discuss rosser's theorem and prime counting function

notebooks/problem0007.ipynb

@@ -30,8 +30,7 @@
     }
    ],
    "source": [
-    "# Primes.unrank is 0-indexed, so the 10001st prime is at index 10000\n",
+    "nth_prime(10001)"
-    "Primes().unrank(10000)"
    ]
   },
   {
@@ -97,7 +96,7 @@
     "\n",
     "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",
     "\n",
-    "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",
+    "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",
     "\n",
     "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."
    ]
@@ -153,6 +152,149 @@
     "    return True"
    ]
   },
+  {
+   "cell_type": "markdown",
+   "id": "45c9ca5f-97f7-4091-8a6b-6851a84b39a2",
+   "metadata": {},
+   "source": [
+    "With this implementation, we can generate the primes by iterating over the positive integers and testing if each is prime."
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": 5,
+   "id": "9ea628f0-79f0-447d-a444-6e66565a3620",
+   "metadata": {},
+   "outputs": [],
+   "source": [
+    "from itertools import count\n",
+    "\n",
+    "def primes(start=1):\n",
+    "    for n in count(start):\n",
+    "        if is_prime(n):\n",
+    "            yield n"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "id": "f8e9fad3-8ef0-469e-b381-43315b915559",
+   "metadata": {},
+   "source": [
+    "Then to solve the problem, we can just count the primes until we reach 10001.\n",
+    "\n",
+    "## Rosser's theorem\n",
+    "This strategy works... but we can go even deeper.\n",
+    "\n",
+    "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."
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": 6,
+   "id": "d7f5255b-d21e-4cfa-837d-9289c44c156a",
+   "metadata": {},
+   "outputs": [
+    {
+     "data": {
+      "text/plain": [
+       "92114"
+      ]
+     },
+     "execution_count": 6,
+     "metadata": {},
+     "output_type": "execute_result"
+    }
+   ],
+   "source": [
+    "lower = ceil(10001 * log(10001))\n",
+    "lower"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "id": "48cbf855-fe03-4f7f-9c07-45a1c1c0295f",
+   "metadata": {},
+   "source": [
+    "## Prime-counting function\n",
+    "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",
+    "\n",
+    "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."
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": 7,
+   "id": "43b0b9df-d745-4890-a835-aa0c3e3bced8",
+   "metadata": {},
+   "outputs": [],
+   "source": [
+    "def prime_counting_function(n):\n",
+    "    return len(prime_range(n+1))"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "id": "0c1e85af-74ba-48ee-9f5e-9aab35cc2a55",
+   "metadata": {},
+   "source": [
+    "This will tell us where to start our tally if we start checking for primes at 92114."
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": 8,
+   "id": "1636b854-6d24-4373-987a-8a66e3b91fe4",
+   "metadata": {},
+   "outputs": [
+    {
+     "data": {
+      "text/plain": [
+       "8896"
+      ]
+     },
+     "execution_count": 8,
+     "metadata": {},
+     "output_type": "execute_result"
+    }
+   ],
+   "source": [
+    "tally = prime_counting_function(lower)\n",
+    "tally"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "id": "13c91512-6a23-4569-bdea-2e88f4414922",
+   "metadata": {},
+   "source": [
+    "This runs considerably quicker than starting from 1."
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": 9,
+   "id": "1590869a-5d9d-4662-b07b-18511eb25b79",
+   "metadata": {},
+   "outputs": [
+    {
+     "data": {
+      "text/plain": [
+       "104743"
+      ]
+     },
+     "execution_count": 9,
+     "metadata": {},
+     "output_type": "execute_result"
+    }
+   ],
+   "source": [
+    "for (i, p) in enumerate(primes(start=lower), start=tally+1):\n",
+    "    if i == 10001:\n",
+    "        break\n",
+    "\n",
+    "p"
+   ]
+  },
   {
    "cell_type": "markdown",
    "id": "ec384441",
@@ -161,6 +303,7 @@
     "## Relevant sequences\n",
     "* Prime numbers: [A000040](https://oeis.org/A000040)\n",
     "* Carmichael numbers: [A002997](https://oeis.org/A002997)\n",
+    "* Prime-counting function: [A000720](https://oeis.org/A000720)\n",
     "\n",
     "#### Copyright (C) 2025 filifa\n",
     "\n",
@@ -170,7 +313,7 @@
  ],
  "metadata": {
   "kernelspec": {
-   "display_name": "SageMath 9.5",
+   "display_name": "SageMath 10.7",
    "language": "sage",
    "name": "sagemath"
   },
@@ -184,7 +327,7 @@
    "name": "python",
    "nbconvert_exporter": "python",
    "pygments_lexer": "ipython3",
-   "version": "3.11.2"
+   "version": "3.12.5"
   }
  },
  "nbformat": 4,