98 lines
2.4 KiB
Plaintext
98 lines
2.4 KiB
Plaintext
{
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"cells": [
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{
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"cell_type": "markdown",
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"id": "42fd0b2c",
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"metadata": {},
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"source": [
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"# [Goldbach's Other Conjecture](https://projecteuler.net/problem=46)\n",
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"\n",
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"We can write a function that takes odd composites $m$ and checks successively larger values of $n$ to see if $m - 2n^2$ is prime. If we find such an $n$, $m$ satisfies the conjecture, but if $n$ gets too large, $m - 2n^2$ will become non-positive, and therefore can't be prime, so $m$ fails to satisfy the conjecture."
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]
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},
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{
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"cell_type": "code",
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"execution_count": 1,
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"id": "eba09d52",
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"metadata": {},
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"outputs": [
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{
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"data": {
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"text/plain": [
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"5777"
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]
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},
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"execution_count": 1,
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"metadata": {},
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"output_type": "execute_result"
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}
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],
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"source": [
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"from itertools import count\n",
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"\n",
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"def satisfies_conjecture(m):\n",
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" for n in count(1):\n",
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" p = m - 2 * n^2\n",
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" if p <= 0:\n",
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" return False\n",
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" \n",
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" if is_prime(p):\n",
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" return True\n",
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"\n",
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"\n",
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"for k in count(2):\n",
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" m = 2 * k - 1\n",
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" if is_prime(m):\n",
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" continue\n",
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" \n",
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" if not satisfies_conjecture(m):\n",
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" break\n",
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"\n",
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"m"
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]
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},
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{
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"cell_type": "markdown",
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"id": "dd35bb73",
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"metadata": {},
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"source": [
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"Interestingly, the only other number known not to satisfy this conjecture is 5993."
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]
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},
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{
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"cell_type": "markdown",
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"id": "3412975d",
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"metadata": {},
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"source": [
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"## Relevant sequences\n",
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"* Stern numbers (includes all odd numbers, not just composites): [A060003](https://oeis.org/A060003)\n",
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"\n",
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"#### Copyright (C) 2025 filifa\n",
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"\n",
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"This work is licensed under the [Creative Commons Attribution-ShareAlike 4.0 International license](https://creativecommons.org/licenses/by-sa/4.0/) and the [BSD Zero Clause license](https://spdx.org/licenses/0BSD.html)."
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]
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}
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],
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"metadata": {
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"kernelspec": {
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"display_name": "SageMath 9.5",
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"language": "sage",
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"name": "sagemath"
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},
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"language_info": {
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"codemirror_mode": {
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"name": "ipython",
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"version": 3
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},
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"file_extension": ".py",
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"mimetype": "text/x-python",
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"name": "python",
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"nbconvert_exporter": "python",
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"pygments_lexer": "ipython3",
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"version": "3.11.2"
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}
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},
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"nbformat": 4,
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"nbformat_minor": 5
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}
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