2020-09-21-MATH 3600 Goldbach

436 days ago by calkin

Investigations around Goldbach's conjecture.  

Goldbach, in a letter to Euler in 1742, conjectured that every even number greater than 2 is a sum of two prime numbers.

Let's start by making a list of odd primes less than $N$ (we'll start with $N=100$).

Then we can walk through the list twice, and for each pair $p$ and $q$ of primes, compute $p+q$ and increment the count 

N=10000 i=3 prime_list=[] while i<N: if is_prime(i): prime_list.append(i) i+=2 
       
goldbach_count = [0 for i in srange(2*N)] 
       
for p in prime_list: for q in prime_list: goldbach_count[p+q]+=1 
       
better_goldbach_list=[] for i in srange(3,N): better_goldbach_list.append([2*i,goldbach_count[2*i]]) 
       
list_plot(better_goldbach_list) 
       
print(better_goldbach_list) 
       
WARNING: Output truncated!  
full_output.txt



[[6, 1], [8, 2], [10, 3], [12, 2], [14, 3], [16, 4], [18, 4], [20, 4],
[22, 5], [24, 6], [26, 5], [28, 4], [30, 6], [32, 4], [34, 7], [36, 8],
[38, 3], [40, 6], [42, 8], [44, 6], [46, 7], [48, 10], [50, 8], [52, 6],
[54, 10], [56, 6], [58, 7], [60, 12], [62, 5], [64, 10], [66, 12], [68,
4], [70, 10], [72, 12], [74, 9], [76, 10], [78, 14], [80, 8], [82, 9],
[84, 16], [86, 9], [88, 8], [90, 18], [92, 8], [94, 9], [96, 14], [98,
6], [100, 12], [102, 16], [104, 10], [106, 11], [108, 16], [110, 12],
[112, 14], [114, 20], [116, 12], [118, 11], [120, 24], [122, 7], [124,
10], [126, 20], [128, 6], [130, 14], [132, 18], [134, 11], [136, 10],
[138, 16], [140, 14], [142, 15], [144, 22], [146, 11], [148, 10], [150,
24], [152, 8], [154, 16], [156, 22], [158, 9], [160, 16], [162, 20],
[164, 10], [166, 11], [168, 26], [170, 18], [172, 12], [174, 22], [176,
14], [178, 13], [180, 28], [182, 12], [184, 16], [186, 26], [188, 10],
[190, 16], [192, 22], [194, 13], [196, 18], [198, 26], [200, 16], [202,
17], [204, 28], [206, 13], [208, 14], [210, 38], [212, 12], [214, 15],
[216, 26], [218, 13], [220, 18], [222, 22], [224, 14], [226, 13], [228,
24], [230, 18], [232, 14], [234, 30], [236, 18], [238, 18], [240, 36],
[242, 16], [244, 18], [246, 32], [248, 12], [250, 18], [252, 32], [254,
17], [256, 16], [258, 28], [260, 20], [262, 17], [264, 32], [266, 16],
[268, 18], [270, 38], [272, 14], [274, 21], [276, 32], [278, 13], [280,
28], [282, 32], [284, 16], [286, 24], [288, 34], [290, 20], [292, 16],
[294, 38], [296, 16], [298, 21], [300, 42], [302, 17], [304, 20], [306,
30], [308, 16], [310, 24], [312, 34], [314, 17], [316, 20], [318, 30],
[320, 22], [322, 22], [324, 40], [326, 13], [328, 20], [330, 48], [332,
12], [334, 21], [336, 38], [338, 18], [340, 26], [342, 34], [344, 20],
[346, 17], [348, 32], [350, 26], [352, 20], [354, 40], [356, 18], [358,
19], [360, 44], [362, 15], [364, 28], [366, 36], [368, 16], [370, 28],
[372, 36], [374, 20], [376, 22], [378, 44], [380, 26], [382, 19], [384,
38], [386, 23], [388, 18], [390, 54], [392, 22], [394, 21], [396, 42],
[398, 13], [400, 28], [402, 34], [404, 22], [406, 26], [408, 40], [410,
26], [412, 22], [414, 42], [416, 20], [418, 22], [420, 60], [422, 21],
[424, 24], [426, 42], [428, 18], [430, 28], [432, 38], [434, 26], [436,
22], [438, 42], [440, 28], [442, 26], [444, 42], [446, 23], [448, 26],
[450, 54], [452, 24], [454, 23], [456, 48], [458, 17], [460, 32], [462,
56], [464, 24], [466, 25], [468, 48], [470, 30], [472, 26], [474, 46],
[476, 28], [478, 21], [480, 58], [482, 21], [484, 28], [486, 46], [488,
18], [490, 38], [492, 44], [494, 26], [496, 26], [498, 46], [500, 26],
[502, 29], [504, 54], [506, 30], [508, 28], [510, 64], [512, 22], [514,
27], [516, 46], [518, 22], [520, 34], [522, 48], [524, 22], [526, 29],
[528, 50], [530, 28], [532, 34], [534, 44], [536, 26], [538, 27], [540,
60], [542, 19], [544, 26], [546, 60], [548, 22], [550, 38], [552, 46],
[554, 21], [556, 22], [558, 46], [560, 36], [562, 27], [564, 48], [566,
25], [568, 26], [570, 62], [572, 22], [574, 32], [576, 52], [578, 24],
[580, 38], [582, 50], [584, 24], [586, 25], [588, 58], [590, 32], [592,
30], [594, 54], [596, 24], [598, 30], [600, 64], [602, 24], [604, 28],
[606, 54], [608, 26], [610, 40], [612, 52], [614, 29], [616, 38], [618,
52], [620, 36], [622, 33], [624, 62], [626, 23], [628, 32], [630, 82],
[632, 20], [634, 27], [636, 56], [638, 30], [640, 36], [642, 50], [644,
34], [646, 32], [648, 54], [650, 42], [652, 30], [654, 58], [656, 26],
[658, 38], [660, 82], [662, 27], [664, 32], [666, 62], [668, 22], [670,
42], [672, 66], [674, 29], [676, 34], [678, 56], [680, 42], [682, 32],
[684, 60], [686, 32], [688, 32], [690, 78], [692, 22], [694, 37], [696,
60], [698, 27], [700, 48], [702, 62], [704, 36], [706, 37], [708, 48],
[710, 32], [712, 34], [714, 74], [716, 28], [718, 29], [720, 78], [722,
28], [724, 30], [726, 62], [728, 30], [730, 42], [732, 62], [734, 29],
[736, 38], [738, 58], [740, 36], [742, 38], [744, 62], [746, 35], [748,
38], [750, 78], [752, 28], [754, 34], [756, 70], [758, 29], [760, 42],
[762, 60], [764, 34], [766, 33], [768, 62], [770, 52], [772, 36], [774,
64], [776, 32], [778, 29], [780, 88], [782, 28], [784, 36], [786, 60],
[788, 30], [790, 44], [792, 68], [794, 33], [796, 28], [798, 76], [800,
42], [802, 31], [804, 64], [806, 32], [808, 28], [810, 78], [812, 36],
[814, 40], [816, 68], [818, 33], [820, 40], [822, 58], [824, 32], [826,
42], [828, 68], [830, 44], [832, 44], [834, 66], [836, 36], [838, 33],
[840, 102], [842, 35], [844, 34], [846, 64], [848, 30], [850, 50], [852,
62], [854, 40], [856, 38], [858, 78], [860, 36], [862, 33], [864, 66],
[866, 33], [868, 42], [870, 92], [872, 36], [874, 38], [876, 72], [878,
27], [880, 50], [882, 78], [884, 42], [886, 35], [888, 74], [890, 46],
[892, 38], [894, 68], [896, 40], [898, 37], [900, 96], [902, 30], [904,
34], [906, 68], [908, 30], [910, 62], [912, 62], [914, 39], [916, 36],
[918, 70], [920, 46], [922, 39], [924, 94], [926, 35], [928, 36], [930,
86], [932, 34], [934, 39], [936, 72], [938, 36], [940, 48], [942, 68],
[944, 36], [946, 40], [948, 66], [950, 50], [952, 46], [954, 74], [956,
38], [958, 43], [960, 90], [962, 32], [964, 36], [966, 90], [968, 34],
[970, 54], [972, 64], [974, 33], [976, 38], [978, 70], [980, 52], [982,
33], [984, 78], [986, 40], [988, 46], [990, 104], [992, 26], [994, 50],
[996, 74], [998, 33], [1000, 56], [1002, 72], [1004, 36], [1006, 35],
[1008, 84], [1010, 50], [1012, 46], [1014, 78], [1016, 36], [1018, 39],
[1020, 102], [1022, 36], [1024, 44], [1026, 84], [1028, 36], [1030, 50],
[1032, 72], [1034, 42], [1036, 54], [1038, 80], [1040, 52], [1042, 43],
[1044, 78], [1046, 37], [1048, 38], [1050, 114], [1052, 36], [1054, 48],
[1056, 88], [1058, 38], [1060, 54], [1062, 74], [1064, 48], [1066, 48],
[1068, 78], [1070, 50], [1072, 42], [1074, 80], [1076, 40], [1078, 54],
[1080, 108], [1082, 39], [1084, 42], [1086, 78], [1088, 36], [1090, 52],
[1092, 96], [1094, 45], [1096, 36], [1098, 80], [1100, 56], [1102, 48],
[1104, 88], [1106, 50], [1108, 50], [1110, 108], [1112, 32], [1114, 45],
[1116, 82], [1118, 44], [1120, 68], [1122, 94], [1124, 38], [1126, 45],
[1128, 78], [1130, 52], [1132, 44], [1134, 98], [1136, 46], [1138, 39],
[1140, 116], [1142, 35], [1144, 48], [1146, 76], [1148, 52], [1150, 54],
[1152, 72], [1154, 37], [1156, 44], [1158, 84], [1160, 58], [1162, 50],
[1164, 86], [1166, 48], [1168, 44], [1170, 116], [1172, 36], [1174, 43],
[1176, 98], [1178, 38], [1180, 52], [1182, 80], [1184, 40], [1186, 39],
[1188, 86], [1190, 66], [1192, 46], [1194, 90], [1196, 48], [1198, 47],
[1200, 108], [1202, 37], [1204, 56], [1206, 86], [1208, 40], [1210, 64],
[1212, 84], [1214, 43], [1216, 42], [1218, 98], [1220, 54], [1222, 50],
[1224, 90], [1226, 43], [1228, 44], [1230, 110], [1232, 56], [1234, 49],
[1236, 84], [1238, 35], [1240, 68], [1242, 88], [1244, 46], [1246, 52],
[1248, 90], [1250, 56], [1252, 46], [1254, 102], [1256, 40], [1258, 42],
[1260, 136], [1262, 43], [1264, 52], [1266, 84], [1268, 42], [1270, 54],
[1272, 80], [1274, 52], [1276, 50], [1278, 84], [1280, 54], [1282, 51],
[1284, 92], [1286, 43], [1288, 58], [1290, 120], [1292, 46], [1294, 51],
[1296, 98], [1298, 40], [1300, 66], [1302, 106], [1304, 48], [1306, 47],
[1308, 92], [1310, 60], [1312, 46], [1314, 92], [1316, 54], [1318, 51],
[1320, 132], [1322, 41], [1324, 52], [1326, 106], [1328, 44], [1330,
82], [1332, 94], [1334, 50], [1336, 50], [1338, 90], [1340, 54], [1342,
54], [1344, 108], [1346, 49], [1348, 44], [1350, 120], [1352, 44],
[1354, 41], [1356, 82], [1358, 48], [1360, 66], [1362, 88], [1364, 54],
[1366, 53], [1368, 96], [1370, 56], [1372, 54], [1374, 94], [1376, 46],
[1378, 54], [1380, 122], [1382, 39], [1384, 48], [1386, 118], [1388,
40], [1390, 60], [1392, 88], [1394, 48], [1396, 48], [1398, 90], [1400,
68], [1402, 53], [1404, 96], [1406, 46], [1408, 50], [1410, 116], [1412,
36], [1414, 60], [1416, 94], [1418, 39], [1420, 68], [1422, 82], [1424,
44], [1426, 46], [1428, 114], [1430, 70], [1432, 50], [1434, 100],
[1436, 44], [1438, 45], [1440, 120], [1442, 54], [1444, 56], [1446, 90],
[1448, 40], [1450, 72], [1452, 98], [1454, 51], [1456, 68], [1458, 96],
[1460, 66], [1462, 58], [1464, 94], [1466, 49], [1468, 48], [1470, 146],
[1472, 44], [1474, 60], [1476, 102], [1478, 51], [1480, 68], [1482,
104], [1484, 58], [1486, 49], [1488, 104], [1490, 62], [1492, 54],
[1494, 100], [1496, 56], [1498, 64], [1500, 134], [1502, 53], [1504,
54], [1506, 102], [1508, 50], [1510, 66], [1512, 118], [1514, 45],
[1516, 52], [1518, 112], [1520, 62], [1522, 53], [1524, 96], [1526, 56],
[1528, 54], [1530, 138], [1532, 44], [1534, 64], [1536, 94], [1538, 51],
[1540, 92], [1542, 92], [1544, 42], [1546, 59], [1548, 102], [1550, 62],
[1552, 56], [1554, 124], [1556, 48], [1558, 50], [1560, 144], [1562,
58], [1564, 56], [1566, 102], [1568, 50], [1570, 76], [1572, 114],
[1574, 45], [1576, 52], [1578, 94], [1580, 62], [1582, 68], [1584, 116],
[1586, 56], [1588, 52], [1590, 142], [1592, 50], [1594, 53], [1596,
128], [1598, 48], [1600, 72], [1602, 106], [1604, 46], [1606, 60],
[1608, 94], [1610, 84], [1612, 70], [1614, 106], [1616, 54], [1618, 63],
[1620, 130], [1622, 47], [1624, 70], [1626, 110], [1628, 58], [1630,
80], [1632, 120], [1634, 54], [1636, 54], [1638, 134], [1640, 70],
[1642, 53], [1644, 104], [1646, 51], [1648, 56], [1650, 152], [1652,
56], [1654, 61], [1656, 110], [1658, 53], [1660, 78], [1662, 108],
[1664, 56], [1666, 68], [1668, 106], [1670, 74], [1672, 70], [1674,
112], [1676, 56], [1678, 61], [1680, 166], [1682, 48], [1684, 62],
[1686, 112], [1688, 52], [1690, 74], [1692, 114], [1694, 64], [1696,
56], [1698, 104], [1700, 68], [1702, 60], [1704, 110], [1706, 59],
[1708, 64], [1710, 156], [1712, 52], [1714, 53], [1716, 136], [1718,
41], [1720, 76], [1722, 128], [1724, 58], [1726, 61], [1728, 106],
[1730, 72], [1732, 50], [1734, 110], [1736, 70], [1738, 66], [1740,
152], [1742, 50], [1744, 66], [1746, 110], [1748, 50], [1750, 96],
[1752, 104], [1754, 53], [1756, 60], [1758, 110], [1760, 82], [1762,
57], [1764, 138], [1766, 63], [1768, 62], [1770, 146], [1772, 56],
[1774, 53], [1776, 106], [1778, 66], [1780, 74], [1782, 118], [1784,
50], [1786, 60], [1788, 104], [1790, 72], [1792, 72], [1794, 132],
[1796, 62], [1798, 54], [1800, 150], [1802, 62], [1804, 62], [1806,
144], [1808, 56], [1810, 72], [1812, 106], [1814, 55], [1816, 54],
[1818, 106], [1820, 92], [1822, 53], [1824, 116], [1826, 66], [1828,
58], [1830, 152], [1832, 50], [1834, 68], [1836, 124], [1838, 55],
[1840, 74], [1842, 108], [1844, 64], [1846, 56], [1848, 140], [1850,
76], [1852, 56], [1854, 108], [1856, 64], [1858, 57], [1860, 152],
[1862, 68], [1864, 64], [1866, 106], [1868, 56], [1870, 86], [1872,
116], [1874, 59], [1876, 72], [1878, 122], [1880, 76], [1882, 65],
[1884, 122], [1886, 60], [1888, 56], [1890, 182], [1892, 70], [1894,
63], [1896, 126], [1898, 66], [1900, 72], [1902, 110], [1904, 72],
[1906, 59], [1908, 116], [1910, 72], [1912, 60], [1914, 132], [1916,
56], [1918, 70], [1920, 162], [1922, 60], [1924, 68], [1926, 116],
[1928, 60], [1930, 78], [1932, 136], [1934, 55], [1936, 66], [1938,
130], [1940, 82], [1942, 59], [1944, 116], [1946, 68], [1948, 62],
[1950, 166], [1952, 58], [1954, 69], [1956, 106], [1958, 54], [1960,
96], [1962, 112], [1964, 52], [1966, 57], [1968, 124], [1970, 76],
[1972, 64], [1974, 142], [1976, 66], [1978, 64], [1980, 164], [1982,
59], [1984, 64], [1986, 116], [1988, 60], [1990, 84], [1992, 118],
[1994, 53], [1996, 56], [1998, 112], [2000, 74], [2002, 88], [2004,
118], [2006, 70], [2008, 56], [2010, 168], [2012, 54], [2014, 70],
[2016, 146], [2018, 55], [2020, 82], [2022, 118], [2024, 64], [2026,
63], [2028, 124], [2030, 98], [2032, 58], [2034, 128], [2036, 66],
[2038, 65], [2040, 170], [2042, 59], [2044, 76], [2046, 150], [2048,
50], [2050, 84], [2052, 130], [2054, 68], [2056, 64], [2058, 150],
[2060, 90], [2062, 65], [2064, 116], [2066, 67], [2068, 70], [2070,
164], [2072, 72], [2074, 66], [2076, 116], [2078, 53], [2080, 96],
[2082, 118], [2084, 60], [2086, 78], [2088, 124], [2090, 92], [2092,
68], [2094, 124], [2096, 64], [2098, 63], [2100, 194], [2102, 61],
[2104, 70], [2106, 142], [2108, 64], [2110, 82], [2112, 134], [2114,
68], [2116, 74], [2118, 126], [2120, 74], [2122, 65], [2124, 134],
[2126, 61], [2128, 84], [2130, 164], [2132, 66], [2134, 70], [2136,
132], [2138, 57], [2140, 94], [2142, 162], [2144, 64], [2146, 72],
[2148, 126], [2150, 86], [2152, 64], [2154, 128], [2156, 86], [2158,
74], [2160, 158], [2162, 68], [2164, 76], [2166, 134], [2168, 62],
[2170, 104], [2172, 130], [2174, 61], [2176, 70], [2178, 136], [2180,
82], [2182, 67], [2184, 170], [2186, 67], [2188, 62], [2190, 164],
[2192, 66], [2194, 67], [2196, 124], [2198, 68], [2200, 92], [2202,
120], [2204, 64], [2206, 69], [2208, 120], [2210, 94], [2212, 76],
[2214, 136], [2216, 68], [2218, 57], [2220, 180], [2222, 70], [2224,
66], [2226, 156], [2228, 56], [2230, 82], [2232, 134], [2234, 63],
[2236, 72], [2238, 116], [2240, 102], [2242, 70], [2244, 154], [2246,
65], [2248, 66], [2250, 172], [2252, 52], [2254, 88], [2256, 140],
[2258, 63], [2260, 88], [2262, 150], [2264, 62], [2266, 76], [2268,
150], [2270, 82], [2272, 66], [2274, 122], [2276, 64], [2278, 72],
[2280, 184], [2282, 76], [2284, 66], [2286, 136], [2288, 74], [2290,
94], [2292, 136], [2294, 64], [2296, 80], [2298, 126], [2300, 98],
[2302, 63], [2304, 136], [2306, 67], [2308, 68], [2310, 228], [2312,
70], [2314, 80], [2316, 132], [2318, 76], [2320, 96], [2322, 134],
[2324, 76], [2326, 69], [2328, 128], [2330, 86], [2332, 70], [2334,
136], [2336, 70], [2338, 76], [2340, 188], [2342, 67], [2344, 72],
[2346, 152], [2348, 62], [2350, 90], [2352, 166], [2354, 82], [2356,
74], [2358, 124], [2360, 90], [2362, 65], [2364, 132], [2366, 88],
[2368, 74], [2370, 184], [2372, 66], [2374, 75], [2376, 152], [2378,
64], [2380, 120], [2382, 134], [2384, 70], [2386, 75], [2388, 134],
[2390, 88], [2392, 82], [2394, 168], [2396, 74], [2398, 74], [2400,
180], [2402, 73], [2404, 72], [2406, 144], [2408, 78], [2410, 100],
[2412, 138], [2414, 74], [2416, 70], [2418, 156], [2420, 96], [2422,
82], [2424, 144], [2426, 63], [2428, 70], [2430, 188], [2432, 72],
[2434, 75], [2436, 172], [2438, 62], [2440, 100], [2442, 150], [2444,
74], [2446, 79], [2448, 146], [2450, 112], [2452, 72], [2454, 142],
[2456, 60], [2458, 77], [2460, 186], [2462, 69], [2464, 104], [2466,
138], [2468, 64], [2470, 114], [2472, 142], [2474, 77], [2476, 74],
[2478, 174], [2480, 92], [2482, 78], [2484, 152], [2486, 76], [2488,
80], [2490, 188], [2492, 84], [2494, 82], [2496, 150], [2498, 65],
[2500, 94], [2502, 136], [2504, 64], [2506, 92], [2508, 148], [2510,
90], [2512, 68], [2514, 144], [2516, 76], [2518, 73], [2520, 224],
[2522, 70], [2524, 82], [2526, 136], [2528, 68], [2530, 110], [2532,
140], [2534, 82], [2536, 72], [2538, 140], [2540, 86], [2542, 76],
[2544, 154], [2546, 72], [2548, 96], [2550, 198], [2552, 86], [2554,
81], [2556, 142], [2558, 61], [2560, 96], [2562, 164], [2564, 68],
[2566, 73], [2568, 128], [2570, 88], [2572, 78], [2574, 172], [2576,
86], [2578, 71], [2580, 190], [2582, 71], [2584, 86], [2586, 146],
[2588, 66], [2590, 120], [2592, 136], [2594, 69], [2596, 90], [2598,
142], [2600, 98], [2602, 71], [2604, 182], [2606, 69], [2608, 72],
[2610, 196], [2612, 78], [2614, 73], [2616, 144], [2618, 90], [2620,
96], [2622, 154], [2624, 76], [2626, 70], [2628, 138], [2630, 92],
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WARNING: Output truncated!  
full_output.txt



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...

[17506, 49], [17508, 78], [17510, 66], [17512, 40], [17514, 112], [17516, 36], [17518, 30], [17520, 102], [17522, 47], [17524, 40], [17526, 84], [17528, 54], [17530, 58], [17532, 86], [17534, 44], [17536, 38], [17538, 84], [17540, 52], [17542, 44], [17544, 82], [17546, 34], [17548, 40], [17550, 118], [17552, 52], [17554, 30], [17556, 110], [17558, 39], [17560, 58], [17562, 84], [17564, 46], [17566, 37], [17568, 76], [17570, 72], [17572, 42], [17574, 82], [17576, 46], [17578, 40], [17580, 106], [17582, 40], [17584, 46], [17586, 78], [17588, 38], [17590, 46], [17592, 82], [17594, 42], [17596, 42], [17598, 78], [17600, 54], [17602, 36], [17604, 78], [17606, 47], [17608, 40], [17610, 104], [17612, 56], [17614, 45], [17616, 62], [17618, 42], [17620, 38], [17622, 86], [17624, 44], [17626, 40], [17628, 84], [17630, 58], [17632, 42], [17634, 76], [17636, 34], [17638, 33], [17640, 122], [17642, 43], [17644, 42], [17646, 78], [17648, 52], [17650, 46], [17652, 68], [17654, 52], [17656, 40], 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82], [18272, 24], [18274, 25], [18276, 52], [18278, 40], [18280, 36], [18282, 58], [18284, 40], [18286, 36], [18288, 46], [18290, 46], [18292, 32], [18294, 64], [18296, 24], [18298, 28], [18300, 68], [18302, 29], [18304, 32], [18306, 58], [18308, 32], [18310, 28], [18312, 64], [18314, 23], [18316, 24], [18318, 66], [18320, 40], [18322, 25], [18324, 54], [18326, 42], [18328, 24], [18330, 72], [18332, 36], [18334, 24], [18336, 44], [18338, 32], [18340, 34], [18342, 60], [18344, 32], [18346, 25], [18348, 56], [18350, 48], [18352, 28], [18354, 66], [18356, 26], [18358, 22], [18360, 78], [18362, 35], [18364, 24], [18366, 56], [18368, 38], [18370, 40], [18372, 58], [18374, 27], [18376, 28], [18378, 52], [18380, 30], [18382, 30], [18384, 64], [18386, 30], [18388, 26], [18390, 68], [18392, 36], [18394, 18], [18396, 62], [18398, 31], [18400, 30], [18402, 60], [18404, 28], [18406, 17], [18408, 58], [18410, 46], [18412, 26], [18414, 62], [18416, 24], [18418, 23], [18420, 58], [18422, 24], [18424, 22], [18426, 52], [18428, 30], [18430, 42], [18432, 58], [18434, 38], [18436, 24], [18438, 56], [18440, 42], [18442, 29], [18444, 56], [18446, 28], [18448, 26], [18450, 64], [18452, 32], [18454, 21], [18456, 42], [18458, 30], [18460, 36], [18462, 64], [18464, 28], [18466, 30], [18468, 62], [18470, 36], [18472, 26], [18474, 56], [18476, 28], [18478, 19], [18480, 102], [18482, 31], [18484, 32], [18486, 50], [18488, 28], [18490, 24], [18492, 48], [18494, 28], [18496, 32], [18498, 50], [18500, 40], [18502, 30], [18504, 52], [18506, 34], [18508, 20], [18510, 68], [18512, 26], [18514, 25], [18516, 42], [18518, 30], [18520, 30], [18522, 62], [18524, 34], [18526, 20], [18528, 58], [18530, 36], [18532, 28], [18534, 36], [18536, 28], [18538, 30], [18540, 56], [18542, 30], [18544, 16], [18546, 50], [18548, 38], [18550, 42], [18552, 56], [18554, 29], [18556, 22], [18558, 54], [18560, 32], [18562, 15], [18564, 74], [18566, 19], [18568, 16], [18570, 76], [18572, 34], [18574, 20], [18576, 52], 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[18732, 48], [18734, 24], [18736, 18], [18738, 44], [18740, 30], [18742, 19], [18744, 38], [18746, 32], [18748, 26], [18750, 48], [18752, 32], [18754, 25], [18756, 40], [18758, 18], [18760, 32], [18762, 44], [18764, 16], [18766, 14], [18768, 42], [18770, 30], [18772, 22], [18774, 46], [18776, 18], [18778, 22], [18780, 58], [18782, 27], [18784, 18], [18786, 34], [18788, 36], [18790, 32], [18792, 36], [18794, 25], [18796, 16], [18798, 32], [18800, 28], [18802, 26], [18804, 40], [18806, 17], [18808, 22], [18810, 54], [18812, 26], [18814, 18], [18816, 50], [18818, 16], [18820, 14], [18822, 42], [18824, 26], [18826, 13], [18828, 42], [18830, 38], [18832, 22], [18834, 46], [18836, 22], [18838, 15], [18840, 46], [18842, 21], [18844, 20], [18846, 28], [18848, 16], [18850, 26], [18852, 44], [18854, 22], [18856, 16], [18858, 50], [18860, 28], [18862, 21], [18864, 32], [18866, 17], [18868, 14], [18870, 64], [18872, 24], [18874, 15], [18876, 36], [18878, 23], [18880, 22], [18882, 42], [18884, 20], [18886, 16], [18888, 32], [18890, 20], [18892, 24], [18894, 38], [18896, 24], [18898, 22], [18900, 58], [18902, 22], [18904, 18], [18906, 36], [18908, 18], [18910, 24], [18912, 30], [18914, 20], [18916, 18], [18918, 34], [18920, 28], [18922, 13], [18924, 44], [18926, 15], [18928, 22], [18930, 46], [18932, 16], [18934, 13], [18936, 38], [18938, 26], [18940, 18], [18942, 54], [18944, 20], [18946, 15], [18948, 30], [18950, 24], [18952, 22], [18954, 36], [18956, 14], [18958, 19], [18960, 42], [18962, 22], [18964, 20], [18966, 30], [18968, 24], [18970, 28], [18972, 42], [18974, 24], [18976, 14], [18978, 34], [18980, 24], [18982, 13], [18984, 34], [18986, 16], [18988, 16], [18990, 50], [18992, 16], [18994, 15], [18996, 24], [18998, 22], [19000, 26], [19002, 24], [19004, 14], [19006, 16], [19008, 32], [19010, 26], [19012, 20], [19014, 30], [19016, 28], [19018, 14], [19020, 42], [19022, 19], [19024, 16], [19026, 34], [19028, 12], [19030, 18], [19032, 42], [19034, 20], [19036, 8], [19038, 34], [19040, 24], [19042, 17], [19044, 34], [19046, 12], [19048, 16], [19050, 30], [19052, 22], [19054, 14], [19056, 24], [19058, 28], [19060, 30], [19062, 32], [19064, 22], [19066, 15], [19068, 36], [19070, 20], [19072, 12], [19074, 32], [19076, 16], [19078, 13], [19080, 40], [19082, 24], [19084, 16], [19086, 32], [19088, 16], [19090, 22], [19092, 38], [19094, 15], [19096, 18], [19098, 32], [19100, 20], [19102, 13], [19104, 24], [19106, 14], [19108, 14], [19110, 54], [19112, 16], [19114, 18], [19116, 28], [19118, 14], [19120, 14], [19122, 34], [19124, 18], [19126, 16], [19128, 32], [19130, 20], [19132, 12], [19134, 32], [19136, 14], [19138, 12], [19140, 48], [19142, 12], [19144, 18], [19146, 20], [19148, 14], [19150, 18], [19152, 42], [19154, 16], [19156, 14], [19158, 30], [19160, 20], [19162, 20], [19164, 26], [19166, 14], [19168, 18], [19170, 38], [19172, 16], [19174, 13], [19176, 26], [19178, 14], [19180, 24], [19182, 36], [19184, 10], [19186, 10], [19188, 34], [19190, 14], [19192, 4], [19194, 32], [19196, 6], [19198, 12], [19200, 40], [19202, 13], [19204, 10], [19206, 32], [19208, 20], [19210, 22], [19212, 26], [19214, 14], [19216, 16], [19218, 18], [19220, 16], [19222, 16], [19224, 28], [19226, 13], [19228, 16], [19230, 38], [19232, 16], [19234, 12], [19236, 24], [19238, 5], [19240, 14], [19242, 30], [19244, 18], [19246, 9], [19248, 30], [19250, 26], [19252, 14], [19254, 28], [19256, 10], [19258, 11], [19260, 30], [19262, 13], [19264, 20], [19266, 24], [19268, 12], [19270, 18], [19272, 32], [19274, 12], [19276, 10], [19278, 34], [19280, 18], [19282, 12], [19284, 24], [19286, 9], [19288, 10], [19290, 34], [19292, 20], [19294, 10], [19296, 18], [19298, 15], [19300, 18], [19302, 22], [19304, 12], [19306, 14], [19308, 26], [19310, 14], [19312, 12], [19314, 18], [19316, 10], [19318, 14], [19320, 36], [19322, 17], [19324, 12], [19326, 22], [19328, 12], [19330, 12], [19332, 18], [19334, 12], [19336, 10], [19338, 24], [19340, 16], [19342, 12], [19344, 24], [19346, 10], [19348, 10], [19350, 32], [19352, 10], [19354, 15], [19356, 16], [19358, 9], [19360, 10], [19362, 32], [19364, 14], [19366, 10], [19368, 30], [19370, 18], [19372, 12], [19374, 14], [19376, 8], [19378, 13], [19380, 24], [19382, 14], [19384, 8], [19386, 18], [19388, 10], [19390, 14], [19392, 20], [19394, 11], [19396, 10], [19398, 26], [19400, 12], [19402, 8], [19404, 26], [19406, 8], [19408, 10], [19410, 24], [19412, 14], [19414, 6], [19416, 16], [19418, 14], [19420, 14], [19422, 18], [19424, 4], [19426, 10], [19428, 14], [19430, 18], [19432, 6], [19434, 20], [19436, 8], [19438, 9], [19440, 24], [19442, 9], [19444, 6], [19446, 22], [19448, 10], [19450, 2], [19452, 22], [19454, 6], [19456, 6], [19458, 18], [19460, 14], [19462, 12], [19464, 18], [19466, 7], [19468, 8], [19470, 22], [19472, 10], [19474, 8], [19476, 10], [19478, 13], [19480, 12], [19482, 18], [19484, 8], [19486, 5], [19488, 26], [19490, 10], [19492, 6], [19494, 12], [19496, 6], [19498, 1], [19500, 22], [19502, 12], [19504, 0], [19506, 18], [19508, 12], [19510, 10], [19512, 12], [19514, 12], [19516, 8], [19518, 14], [19520, 14], [19522, 4], [19524, 12], [19526, 8], [19528, 6], [19530, 20], [19532, 10], [19534, 5], [19536, 22], [19538, 7], [19540, 4], [19542, 8], [19544, 8], [19546, 6], [19548, 14], [19550, 18], [19552, 8], [19554, 16], [19556, 8], [19558, 6], [19560, 20], [19562, 11], [19564, 4], [19566, 8], [19568, 12], [19570, 6], [19572, 22], [19574, 5], [19576, 6], [19578, 18], [19580, 14], [19582, 5], [19584, 14], [19586, 8], [19588, 2], [19590, 16], [19592, 12], [19594, 4], [19596, 10], [19598, 14], [19600, 8], [19602, 14], [19604, 10], [19606, 9], [19608, 12], [19610, 10], [19612, 2], [19614, 6], [19616, 8], [19618, 6], [19620, 22], [19622, 7], [19624, 4], [19626, 16], [19628, 12], [19630, 6], [19632, 6], [19634, 5], [19636, 4], [19638, 10], [19640, 10], [19642, 6], [19644, 10], [19646, 10], [19648, 4], [19650, 18], [19652, 8], [19654, 4], [19656, 10], [19658, 3], [19660, 4], [19662, 16], [19664, 6], [19666, 3], [19668, 14], [19670, 12], [19672, 6], [19674, 14], [19676, 4], [19678, 5], [19680, 6], [19682, 6], [19684, 4], [19686, 6], [19688, 12], [19690, 10], [19692, 10], [19694, 6], [19696, 4], [19698, 12], [19700, 8], [19702, 1], [19704, 8], [19706, 4], [19708, 4], [19710, 14], [19712, 8], [19714, 3], [19716, 14], [19718, 9], [19720, 4], [19722, 10], [19724, 2], [19726, 4], [19728, 4], [19730, 6], [19732, 4], [19734, 10], [19736, 6], [19738, 2], [19740, 14], [19742, 7], [19744, 4], [19746, 6], [19748, 4], [19750, 0], [19752, 8], [19754, 6], [19756, 2], [19758, 12], [19760, 8], [19762, 4], [19764, 6], [19766, 5], [19768, 2], [19770, 8], [19772, 2], [19774, 5], [19776, 2], [19778, 6], [19780, 6], [19782, 6], [19784, 6], [19786, 2], [19788, 8], [19790, 6], [19792, 2], [19794, 4], [19796, 2], [19798, 2], [19800, 8], [19802, 5], [19804, 0], [19806, 8], [19808, 4], [19810, 2], [19812, 6], [19814, 3], [19816, 2], [19818, 4], [19820, 2], [19822, 0], [19824, 8], [19826, 2], [19828, 2], [19830, 6], [19832, 6], [19834, 0], [19836, 4], [19838, 4], [19840, 0], [19842, 2], [19844, 2], [19846, 1], [19848, 2], [19850, 4], [19852, 2], [19854, 4], [19856, 4], [19858, 1], [19860, 4], [19862, 1], [19864, 2], [19866, 0], [19868, 2], [19870, 2], [19872, 4], [19874, 4], [19876, 0], [19878, 2], [19880, 4], [19882, 1], [19884, 0], [19886, 0], [19888, 0], [19890, 4], [19892, 0], [19894, 0], [19896, 4], [19898, 3], [19900, 0], [19902, 2], [19904, 2], [19906, 0], [19908, 2], [19910, 0], [19912, 0], [19914, 2], [19916, 2], [19918, 0], [19920, 0], [19922, 2], [19924, 0], [19926, 0], [19928, 0], [19930, 0], [19932, 0], [19934, 1], [19936, 0], [19938, 0], [19940, 2], [19942, 0], [19944, 0], [19946, 1], [19948, 0], [19950, 0], [19952, 0], [19954, 0], [19956, 0], [19958, 0], [19960, 0], [19962, 0], [19964, 0], [19966, 0], [19968, 0], [19970, 0], [19972, 0], [19974, 0], [19976, 0], [19978, 0], [19980, 0], [19982, 0], [19984, 0], [19986, 0], [19988, 0], [19990, 0], [19992, 0], [19994, 0], [19996, 0], [19998, 0]]
better_goldbach_list0=[] better_goldbach_list1=[] better_goldbach_list2=[] for i in srange(3,N): if 2*i %3 ==0: better_goldbach_list0.append([2*i,goldbach_count[2*i]]) elif 2*i % 3 ==1: better_goldbach_list1.append([2*i,goldbach_count[2*i]]) elif 2*i %3 == 2: better_goldbach_list2.append([2*i,goldbach_count[2*i]]) 
       
A=list_plot(better_goldbach_list0, color='red') B=list_plot(better_goldbach_list1, color='blue') C=list_plot(better_goldbach_list2, color='green') 
       
show(C+B+A) 
       
better_goldbach_list3and5=[] better_goldbach_list3not5=[] better_goldbach_list5not3=[] better_goldbach_listnot3not5=[] for i in srange(3,N): if 2*i %15 ==0: better_goldbach_list3and5.append([2*i,goldbach_count[2*i]]) elif 2*i % 3 ==0 and 2*i%5 !=0 : better_goldbach_list3not5.append([2*i,goldbach_count[2*i]]) elif 2*i %3 != 0 and 2*i%5 ==0 : better_goldbach_list5not3.append([2*i,goldbach_count[2*i]]) else: better_goldbach_listnot3not5.append([2*i,goldbach_count[2*i]]) 
       
A=list_plot(better_goldbach_list3and5, color='red') B=list_plot(better_goldbach_list3not5, color='blue') C=list_plot(better_goldbach_list5not3, color='green') D=list_plot(better_goldbach_listnot3not5, color='black') show(A+B+C+D) 
       

Aside: overloading of operators: the + operator depends on the type of objects being added.  Provided the 

var('a,b,c,d') list1=[1,2,3] list2=[a,b,c,d] print(list1+list2) 
       
[1, 2, 3, a, b, c, d]
[1, 2, 3, a, b, c, d]
print(list2+list1) 
       
[a, b, c, d, 1, 2, 3]
[a, b, c, d, 1, 2, 3]
print(list1 + 5) 
       
Traceback (click to the left of this block for traceback)
...
TypeError: unsupported operand parent(s) for '+': '<type 'list'>'
and 'Integer Ring'
Traceback (most recent call last):
  File "<stdin>", line 1, in <module>
  File "_sage_input_47.py", line 10, in <module>
    exec compile(u'open("___code___.py","w").write("# -*- coding: utf-8 -*-\\n" + _support_.preparse_worksheet_cell(base64.b64decode("cHJpbnQobGlzdDEgKyA1KQ=="),globals())+"\\n"); execfile(os.path.abspath("___code___.py"))
  File "", line 1, in <module>
    
  File "/tmp/tmppmAIWK/___code___.py", line 3, in <module>
    exec compile(u'print(list1 + _sage_const_5 )
  File "", line 1, in <module>
    
  File "sage/structure/element.pyx", line 1651, in sage.structure.element.RingElement.__add__ (/usr/local/sage-6.10/src/build/cythonized/sage/structure/element.c:15853)
  File "sage/structure/coerce.pyx", line 1069, in sage.structure.coerce.CoercionModel_cache_maps.bin_op (/usr/local/sage-6.10/src/build/cythonized/sage/structure/coerce.c:9737)
TypeError: unsupported operand parent(s) for '+': '<type 'list'>' and 'Integer Ring'

We have seen heuristically that there ought to be a "Hardy-Littlewood correction factor" hl(n) which tells us how many extra times we ought to see a number appear as a sum, given its prime factors

$hl(n):= \prod_{p|n} \frac{p-1}{p-2}$

We'll discover next time how to code $hl(n)$.

def HL(n): if n <2: return(1) cf=1 a=list(factor(2*n))[1:] for i in a: p=i[0] cf*=(p-1)/(p-2) return(cf) 
       
factor(24) 
       
2^3 * 3
2^3 * 3
type(factor(24)) 
       
<class
'sage.structure.factorization_integer.IntegerFactorization'>
<class 'sage.structure.factorization_integer.IntegerFactorization'>
factor(24)[0] 
       
(2, 3)
(2, 3)
factor(24)[1] 
       
(3, 1)
(3, 1)
list(factor(210))[1:] 
       
[(3, 1), (5, 1), (7, 1)]
[(3, 1), (5, 1), (7, 1)]
HL(240) 
       
8/3
8/3

We want to take the better_goldbach_list, and divide the number of representations of $n$ as a sum of two primes by the HL correction factor.

HL_corrected_Goldbach_list=[] for i in better_goldbach_list: a=[i[0], i[1]/HL(i[0])] HL_corrected_Goldbach_list.append(a) 
       
list_plot(HL_corrected_Goldbach_list) 
       

What sort of questions could we ask about the "corrected" number of representations?

We could attempt to find a curve of best fit for these points, and then test if it still fits for points further out.

For lines or curves of best fit, we want the average distance of points from the curve to be zero, and we want the sum of the squares of the differences between the points and the curves to be minimized.


How big should the curve be?  If we think of primes as behaving a bit like random numbers, can we predict how many representations a typical number should have?

Big theorem in number theory: conjectured by Gauss, Legendre, Dirichlet, around 1792, and proved in 1896, independently by Hadamard and de la Valleee Poussin:  The Prime Number Theorem.

Define $Li(x) = \int_2^{x} \frac{1}{\log(x)} dx$.  Then $\pi(x) \sim \frac{x}{Li(x)}$.   Here, "asymptotic" means 

\[ \frac{Li(x)\pi(x)}{x} \rightarrow 1 \mbox{ as } x \rightarrow \infty \]

list_prime_pi = [ [i, i/prime_pi(i)] for i in srange(2,100000)] 
       
A=list_plot(list_prime_pi) 
       
B=plot(x/Li(x),x,3,100000,color='red') 
       
show(A+B) 
       
Li(2) 
       
0
0
Li(10) 
       
log_integral(10) - log_integral(2)
log_integral(10) - log_integral(2)

We "expect" that the number of Goldbach representations of an even integer should be around $\frac{n}{\log(n)^2}$

plot(x/(log(x)^2),x,2,20000000000000) 
       
log(20000000000000.) 
       
30.6267533894825
30.6267533894825

What is the Riemann Hypothesis? 

Easy answer: we know (from the Prime number theorem) that $\pi(x) \sim \frac{x}{Li(x)}$.

One form of the Riemann Hypothesis says that $|\pi(x) -\frac{x}{Li(x)}|$ grows no faster than about $x^{1/2}$.  More precisely, for every $\epsilon>0$, that 

\[ \frac{|\pi(x)-\frac{x}{Li(x)}|}{x^{1/2 + \epsilon}} \rightarrow 0\]

as $x \rightarrow \infty$.

The exact statement of the RH involves functions of a complex variable, and where the roots of the Riemann Zeta function lie.

list_prime_pi_minus_li = [ [i, prime_pi(i)-i/Li(i).n()] for i in srange(3,10000)] 
       
C=list_plot(list_prime_pi_minus_li) 
       
D=plot(sqrt(x)*log(x)*sqrt(log(log(x))),x,3,10000,color='red') 
       
show(C+D)