From 8a3efb8fe4dd942185ada009d83a894dd7492042 Mon Sep 17 00:00:00 2001
From: Neil Smith <neil.git@njae.me.uk>
Date: Fri, 17 Feb 2017 10:37:45 +0000
Subject: [PATCH] More tweakes to prime generation and testing

---
 euler35.ipynb | 198 +++++++++++++-------------------------------------
 euler35.rb    |  30 ++++----
 primes.rb     |  14 +++-
 3 files changed, 80 insertions(+), 162 deletions(-)

diff --git a/euler35.ipynb b/euler35.ipynb
index dcd6039..f3bda1b 100644
--- a/euler35.ipynb
+++ b/euler35.ipynb
@@ -2,7 +2,7 @@
  "cells": [
   {
    "cell_type": "code",
-   "execution_count": 1,
+   "execution_count": 25,
    "metadata": {
     "collapsed": false,
     "scrolled": true
@@ -11,78 +11,23 @@
     {
      "data": {
       "text/plain": [
-       "#<Primes:0x0055fc1b2516c0 @primes=[2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97, 101, 103, 107, 109, 113, 127, 131, 137, 139, 149, 151, 157, 163, 167, 173, 179, 181, 191, 193, 197, 199, 211, 223, 227, 229, 233, 239, 241, 251, 257, 263, 269, 271, 277, 281, 283, 293, 307, 311, 313, 317, 331, 337, 347, 349, 353, 359, 367, 373, 379, 383, 389, 397, 401, 409, 419, 421, 431, 433, 439, 443, 449, 457, 461, 463, 467, 479, 487, 491, 499, 503, 509, 521, 523, 541, 547, 557, 563, 569, 571, 577, 587, 593, 599, 601, 607, 613, 617, 619, 631, 641, 643, 647, 653, 659, 661, 673, 677, 683, 691, 701, 709, 719, 727, 733, 739, 743, 751, 757, 761, 769, 773, 787, 797, 809, 811, 821, 823, 827, 829, 839, 853, 857, 859, 863, 877, 881, 883, 887, 907, 911, 919, 929, 937, 941, 947, 953, 967, 971, 977, 983, 991, 997, 1009, 1013, 1019, 1021, 1031, 1033, 1039, 1049, 1051, 1061, 1063, 1069, 1087, 1091, 1093, 1097, 1103, 1109, 1117, 1123, 1129, 1151, 1153, 1163, 1171, 1181, 1187, 1193, 1201, 1213, 1217, 1223, 1229, 1231, 1237, 1249, 1259, 1277, 1279, 1283, 1289, 1291, 1297, 1301, 1303, 1307, 1319, 1321, 1327, 1361, 1367, 1373, 1381, 1399, 1409, 1423, 1427, 1429, 1433, 1439, 1447, 1451, 1453, 1459, 1471, 1481, 1483, 1487, 1489, 1493, 1499, 1511, 1523, 1531, 1543, 1549, 1553, 1559, 1567, 1571, 1579, 1583, 1597, 1601, 1607, 1609, 1613, 1619, 1621, 1627, 1637, 1657, 1663, 1667, 1669, 1693, 1697, 1699, 1709, 1721, 1723, 1733, 1741, 1747, 1753, 1759, 1777, 1783, 1787, 1789, 1801, 1811, 1823, 1831, 1847, 1861, 1867, 1871, 1873, 1877, 1879, 1889, 1901, 1907, 1913, 1931, 1933, 1949, 1951, 1973, 1979, 1987, 1993, 1997, 1999, 2003, 2011, 2017, 2027, 2029, 2039, 2053, 2063, 2069, 2081, 2083, 2087, 2089, 2099, 2111, 2113, 2129, 2131, 2137, 2141, 2143, 2153, 2161, 2179, 2203, 2207, 2213, 2221, 2237, 2239, 2243, 2251, 2267, 2269, 2273, 2281, 2287, 2293, 2297, 2309, 2311, 2333, 2339, 2341, 2347, 2351, 2357, 2371, 2377, 2381, 2383, 2389, 2393, 2399, 2411, 2417, 2423, 2437, 2441, 2447, 2459, 2467, 2473, 2477, 2503, 2521, 2531, 2539, 2543, 2549, 2551, 2557, 2579, 2591, 2593, 2609, 2617, 2621, 2633, 2647, 2657, 2659, 2663, 2671, 2677, 2683, 2687, 2689, 2693, 2699, 2707, 2711, 2713, 2719, 2729, 2731, 2741, 2749, 2753, 2767, 2777, 2789, 2791, 2797, 2801, 2803, 2819, 2833, 2837, 2843, 2851, 2857, 2861, 2879, 2887, 2897, 2903, 2909, 2917, 2927, 2939, 2953, 2957, 2963, 2969, 2971, 2999, 3001, 3011, 3019, 3023, 3037, 3041, 3049, 3061, 3067, 3079, 3083, 3089, 3109, 3119, 3121, 3137, 3163, 3167, 3169, 3181, 3187, 3191, 3203, 3209, 3217, 3221, 3229, 3251, 3253, 3257, 3259, 3271, 3299, 3301, 3307, 3313, 3319, 3323, 3329, 3331, 3343, 3347, 3359, 3361, 3371, 3373, 3389, 3391, 3407, 3413, 3433, 3449, 3457, 3461, 3463, 3467, 3469, 3491, 3499, 3511, 3517, 3527, 3529, 3533, 3539, 3541, 3547, 3557, 3559, 3571, 3581, 3583, 3593, 3607, 3613, 3617, 3623, 3631, 3637, 3643, 3659, 3671, 3673, 3677, 3691, 3697, 3701, 3709, 3719, 3727, 3733, 3739, 3761, 3767, 3769, 3779, 3793, 3797, 3803, 3821, 3823, 3833, 3847, 3851, 3853, 3863, 3877, 3881, 3889, 3907, 3911, 3917, 3919, 3923, 3929, 3931, 3943, 3947, 3967, 3989, 4001, 4003, 4007, 4013, 4019, 4021, 4027, 4049, 4051, 4057, 4073, 4079, 4091, 4093, 4099, 4111, 4127, 4129, 4133, 4139, 4153, 4157, 4159, 4177, 4201, 4211, 4217, 4219, 4229, 4231, 4241, 4243, 4253, 4259, 4261, 4271, 4273, 4283, 4289, 4297, 4327, 4337, 4339, 4349, 4357, 4363, 4373, 4391, 4397, 4409, 4421, 4423, 4441, 4447, 4451, 4457, 4463, 4481, 4483, 4493, 4507, 4513, 4517, 4519, 4523, 4547, 4549, 4561, 4567, 4583, 4591, 4597, 4603, 4621, 4637, 4639, 4643, 4649, 4651, 4657, 4663, 4673, 4679, 4691, 4703, 4721, 4723, 4729, 4733, 4751, 4759, 4783, 4787, 4789, 4793, 4799, 4801, 4813, 4817, 4831, 4861, 4871, 4877, 4889, 4903, 4909, 4919, 4931, 4933, 4937, 4943, 4951, 4957, 4967, 4969, 4973, 4987, 4993, 4999, 5003, 5009, 5011, 5021, 5023, 5039, 5051, 5059, 5077, 5081, 5087, 5099, 5101, 5107, 5113, 5119, 5147, 5153, 5167, 5171, 5179, 5189, 5197, 5209, 5227, 5231, 5233, 5237, 5261, 5273, 5279, 5281, 5297, 5303, 5309, 5323, 5333, 5347, 5351, 5381, 5387, 5393, 5399, 5407, 5413, 5417, 5419, 5431, 5437, 5441, 5443, 5449, 5471, 5477, 5479, 5483, 5501, 5503, 5507, 5519, 5521, 5527, 5531, 5557, 5563, 5569, 5573, 5581, 5591, 5623, 5639, 5641, 5647, 5651, 5653, 5657, 5659, 5669, 5683, 5689, 5693, 5701, 5711, 5717, 5737, 5741, 5743, 5749, 5779, 5783, 5791, 5801, 5807, 5813, 5821, 5827, 5839, 5843, 5849, 5851, 5857, 5861, 5867, 5869, 5879, 5881, 5897, 5903, 5923, 5927, 5939, 5953, 5981, 5987, 6007, 6011, 6029, 6037, 6043, 6047, 6053, 6067, 6073, 6079, 6089, 6091, 6101, 6113, 6121, 6131, 6133, 6143, 6151, 6163, 6173, 6197, 6199, 6203, 6211, 6217, 6221, 6229, 6247, 6257, 6263, 6269, 6271, 6277, 6287, 6299, 6301, 6311, 6317, 6323, 6329, 6337, 6343, 6353, 6359, 6361, 6367, 6373, 6379, 6389, 6397, 6421, 6427, 6449, 6451, 6469, 6473, 6481, 6491, 6521, 6529, 6547, 6551, 6553, 6563, 6569, 6571, 6577, 6581, 6599, 6607, 6619, 6637, 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8147, 8161, 8167, 8171, 8179, 8191, 8209, 8219, 8221, 8231, 8233, 8237, 8243, 8263, 8269, 8273, 8287, 8291, 8293, 8297, 8311, 8317, 8329, 8353, 8363, 8369, 8377, 8387, 8389, 8419, 8423, 8429, 8431, 8443, 8447, 8461, 8467, 8501, 8513, 8521, 8527, 8537, 8539, 8543, 8563, 8573, 8581, 8597, 8599, 8609, 8623, 8627, 8629, 8641, 8647, 8663, 8669, 8677, 8681, 8689, 8693, 8699, 8707, 8713, 8719, 8731, 8737, 8741, 8747, 8753, 8761, 8779, 8783, 8803, 8807, 8819, 8821, 8831, 8837, 8839, 8849, 8861, 8863, 8867, 8887, 8893, 8923, 8929, 8933, 8941, 8951, 8963, 8969, 8971, 8999, 9001, 9007, 9011, 9013, 9029, 9041, 9043, 9049, 9059, 9067, 9091, 9103, 9109, 9127, 9133, 9137, 9151, 9157, 9161, 9173, 9181, 9187, 9199, 9203, 9209, 9221, 9227, 9239, 9241, 9257, 9277, 9281, 9283, 9293, 9311, 9319, 9323, 9337, 9341, 9343, 9349, 9371, 9377, 9391, 9397, 9403, 9413, 9419, 9421, 9431, 9433, 9437, 9439, 9461, 9463, 9467, 9473, 9479, 9491, 9497, 9511, 9521, 9533, 9539, 9547, 9551, 9587, 9601, 9613, 9619, 9623, 9629, 9631, 9643, 9649, 9661, 9677, 9679, 9689, 9697, 9719, 9721, 9733, 9739, 9743, 9749, 9767, 9769, 9781, 9787, 9791, 9803, 9811, 9817, 9829, 9833, 9839, 9851, 9857, 9859, 9871, 9883, 9887, 9901, 9907, 9923, 9929, 9931, 9941, 9949, 9967, 9973], @cache_up_to=30, @mod_candidates=[1, 7, 11, 13, 17, 19, 23, 29]>"
+       "#<Primes:0x00557898fcb500 @primes=[2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97, 101, 103, 107, 109, 113, 127, 131, 137, 139, 149, 151, 157, 163, 167, 173, 179, 181, 191, 193, 197, 199, 211, 223, 227, 229, 233, 239, 241, 251, 257, 263, 269, 271, 277, 281, 283, 293, 307, 311, 313, 317, 331, 337, 347, 349, 353, 359, 367, 373, 379, 383, 389, 397, 401, 409, 419, 421, 431, 433, 439, 443, 449, 457, 461, 463, 467, 479, 487, 491, 499, 503, 509, 521, 523, 541, 547, 557, 563, 569, 571, 577, 587, 593, 599, 601, 607, 613, 617, 619, 631, 641, 643, 647, 653, 659, 661, 673, 677, 683, 691, 701, 709, 719, 727, 733, 739, 743, 751, 757, 761, 769, 773, 787, 797, 809, 811, 821, 823, 827, 829, 839, 853, 857, 859, 863, 877, 881, 883, 887, 907, 911, 919, 929, 937, 941, 947, 953, 967, 971, 977, 983, 991, 997, 1009, 1013, 1019, 1021, 1031, 1033, 1039, 1049, 1051, 1061, 1063, 1069, 1087, 1091, 1093, 1097, 1103, 1109, 1117, 1123, 1129, 1151, 1153, 1163, 1171, 1181, 1187, 1193, 1201, 1213, 1217, 1223, 1229, 1231, 1237, 1249, 1259, 1277, 1279, 1283, 1289, 1291, 1297, 1301, 1303, 1307, 1319, 1321, 1327, 1361, 1367, 1373, 1381, 1399, 1409, 1423, 1427, 1429, 1433, 1439, 1447, 1451, 1453, 1459, 1471, 1481, 1483, 1487, 1489, 1493, 1499, 1511, 1523, 1531, 1543, 1549, 1553, 1559, 1567, 1571, 1579, 1583, 1597, 1601, 1607, 1609, 1613, 1619, 1621, 1627, 1637, 1657, 1663, 1667, 1669, 1693, 1697, 1699, 1709, 1721, 1723, 1733, 1741, 1747, 1753, 1759, 1777, 1783, 1787, 1789, 1801, 1811, 1823, 1831, 1847, 1861, 1867, 1871, 1873, 1877, 1879, 1889, 1901, 1907, 1913, 1931, 1933, 1949, 1951, 1973, 1979, 1987, 1993, 1997, 1999, 2003, 2011, 2017, 2027, 2029, 2039, 2053, 2063, 2069, 2081, 2083, 2087, 2089, 2099, 2111, 2113, 2129, 2131, 2137, 2141, 2143, 2153, 2161, 2179, 2203, 2207, 2213, 2221, 2237, 2239, 2243, 2251, 2267, 2269, 2273, 2281, 2287, 2293, 2297, 2309, 2311, 2333, 2339, 2341, 2347, 2351, 2357, 2371, 2377, 2381, 2383, 2389, 2393, 2399, 2411, 2417, 2423, 2437, 2441, 2447, 2459, 2467, 2473, 2477, 2503, 2521, 2531, 2539, 2543, 2549, 2551, 2557, 2579, 2591, 2593, 2609, 2617, 2621, 2633, 2647, 2657, 2659, 2663, 2671, 2677, 2683, 2687, 2689, 2693, 2699, 2707, 2711, 2713, 2719, 2729, 2731, 2741, 2749, 2753, 2767, 2777, 2789, 2791, 2797, 2801, 2803, 2819, 2833, 2837, 2843, 2851, 2857, 2861, 2879, 2887, 2897, 2903, 2909, 2917, 2927, 2939, 2953, 2957, 2963, 2969, 2971, 2999, 3001, 3011, 3019, 3023, 3037, 3041, 3049, 3061, 3067, 3079, 3083, 3089, 3109, 3119, 3121, 3137, 3163, 3167, 3169, 3181, 3187, 3191, 3203, 3209, 3217, 3221, 3229, 3251, 3253, 3257, 3259, 3271, 3299, 3301, 3307, 3313, 3319, 3323, 3329, 3331, 3343, 3347, 3359, 3361, 3371, 3373, 3389, 3391, 3407, 3413, 3433, 3449, 3457, 3461, 3463, 3467, 3469, 3491, 3499, 3511, 3517, 3527, 3529, 3533, 3539, 3541, 3547, 3557, 3559, 3571, 3581, 3583, 3593, 3607, 3613, 3617, 3623, 3631, 3637, 3643, 3659, 3671, 3673, 3677, 3691, 3697, 3701, 3709, 3719, 3727, 3733, 3739, 3761, 3767, 3769, 3779, 3793, 3797, 3803, 3821, 3823, 3833, 3847, 3851, 3853, 3863, 3877, 3881, 3889, 3907, 3911, 3917, 3919, 3923, 3929, 3931, 3943, 3947, 3967, 3989, 4001, 4003, 4007, 4013, 4019, 4021, 4027, 4049, 4051, 4057, 4073, 4079, 4091, 4093, 4099, 4111, 4127, 4129, 4133, 4139, 4153, 4157, 4159, 4177, 4201, 4211, 4217, 4219, 4229, 4231, 4241, 4243, 4253, 4259, 4261, 4271, 4273, 4283, 4289, 4297, 4327, 4337, 4339, 4349, 4357, 4363, 4373, 4391, 4397, 4409, 4421, 4423, 4441, 4447, 4451, 4457, 4463, 4481, 4483, 4493, 4507, 4513, 4517, 4519, 4523, 4547, 4549, 4561, 4567, 4583, 4591, 4597, 4603, 4621, 4637, 4639, 4643, 4649, 4651, 4657, 4663, 4673, 4679, 4691, 4703, 4721, 4723, 4729, 4733, 4751, 4759, 4783, 4787, 4789, 4793, 4799, 4801, 4813, 4817, 4831, 4861, 4871, 4877, 4889, 4903, 4909, 4919, 4931, 4933, 4937, 4943, 4951, 4957, 4967, 4969, 4973, 4987, 4993, 4999, 5003, 5009, 5011, 5021, 5023, 5039, 5051, 5059, 5077, 5081, 5087, 5099, 5101, 5107, 5113, 5119, 5147, 5153, 5167, 5171, 5179, 5189, 5197, 5209, 5227, 5231, 5233, 5237, 5261, 5273, 5279, 5281, 5297, 5303, 5309, 5323, 5333, 5347, 5351, 5381, 5387, 5393, 5399, 5407, 5413, 5417, 5419, 5431, 5437, 5441, 5443, 5449, 5471, 5477, 5479, 5483, 5501, 5503, 5507, 5519, 5521, 5527, 5531, 5557, 5563, 5569, 5573, 5581, 5591, 5623, 5639, 5641, 5647, 5651, 5653, 5657, 5659, 5669, 5683, 5689, 5693, 5701, 5711, 5717, 5737, 5741, 5743, 5749, 5779, 5783, 5791, 5801, 5807, 5813, 5821, 5827, 5839, 5843, 5849, 5851, 5857, 5861, 5867, 5869, 5879, 5881, 5897, 5903, 5923, 5927, 5939, 5953, 5981, 5987, 6007, 6011, 6029, 6037, 6043, 6047, 6053, 6067, 6073, 6079, 6089, 6091, 6101, 6113, 6121, 6131, 6133, 6143, 6151, 6163, 6173, 6197, 6199, 6203, 6211, 6217, 6221, 6229, 6247, 6257, 6263, 6269, 6271, 6277, 6287, 6299, 6301, 6311, 6317, 6323, 6329, 6337, 6343, 6353, 6359, 6361, 6367, 6373, 6379, 6389, 6397, 6421, 6427, 6449, 6451, 6469, 6473, 6481, 6491, 6521, 6529, 6547, 6551, 6553, 6563, 6569, 6571, 6577, 6581, 6599, 6607, 6619, 6637, 6653, 6659, 6661, 6673, 6679, 6689, 6691, 6701, 6703, 6709, 6719, 6733, 6737, 6761, 6763, 6779, 6781, 6791, 6793, 6803, 6823, 6827, 6829, 6833, 6841, 6857, 6863, 6869, 6871, 6883, 6899, 6907, 6911, 6917, 6947, 6949, 6959, 6961, 6967, 6971, 6977, 6983, 6991, 6997, 7001, 7013, 7019, 7027, 7039, 7043, 7057, 7069, 7079, 7103, 7109, 7121, 7127, 7129, 7151, 7159, 7177, 7187, 7193, 7207, 7211, 7213, 7219, 7229, 7237, 7243, 7247, 7253, 7283, 7297, 7307, 7309, 7321, 7331, 7333, 7349, 7351, 7369, 7393, 7411, 7417, 7433, 7451, 7457, 7459, 7477, 7481, 7487, 7489, 7499, 7507, 7517, 7523, 7529, 7537, 7541, 7547, 7549, 7559, 7561, 7573, 7577, 7583, 7589, 7591, 7603, 7607, 7621, 7639, 7643, 7649, 7669, 7673, 7681, 7687, 7691, 7699, 7703, 7717, 7723, 7727, 7741, 7753, 7757, 7759, 7789, 7793, 7817, 7823, 7829, 7841, 7853, 7867, 7873, 7877, 7879, 7883, 7901, 7907, 7919, 7927, 7933, 7937, 7949, 7951, 7963, 7993, 8009, 8011, 8017, 8039, 8053, 8059, 8069, 8081, 8087, 8089, 8093, 8101, 8111, 8117, 8123, 8147, 8161, 8167, 8171, 8179, 8191, 8209, 8219, 8221, 8231, 8233, 8237, 8243, 8263, 8269, 8273, 8287, 8291, 8293, 8297, 8311, 8317, 8329, 8353, 8363, 8369, 8377, 8387, 8389, 8419, 8423, 8429, 8431, 8443, 8447, 8461, 8467, 8501, 8513, 8521, 8527, 8537, 8539, 8543, 8563, 8573, 8581, 8597, 8599, 8609, 8623, 8627, 8629, 8641, 8647, 8663, 8669, 8677, 8681, 8689, 8693, 8699, 8707, 8713, 8719, 8731, 8737, 8741, 8747, 8753, 8761, 8779, 8783, 8803, 8807, 8819, 8821, 8831, 8837, 8839, 8849, 8861, 8863, 8867, 8887, 8893, 8923, 8929, 8933, 8941, 8951, 8963, 8969, 8971, 8999, 9001, 9007, 9011, 9013, 9029, 9041, 9043, 9049, 9059, 9067, 9091, 9103, 9109, 9127, 9133, 9137, 9151, 9157, 9161, 9173, 9181, 9187, 9199, 9203, 9209, 9221, 9227, 9239, 9241, 9257, 9277, 9281, 9283, 9293, 9311, 9319, 9323, 9337, 9341, 9343, 9349, 9371, 9377, 9391, 9397, 9403, 9413, 9419, 9421, 9431, 9433, 9437, 9439, 9461, 9463, 9467, 9473, 9479, 9491, 9497, 9511, 9521, 9533, 9539, 9547, 9551, 9587, 9601, 9613, 9619, 9623, 9629, 9631, 9643, 9649, 9661, 9677, 9679, 9689, 9697, 9719, 9721, 9733, 9739, 9743, 9749, 9767, 9769, 9781, 9787, 9791, 9803, 9811, 9817, 9829, 9833, 9839, 9851, 9857, 9859, 9871, 9883, 9887, 9901, 9907, 9923, 9929, 9931, 9941, 9949, 9967, 9973], @cache_up_to=30, @mod_candidates=[1, 7, 11, 13, 17, 19, 23, 29]>"
       ]
      },
-     "execution_count": 1,
+     "execution_count": 25,
      "metadata": {},
      "output_type": "execute_result"
     }
    ],
    "source": [
+    "load 'array-numbers.rb'\n",
     "load 'primes.rb'\n",
     "$primes = Primes.instance"
    ]
   },
   {
    "cell_type": "code",
-   "execution_count": 2,
-   "metadata": {
-    "collapsed": false
-   },
-   "outputs": [
-    {
-     "data": {
-      "text/plain": [
-       ":to_r"
-      ]
-     },
-     "execution_count": 2,
-     "metadata": {},
-     "output_type": "execute_result"
-    }
-   ],
-   "source": [
-    "class Array\n",
-    "  def to_i\n",
-    "    self.map {|d| d.to_s}.join.to_i\n",
-    "  end\n",
-    "  \n",
-    "  def to_r\n",
-    "    Rational(self[0].to_i, self[1].to_i)\n",
-    "  end\n",
-    "end"
-   ]
-  },
-  {
-   "cell_type": "code",
-   "execution_count": 3,
-   "metadata": {
-    "collapsed": false
-   },
-   "outputs": [
-    {
-     "data": {
-      "text/plain": [
-       ":to_digits"
-      ]
-     },
-     "execution_count": 3,
-     "metadata": {},
-     "output_type": "execute_result"
-    }
-   ],
-   "source": [
-    "class Integer\n",
-    "  def to_digits\n",
-    "    self.to_s.split('').map {|d| d.to_i}\n",
-    "  end\n",
-    "end "
-   ]
-  },
-  {
-   "cell_type": "code",
-   "execution_count": 4,
+   "execution_count": 26,
    "metadata": {
     "collapsed": false
    },
@@ -93,7 +38,7 @@
        ":all_rotations"
       ]
      },
-     "execution_count": 4,
+     "execution_count": 26,
      "metadata": {},
      "output_type": "execute_result"
     }
@@ -108,7 +53,7 @@
   },
   {
    "cell_type": "code",
-   "execution_count": 5,
+   "execution_count": 27,
    "metadata": {
     "collapsed": false
    },
@@ -119,7 +64,7 @@
        "[[1, 2, 3], [2, 3, 1], [3, 1, 2]]"
       ]
      },
-     "execution_count": 5,
+     "execution_count": 27,
      "metadata": {},
      "output_type": "execute_result"
     }
@@ -130,7 +75,7 @@
   },
   {
    "cell_type": "code",
-   "execution_count": 6,
+   "execution_count": 28,
    "metadata": {
     "collapsed": false
    },
@@ -141,7 +86,7 @@
        "[123, 231, 312]"
       ]
      },
-     "execution_count": 6,
+     "execution_count": 28,
      "metadata": {},
      "output_type": "execute_result"
     }
@@ -152,7 +97,7 @@
   },
   {
    "cell_type": "code",
-   "execution_count": 7,
+   "execution_count": 29,
    "metadata": {
     "collapsed": false
    },
@@ -163,7 +108,7 @@
        "[false, false, false]"
       ]
      },
-     "execution_count": 7,
+     "execution_count": 29,
      "metadata": {},
      "output_type": "execute_result"
     }
@@ -174,7 +119,7 @@
   },
   {
    "cell_type": "code",
-   "execution_count": 8,
+   "execution_count": 30,
    "metadata": {
     "collapsed": false
    },
@@ -185,7 +130,7 @@
        "[true, true]"
       ]
      },
-     "execution_count": 8,
+     "execution_count": 30,
      "metadata": {},
      "output_type": "execute_result"
     }
@@ -196,7 +141,7 @@
   },
   {
    "cell_type": "code",
-   "execution_count": 9,
+   "execution_count": 31,
    "metadata": {
     "collapsed": false
    },
@@ -207,7 +152,7 @@
        "false"
       ]
      },
-     "execution_count": 9,
+     "execution_count": 31,
      "metadata": {},
      "output_type": "execute_result"
     }
@@ -218,7 +163,7 @@
   },
   {
    "cell_type": "code",
-   "execution_count": 10,
+   "execution_count": 32,
    "metadata": {
     "collapsed": false
    },
@@ -229,7 +174,7 @@
        "true"
       ]
      },
-     "execution_count": 10,
+     "execution_count": 32,
      "metadata": {},
      "output_type": "execute_result"
     }
@@ -240,7 +185,7 @@
   },
   {
    "cell_type": "code",
-   "execution_count": 11,
+   "execution_count": 33,
    "metadata": {
     "collapsed": false
    },
@@ -251,19 +196,14 @@
        ":circular_prime?"
       ]
      },
-     "execution_count": 11,
+     "execution_count": 33,
      "metadata": {},
      "output_type": "execute_result"
     }
    ],
    "source": [
     "class Integer\n",
-    "  @@primes = Primes.instance\n",
-    "  \n",
-    "  def prime?\n",
-    "    @@primes.include? self\n",
-    "  end\n",
-    "  \n",
+    " \n",
     "  def pl\n",
     "    @@primes.primes.length\n",
     "  end\n",
@@ -276,43 +216,7 @@
   },
   {
    "cell_type": "code",
-   "execution_count": 11,
-   "metadata": {
-    "collapsed": false
-   },
-   "outputs": [
-    {
-     "data": {
-      "text/plain": [
-       ":circular_prime?"
-      ]
-     },
-     "execution_count": 11,
-     "metadata": {},
-     "output_type": "execute_result"
-    }
-   ],
-   "source": [
-    "class Integer\n",
-    "  @@primes = Primes.instance\n",
-    "  \n",
-    "  def prime?\n",
-    "    @@primes.include? self\n",
-    "  end\n",
-    "  \n",
-    "  def pl\n",
-    "    @@primes.primes.length\n",
-    "  end\n",
-    "  \n",
-    "  def circular_prime?\n",
-    "    self.to_digits.all_rotations.map {|n| n.to_i}.all? {|n| n.prime?} if self.prime?\n",
-    "  end\n",
-    "end"
-   ]
-  },
-  {
-   "cell_type": "code",
-   "execution_count": 12,
+   "execution_count": 34,
    "metadata": {
     "collapsed": false
    },
@@ -323,7 +227,7 @@
        "1229"
       ]
      },
-     "execution_count": 12,
+     "execution_count": 34,
      "metadata": {},
      "output_type": "execute_result"
     }
@@ -334,7 +238,7 @@
   },
   {
    "cell_type": "code",
-   "execution_count": 13,
+   "execution_count": 35,
    "metadata": {
     "collapsed": false
    },
@@ -345,7 +249,7 @@
        "[2, 3, 5, 7, 11, 13, 17, 31, 37, 71, 73, 79, 97]"
       ]
      },
-     "execution_count": 13,
+     "execution_count": 35,
      "metadata": {},
      "output_type": "execute_result"
     }
@@ -356,7 +260,7 @@
   },
   {
    "cell_type": "code",
-   "execution_count": 14,
+   "execution_count": 36,
    "metadata": {
     "collapsed": false
    },
@@ -367,7 +271,7 @@
        "[2, 3, 5, 7, 11, 13, 17, 31, 37, 71, 73, 79, 97]"
       ]
      },
-     "execution_count": 14,
+     "execution_count": 36,
      "metadata": {},
      "output_type": "execute_result"
     }
@@ -378,7 +282,7 @@
   },
   {
    "cell_type": "code",
-   "execution_count": 15,
+   "execution_count": 37,
    "metadata": {
     "collapsed": false
    },
@@ -389,7 +293,7 @@
        "13"
       ]
      },
-     "execution_count": 15,
+     "execution_count": 37,
      "metadata": {},
      "output_type": "execute_result"
     }
@@ -400,7 +304,7 @@
   },
   {
    "cell_type": "code",
-   "execution_count": 16,
+   "execution_count": 38,
    "metadata": {
     "collapsed": false
    },
@@ -412,7 +316,7 @@
   },
   {
    "cell_type": "code",
-   "execution_count": 17,
+   "execution_count": 39,
    "metadata": {
     "collapsed": false
    },
@@ -423,7 +327,7 @@
   },
   {
    "cell_type": "code",
-   "execution_count": 18,
+   "execution_count": 40,
    "metadata": {
     "collapsed": false
    },
@@ -432,7 +336,7 @@
      "name": "stdout",
      "output_type": "stream",
      "text": [
-      "Time elapsed 0.125950399 seconds, going from 1229 to 1230 primes cached\n"
+      "Time elapsed 0.054970036 seconds, going from 1229 to 1230 primes cached\n"
      ]
     },
     {
@@ -441,7 +345,7 @@
        "33"
       ]
      },
-     "execution_count": 18,
+     "execution_count": 40,
      "metadata": {},
      "output_type": "execute_result"
     }
@@ -458,7 +362,7 @@
   },
   {
    "cell_type": "code",
-   "execution_count": 19,
+   "execution_count": 41,
    "metadata": {
     "collapsed": false
    },
@@ -467,7 +371,7 @@
      "name": "stdout",
      "output_type": "stream",
      "text": [
-      "Time elapsed 0.123380027 seconds, going from 1230 to 1230 primes cached\n"
+      "Time elapsed 0.067191288 seconds, going from 1230 to 1230 primes cached\n"
      ]
     },
     {
@@ -476,7 +380,7 @@
        "33"
       ]
      },
-     "execution_count": 19,
+     "execution_count": 41,
      "metadata": {},
      "output_type": "execute_result"
     }
@@ -493,7 +397,7 @@
   },
   {
    "cell_type": "code",
-   "execution_count": 20,
+   "execution_count": 42,
    "metadata": {
     "collapsed": false
    },
@@ -502,7 +406,7 @@
      "name": "stdout",
      "output_type": "stream",
      "text": [
-      "Time elapsed 9.59874264 seconds, going from 1230 to 9593 primes cached\n"
+      "Time elapsed 0.448493896 seconds, going from 1230 to 9593 primes cached\n"
      ]
     },
     {
@@ -511,7 +415,7 @@
        "43"
       ]
      },
-     "execution_count": 20,
+     "execution_count": 42,
      "metadata": {},
      "output_type": "execute_result"
     }
@@ -528,7 +432,7 @@
   },
   {
    "cell_type": "code",
-   "execution_count": 21,
+   "execution_count": 43,
    "metadata": {
     "collapsed": false
    },
@@ -537,7 +441,7 @@
      "name": "stdout",
      "output_type": "stream",
      "text": [
-      "Time elapsed 7.768858416 seconds, going from 9593 to 9593 primes cached\n"
+      "Time elapsed 0.274737814 seconds, going from 9593 to 9593 primes cached\n"
      ]
     },
     {
@@ -546,7 +450,7 @@
        "43"
       ]
      },
-     "execution_count": 21,
+     "execution_count": 43,
      "metadata": {},
      "output_type": "execute_result"
     }
@@ -563,7 +467,7 @@
   },
   {
    "cell_type": "code",
-   "execution_count": 22,
+   "execution_count": 44,
    "metadata": {
     "collapsed": false
    },
@@ -572,7 +476,7 @@
      "name": "stdout",
      "output_type": "stream",
      "text": [
-      "Time elapsed 710.264030183 seconds, going from 9593 to 78499 primes cached\n"
+      "Time elapsed 10.457163873 seconds, going from 9593 to 78499 primes cached\n"
      ]
     },
     {
@@ -581,7 +485,7 @@
        "55"
       ]
      },
-     "execution_count": 22,
+     "execution_count": 44,
      "metadata": {},
      "output_type": "execute_result"
     }
@@ -598,7 +502,7 @@
   },
   {
    "cell_type": "code",
-   "execution_count": 23,
+   "execution_count": 47,
    "metadata": {
     "collapsed": false
    },
@@ -607,16 +511,16 @@
      "name": "stdout",
      "output_type": "stream",
      "text": [
-      "Time elapsed 597.009405584 seconds, going from 78499 to 78499 primes cached\n"
+      "Time elapsed 2.884361875 seconds, going from 78499 to 78499 primes cached\n"
      ]
     },
     {
      "data": {
       "text/plain": [
-       "55"
+       "[2, 3, 5, 7, 11, 13, 17, 31, 37, 71, 73, 79, 97, 113, 131, 197, 199, 311, 337, 373, 719, 733, 919, 971, 991, 1193, 1931, 3119, 3779, 7793, 7937, 9311, 9377, 11939, 19391, 19937, 37199, 39119, 71993, 91193, 93719, 93911, 99371, 193939, 199933, 319993, 331999, 391939, 393919, 919393, 933199, 939193, 939391, 993319, 999331]"
       ]
      },
-     "execution_count": 23,
+     "execution_count": 47,
      "metadata": {},
      "output_type": "execute_result"
     }
@@ -628,7 +532,7 @@
     "end_time = Time.now\n",
     "pl1 = 0.pl\n",
     "puts \"Time elapsed #{(end_time - beginning_time)} seconds, going from #{pl0} to #{pl1} primes cached\"\n",
-    "cps.length"
+    "cps"
    ]
   },
   {
diff --git a/euler35.rb b/euler35.rb
index d2ba75d..a25edfe 100644
--- a/euler35.rb
+++ b/euler35.rb
@@ -1,14 +1,15 @@
 require 'ruby-prof'
 load 'primes.rb'
+load 'array-numbers.rb'
 
 class Array
-  def to_i
-    self.map {|d| d.to_s}.join.to_i
-  end
+#   def to_i
+#     self.map {|d| d.to_s}.join.to_i
+#   end
   
-  def to_r
-    Rational(self[0].to_i, self[1].to_i)
-  end
+#   def to_r
+#     Rational(self[0].to_i, self[1].to_i)
+#   end
 
   def all_rotations
     (0...self.length).map {|r| self.rotate(r)}
@@ -16,26 +17,27 @@ class Array
 end
 
 class Integer
-  @@primes = Primes.instance
+#   @@primes = Primes.instance
 
-  def prime?
-    @@primes.include? self
-  end
+#   def prime?
+#     @@primes.include? self
+#   end
   
   def pl
     @@primes.primes.length
   end
   
   def circular_prime?
-    self.to_digits.all_rotations.map {|n| n.to_i}.all? {|n| n.prime?} if self.prime?
+    self.prime? && self.to_digits.all_rotations.map {|n| n.to_i}.all? {|n| n.prime?}
   end
 
-  def to_digits
-    self.to_s.split('').map {|d| d.to_i}
-  end
+  # def to_digits
+  #   self.to_s.split('').map {|d| d.to_i}
+  # end
 end 
 
 
+
 RubyProf.start
 
 # (10**6).prime?
diff --git a/primes.rb b/primes.rb
index 7e2cfaa..b254fab 100644
--- a/primes.rb
+++ b/primes.rb
@@ -95,10 +95,21 @@ class Primes
     while n >= @primes.last
       add_next_prime!
     end
-    # @primes.include?(n)
     !!(@primes.bsearch {|i| n - i})
   end
 
+  # Use this if you're only testing a few primes and not needing to generate many more.
+  def is_prime?(n)
+    return true if n == 2
+    return false if n < 2
+    limit = Math.sqrt(n).floor
+    while @primes.last <= limit
+      add_next_prime!
+    end
+    index = @primes.bsearch_index {|i| i > limit} || @primes.length
+    @primes[0..index].none? {|i| n % i == 0}
+  end
+
 end
 
 
@@ -107,5 +118,6 @@ class Integer
   
   def prime?
     @@primes.include? self
+    # @@primes.is_prime? self
   end
 end
-- 
2.34.1