discuss rosser's theorem and prime counting function
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filifa
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ff67099715
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@ -30,8 +30,7 @@
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}
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],
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"source": [
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"# Primes.unrank is 0-indexed, so the 10001st prime is at index 10000\n",
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"Primes().unrank(10000)"
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"nth_prime(10001)"
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]
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},
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{
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@ -97,7 +96,7 @@
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"\n",
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"The test proceeds by repeatedly squaring $a^u$ modulo $n$ until we reach $a^{n-1}$. If at any point in this squaring, we find that $a^{2^{i-1} u}$ does not equal 1 or -1, but the following squaring gives $a^{2^i u} = 1$, then we have found a non-trivial square root of 1 modulo $n$. Therefore, $n$ must be composite.\n",
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"\n",
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"After $t$ squarings, we will reach $a^{n-1}$. If this equals anything other than 1, then by the contrapositive of Fermat's last theorem, this also tells us that $n$ is composite.\n",
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"After $t$ squarings, we will reach $a^{n-1}$. If this equals anything other than 1, then by the contrapositive of Fermat's little theorem, this also tells us that $n$ is composite.\n",
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"\n",
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"If, after $t$ squarings, we have not found a non-trivial square root of 1 modulo $n$ and $a^{n-1} \\equiv 1 \\pmod{n}$, then either $n$ is prime, or is a base $a$ strong pseudoprime."
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]
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@ -153,6 +152,149 @@
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" return True"
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]
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},
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{
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"cell_type": "markdown",
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"id": "45c9ca5f-97f7-4091-8a6b-6851a84b39a2",
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"metadata": {},
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"source": [
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"With this implementation, we can generate the primes by iterating over the positive integers and testing if each is prime."
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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": 5,
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"id": "9ea628f0-79f0-447d-a444-6e66565a3620",
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"metadata": {},
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"outputs": [],
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"source": [
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"from itertools import count\n",
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"\n",
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"def primes(start=1):\n",
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" for n in count(start):\n",
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" if is_prime(n):\n",
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" yield 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": "f8e9fad3-8ef0-469e-b381-43315b915559",
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"metadata": {},
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"source": [
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"Then to solve the problem, we can just count the primes until we reach 10001.\n",
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"\n",
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"## Rosser's theorem\n",
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"This strategy works... but we can go even deeper.\n",
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"\n",
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"Running Miller-Rabin on every integer until we reach the 10001st prime isn't super efficient. To avoid this, we can take advantage of [Rosser's theorem](https://en.wikipedia.org/wiki/Rosser%27s_theorem), which states that $p_n > n \\log n$. In other words, we can start our primality testing at $10001 \\log 10001$ instead of 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": 6,
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"id": "d7f5255b-d21e-4cfa-837d-9289c44c156a",
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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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"92114"
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]
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},
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"execution_count": 6,
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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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"lower = ceil(10001 * log(10001))\n",
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"lower"
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]
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},
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{
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"cell_type": "markdown",
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"id": "48cbf855-fe03-4f7f-9c07-45a1c1c0295f",
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"metadata": {},
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"source": [
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"## Prime-counting function\n",
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"There's one issue preventing us from jumping ahead to 92114: we don't know how many primes are less than 92114, so when we find the next prime (spoiler: it's 92119), we won't know what number prime that is (another spoiler: it's the 8897th prime).\n",
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"\n",
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"To answer this question, we can use the [prime-counting function](https://en.wikipedia.org/wiki/Prime-counting_function). SageMath has the prime-counting function available directly as `prime_pi`, but let's implement it ourselves. As a first pass, we can use a sieve. Check out [problem 10](https://projecteuler.net/problem=10) for a sieve implementation - to avoid duplicating work, we'll use SageMath's sieve here."
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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": 7,
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"id": "43b0b9df-d745-4890-a835-aa0c3e3bced8",
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"metadata": {},
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"outputs": [],
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"source": [
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"def prime_counting_function(n):\n",
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" return len(prime_range(n+1))"
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]
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},
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{
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"cell_type": "markdown",
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"id": "0c1e85af-74ba-48ee-9f5e-9aab35cc2a55",
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"metadata": {},
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"source": [
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"This will tell us where to start our tally if we start checking for primes at 92114."
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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": 8,
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"id": "1636b854-6d24-4373-987a-8a66e3b91fe4",
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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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"8896"
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]
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},
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"execution_count": 8,
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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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"tally = prime_counting_function(lower)\n",
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"tally"
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]
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},
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{
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"cell_type": "markdown",
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"id": "13c91512-6a23-4569-bdea-2e88f4414922",
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"metadata": {},
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"source": [
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"This runs considerably quicker than starting from 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": 9,
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"id": "1590869a-5d9d-4662-b07b-18511eb25b79",
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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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"104743"
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]
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},
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"execution_count": 9,
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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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"for (i, p) in enumerate(primes(start=lower), start=tally+1):\n",
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" if i == 10001:\n",
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" break\n",
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"\n",
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"p"
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]
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},
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{
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"cell_type": "markdown",
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"id": "ec384441",
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@ -161,6 +303,7 @@
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"## Relevant sequences\n",
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"* Prime numbers: [A000040](https://oeis.org/A000040)\n",
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"* Carmichael numbers: [A002997](https://oeis.org/A002997)\n",
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"* Prime-counting function: [A000720](https://oeis.org/A000720)\n",
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"\n",
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"#### Copyright (C) 2025 filifa\n",
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"\n",
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@ -170,7 +313,7 @@
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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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"display_name": "SageMath 10.7",
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"language": "sage",
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"name": "sagemath"
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},
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@ -184,7 +327,7 @@
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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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"version": "3.12.5"
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}
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},
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"nbformat": 4,
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