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ipynb/TruncatablePrimes.ipynb

@@ -40,7 +40,7 @@
     "def left_truncatable_primes(starting_primes=[2, 3, 5, 7]) -> list[int]:\n",
     "    \"\"\"All left-truncatable primes, in ascending order.\"\"\"\n",
     "    candidates = [int(d + str(p)) for d in \"123456789\" for p in starting_primes]\n",
-    "    new_primes = list(filter(isprime, candidates))\n",
+    "    new_primes = [n for n in candidates if isprime(n)]\n",
     "    return starting_primes + (left_truncatable_primes(new_primes) if new_primes else [])"
    ]
   },
@@ -546,12 +546,12 @@
     {
      "data": {
       "text/plain": [
-       "{3: 1.4553237025147137,\n",
-       " 4: 0.5076430372271561,\n",
-       " 5: 0.1345167870334638,\n",
-       " 6: 0.04226359478965734,\n",
-       " 7: 0.00642047704144418,\n",
-       " 8: 0.003178908998431738,\n",
+       "{3: 1.5219868933421978,\n",
+       " 4: 0.44898977301072585,\n",
+       " 5: 0.15980955110048164,\n",
+       " 6: 0.02713174088102189,\n",
+       " 7: 0.007483109552723852,\n",
+       " 8: 0.003178740582981023,\n",
        " 9: 0.0,\n",
        " 10: 0.0,\n",
        " 11: 0.0,\n",
@@ -587,8 +587,8 @@
        "{3: 0.0,\n",
        " 4: 0.0,\n",
        " 5: 0.0,\n",
-       " 6: 0.0,\n",
-       " 7: 0.0010682505261133842,\n",
+       " 6: 0.001082461951462406,\n",
+       " 7: 0.0,\n",
        " 8: 0.0,\n",
        " 9: 0.0,\n",
        " 10: 0.0,\n",
@@ -610,7 +610,7 @@
    "id": "61ee44c3-f963-4fb1-ac20-4a27adc90f83",
    "metadata": {},
    "source": [
-    "In some runs this produces no false primes; in some runs a few get through. \n",
+    "In some runs this produces no false primes; in some runs one or two get through. \n",
     "\n",
     "There are a few numbers, called [**Carmichael numbers**](https://en.wikipedia.org/wiki/Carmichael_number), that have a particularly  high false prime percentage. We can test some of them:"
    ]
@@ -624,17 +624,17 @@
     {
      "data": {
       "text/plain": [
-       "{561: 58.6,\n",
+       "{561: 55.4,\n",
        " 1105: 68.8,\n",
-       " 1729: 74.4,\n",
-       " 2465: 72.7,\n",
-       " 2821: 77.9,\n",
-       " 6601: 81.2,\n",
-       " 8911: 80.6,\n",
-       " 41041: 71.3,\n",
-       " 101101: 71.3,\n",
-       " 825265: 61.4,\n",
-       " 321197185: 66.5}"
+       " 1729: 77.0,\n",
+       " 2465: 76.2,\n",
+       " 2821: 76.6,\n",
+       " 6601: 81.7,\n",
+       " 8911: 80.0,\n",
+       " 41041: 68.2,\n",
+       " 101101: 70.3,\n",
+       " 825265: 59.4,\n",
+       " 321197185: 68.6}"
       ]
      },
      "execution_count": 18,
@@ -653,7 +653,7 @@
    "id": "4ecd2289-66d5-4b0f-ac00-5db0ad92dde9",
    "metadata": {},
    "source": [
-    "That seems seriously bad; they all are above 50% error rate. But if we set `k=100`, we eliminate most (and on some runs all) of the errors:"
+    "That seems seriously bad; they all are above 50% error rate. But if we set `k=50`, we eliminate most (and on some runs all) of the errors:"
    ]
   },
   {
@@ -684,7 +684,7 @@
     }
    ],
    "source": [
-    "{n: false_prime_percent(n, n + 1, k=100, repeat=10_000) for n in carmichael_numbers}"
+    "{n: false_prime_percent(n, n + 1, k=50, repeat=10_000) for n in carmichael_numbers}"
    ]
   },
   {
@@ -752,7 +752,7 @@
    "outputs": [
     {
      "data": {
-      "image/png": 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truncated
+      "image/png": "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 truncated
       "text/plain": [
        "<Figure size 900x600 with 1 Axes>"
       ]
@@ -783,8 +783,8 @@
      "name": "stdout",
      "output_type": "stream",
      "text": [
-      "CPU times: user 1.2 ms, sys: 1 μs, total: 1.2 ms\n",
-      "Wall time: 1.2 ms\n"
+      "CPU times: user 1.25 ms, sys: 0 ns, total: 1.25 ms\n",
+      "Wall time: 1.25 ms\n"
      ]
     }
    ],
@@ -802,8 +802,8 @@
      "name": "stdout",
      "output_type": "stream",
      "text": [
-      "CPU times: user 3.43 ms, sys: 0 ns, total: 3.43 ms\n",
-      "Wall time: 3.43 ms\n"
+      "CPU times: user 3.47 ms, sys: 0 ns, total: 3.47 ms\n",
+      "Wall time: 3.47 ms\n"
      ]
     }
    ],