96 lines
2.6 KiB
Plaintext
96 lines
2.6 KiB
Plaintext
{
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"cells": [
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{
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"cell_type": "markdown",
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"id": "7edea8e5",
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"metadata": {},
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"source": [
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"# [Spiral Primes](https://projecteuler.net/problem=58)\n",
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"\n",
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"It's the return of the Ulam spiral from [problem 28](https://projecteuler.net/problem=28) (this time we're going counter-clockwise, but that doesn't actually affect much).\n",
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"\n",
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"We can handle this problem with a couple of easy-to-derive formulas. First, for a spiral with side length $n$ (note that $n$ must be odd), the number of diagonal entries is $2n-1$. Furthermore, the outermost diagonal entries will be $n^2$, $n^2 - (n-1)$, $n^2 - 2(n-1)$, and $n^2 - 3(n-1)$.\n",
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"\n",
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"With these facts, we can just iterate over odd values of $n$ and calculate the four outermost diagonal entries. We'll keep a running total $p$ of how many primes we see and stop when $\\frac{p}{2n-1} < 0.1$."
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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": "4480be30",
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"metadata": {},
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"outputs": [],
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"source": [
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"def diagonal(n, k): return n^2 - k*(n-1)"
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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": 2,
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"id": "025d7d4b",
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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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"26241"
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]
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},
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"execution_count": 2,
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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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"p = 0\n",
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"for n in count(3, 2):\n",
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" for k in range(0, 4):\n",
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" if is_prime(diagonal(n, k)):\n",
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" p += 1\n",
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" \n",
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" if p / (2*n - 1) < 0.1:\n",
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" break\n",
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"\n",
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"n"
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]
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},
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{
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"cell_type": "markdown",
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"id": "f181c6ae",
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"metadata": {},
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"source": [
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"## Relevant sequences\n",
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"* Numbers on diagonals: [A200975](https://oeis.org/A200975)\n",
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"* Primes at right-angle turns on the Ulam spiral: [A172979](https://oeis.org/A172979)\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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