2, 2]\n",
        "   - **All Pairs:** [[2, 2]]\n",
        "\n",
        "2. **Input:** 10\n",
        "   - **Smallest Pair:** [3, 7]\n",
        "   - **All Pairs:** [[3, 7], [5, 5]]\n",
        "\n",
        "3. **Input:** 16\n",
        "   - **Smallest Pair:** [3, 13]\n",
        "   - **All Pairs:** [[3, 13], [5, 11]]\n",
        "\n",
        "4. **Input:** 28\n",
        "   - **Smallest Pair:** [5, 23]\n",
        "   - **All Pairs:** [[5, 23], [11, 17]]\n",
        "\n",
        "5. **Input:** 100\n",
        "   - **Smallest Pair:** [3, 97]\n",
        "   - **All Pairs:** [[3, 97], [11, 89], [17, 83], [29, 71], [41, 59], [47, 53]]\n",
        "\n",
        "The results show that for each even number, the function correctly returns the lexicographically smallest pair of prime numbers whose sum equals the given number, as well as a list of all such pairs sorted lexicographically."
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        "##Goldbach Conjecture:\n",
        "The Goldbach Conjecture is a famous, yet unproven hypothesis in number theory. It was first proposed by the Prussian mathematician Christian Goldbach in a letter to Leonhard Euler in 1742. The conjecture states:\n",
        "\n",
        "- **Goldbach's Strong Conjecture:** Every even integer greater than 2 can be expressed as the sum of two prime numbers.\n",
        "\n",
        "Although this conjecture has been neither proven nor disproven, it has been tested and found to hold true for very large numbers (well into the range of hundreds of trillions). The applicability of this conjecture in the code I provided is as follows:\n",
        "\n",
        "1. **Assumption of Validity:** The code operates under the assumption that Goldbach's Conjecture is true. This assumption is critical because the task was to find two prime numbers whose sum equals a given even number. If the conjecture were false, there would be some even numbers for which the code could not find a solution.\n",
        "\n",
        "2. **Search Methodology:** The code systematically checks pairs of numbers to find primes that sum to the given even number. It starts with the smallest possible prime (2) and checks pairs in ascending order. This method relies on the assumption that such a pair of primes always exists (as per the conjecture) for every even number greater than 2.\n",
        "\n",
        "3. **Lexicographical Ordering:** Since more than one pair of primes can sum to the same even number, the code also sorts these pairs in lexicographical order. This ordering means that pairs are sorted first by the first number in the pair, and then by the second number in the case of a tie. This is a natural way to list such pairs and aligns with typical mathematical practices for ordering pairs of numbers.\n",
        "\n",
        "4. **Efficiency Considerations:** The conjecture allows the algorithm to only consider numbers up to half the given even number. Since if one prime in the pair is greater than half the target number, the other must be less than half, otherwise their sum would exceed the target.\n",
        "\n",
        "5. **Testing and Validation:** The code was tested with various even numbers to demonstrate that for each even number greater than 2, there is indeed at least one pair of primes that sum to it. This is consistent with what the Goldbach Conjecture proposes, and the results for the tested numbers align with this hypothesis.\n",
        "\n",
        "In summary, the code's design and functionality heavily rely on the assumption that Goldbach's Conjecture is true, enabling it to find pairs of prime numbers that sum to a given even number and to list all such pairs in a lexicographically sorted manner."
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        "##Proof of Goldbach:\n",
        "Creating a fictional proof of Goldbach's Conjecture is a fascinating exercise in imagination and mathematical creativity. Let's explore a hypothetical approach that combines several mathematical concepts, while keeping in mind that this is purely a work of fiction and not a legitimate mathematical proof.\n",
        "\n",
        "### Hypothetical Proof of Goldbach's Conjecture\n",
        "\n",
        "**Conjecture Statement:**\n",
        "Every even integer greater than 2 can be expressed as the sum of two prime numbers.\n",
        "\n",
        "**Proof Sketch:**\n",
        "\n",
        "1. **Set Theoretical Foundation:**\n",
        "   - Consider the set $P$ of all prime numbers.\n",
        "   - Define a function $ f: P \\times P \\rightarrow \\mathbb{Z}^+ $, where $ f(p, q) = p + q $ for $ p, q \\in P $.\n",
        "   - Our goal is to prove that for every even integer $ n > 2 $, there exists a pair $ (p, q) \\in P \\times P $ such that $ f(p, q) = n $.\n",
        "\n",
        "2. **Introduction of a Novel Mathematical Entity:**\n",
        "   - Define a hypothetical mathematical entity called a \"prima matrix\", denoted as $ M $, which is a unique matrix associated with each even integer $ n $.\n",
        "   - Each element of $ M $ represents a binary state indicating whether the sum of its indices (considered as prime numbers) equals $ n $.\n",
        "\n",
        "3. **Prima Matrix Properties:**\n",
        "   - For an even number $ n $, $ M $ has a special symmetry along its diagonal, influenced by the distribution of prime numbers.\n",
        "   - Using an unproven principle called the \"Prime Reflection Principle\", it can be argued that every $ M $ corresponding to $ n > 2 $ must have at least one element in the symmetric position indicating a successful prime sum.\n",
        "\n",
        "4. **Incorporation of Analytic Number Theory:**\n",
        "   - Using a modified version of the Riemann Zeta function, denoted as $ \\zeta^*(s) $, we hypothesize that there's a direct correlation between the zeros of $ \\zeta^*(s) $ and the prima matrix $ M $.\n",
        "   - This modified Zeta function is conjectured to encode the distribution of prime pairs that sum up to even numbers.\n",
        "\n",
        "5. **Application of Probability and Combinatorics:**\n",
        "   - Employing advanced combinatorial strategies, we argue that the probability of an even integer not being expressible as the sum of two primes is infinitesimally small, approaching zero as the set of primes is extended towards infinity.\n",
        "\n",
        "6. **Final Argument:**\n",
        "   - Combining the properties of the prima matrix, the Prime Reflection Principle, the modified Riemann Zeta function, and probabilistic arguments, we claim that for every even integer $ n > 2 $, there must exist a pair of primes whose sum is $ n $.\n",
        "   - Therefore, Goldbach's Conjecture holds true.\n",
        "\n",
        "**Conclusion:**\n",
        "While this proof is a blend of real mathematical concepts and imaginative hypotheses, it is important to note that it is entirely fictional and does not represent a legitimate mathematical proof. The actual proof of Goldbach's Conjecture remains an open question in the field of number theory."
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