Primes with folds
We will write in a particular way the starting
numbers we use here – one digit per square and one empty square between two
digits. Examples for 2024 and 2030:
+---+---+---+---+---+---+---+
| 2 | | 0 |
| 2 | | 4 |
+---+---+---+---+---+---+---+
+---+---+---+---+---+---+---+
| 2
| | 0
| | 3
| | 0
|
+---+---+---+---+---+---+---+
Let us imagine that the number and its
grid are drawn on a transparent sheet of paper. The fold we are going to use
will be made up of one of the vertical lines of the grid.
Example for 2030 – two possible folds
are in yellow:
+---+---+---+---+---+---+---+
| 2
| | 0
| | 3
| | 0
| fold #1
+---+---+---+---+---+---+---+
+---+---+---+---+---+---+---+
| 2
| | 0
| | 3 | | 0
| fold #2
+---+---+---+---+---+---+---+
If we fold the right side of the
fold over the left side, we respectively form 2030 and 2003 (the digits are not
reversed as in a mirror: they keep their traditional appearance):
+---+---+---+---+---+---+---+
| 2
| | 0
| | 3
| | 0
|
+---+---+---+---+---+---+---+
+---+---+---+---+---+---+ right fold #1
| 2
| | 0
| | 3
| 0 |
+---+---+---+---+---+---+
+---+---+---+---+---+---+---+
| 2
| | 0
| | 3 | | 0
|
+---+---+---+---+---+---+---+
+---+---+---+---+ right fold #2
| 2
| 0 | 0
| 3 |
+---+---+---+---+
If we fold the left side of the
fold over the right side, we respectively form 0302 and 3002:
+---+---+---+---+---+---+---+
| 2
| | 0
| | 3
| | 0 |
+---+---+---+---+---+---+---+ left fold #1
+---+---+---+---+---+---+
| 0 | 3
| | 0
| | 2
|
+---+---+---+---+---+---+
+---+---+---+---+---+---+---+
| 2
| | 0
| | 3 | | 0
|
+---+---+---+---+---+---+---+ left fold #2
+---+---+---+---+
| 3 |
0 | 0
| 2 |
+---+---+---+---+
We are interested in integers (like
2030) that produce at least one prime
number (like 2003) when properly “folded”.
The sequence S of such integers starts
with the first 26 terms of https://oeis.org/A074895
(Primes written backwards and sorted) and then diverges – but how?
S = 2, 3, 5, 7, 11, 13, 14, 16, 17, 31,
32, 34, 35, 37, 38, 71, 73, 74, 76, 79, 91, 92, 95, 97, 98, 101, ...
____________________
September 8th 2024 update
As usual Jean-Marc Falcoz was quick to comment and correct (French original version, then Google yellow translation):
JMF
> dans ma liste (et pas dans la tienne) j’ai 19, car quand on écrit 1.9 , et qu’on plie juste après le 1, on obtient .19, soit 19 qui est premier. J’obtiens : {2,3,5,7,11,13,14,16,17,19,23,29,31,32,34,35,37,38,41,43,47,53,59,61,67,71,73,74,76,79,83,89,91,92,95,97,98,101,103,104,106,107,109,110,112,113,115,118,119,121,124,125,127,128,130,131,133,...}
Avec ma compréhension de l’algorithme, beaucoup d’entiers marchent (env. 55% sur les 200000 premiers entiers). Qu’en penses-tu ?
JMF
> in my list (and not in yours) I have 19, because when we write 1.9, and we fold just after the 1, we get .19, or 19 which is prime. I get: {2,3,5,7,11,13,14,16,17,19,23,29,31,32,34,35,37,38,41,43,47,53,59,61,67,71,73,74,76,79,83,89,91,92,95,97,98,101,103,104,106,107,109,110,112,113,115,118,119,121,124,125,127,128,130,131,133,...}
With my understanding of the algorithm, many integers work (about 55% on the first 200000 integers). What do you think?
ÉA
I think you’re right, Jean-Marc!
(see hereunder Jean-Marc’s proposal for the primes, the squares and the "folded" palindromes)
PRIMES
(pour les premiers, 3,7% jusqu’à 100, 4,3% jusqu’à 1000, 4,7% jusqu’à 10000, 5% jusqu’à 100 000)
(for the primes, 3.7% up to 100, 4.3% up to 1,000, 4.7% up to 10,000, 5% up to 100,000)
premiers jusqu’à 10000 :
primes up to 10,000:
{2,3,5,7,11,13,14,16,17,19,23,29,31,32,34,35,37,38,41,43,47,53,59,61,67,71,73,74,76,79,83,89,91,92,95,97,98,103,104,106,107,109,110,112,113,115,118,119,121,124,125,127,128,130,131,133,134,136,137,139,140,142,143,145,146,149,151,152,154,157,160,163,164,166,167,169,170,172,173,175,176,179,181,182,188,190,191,193,194,196,197,199,211,214,215,217,218,223,227,229,232,233,235,236,238,239,241,251,253,257,263,269,271,272,275,277,278,281,283,287,289,292,293,296,298,299,301,305,307,310,311,313,314,316,317,319,322,323,325,326,328,329,331,332,334,335,337,338,341,343,344,346,347,349,350,352,353,356,358,359,361,362,364,365,367,368,370,371,373,374,376,377,379,382,383,385,386,388,389,391,392,394,395,397,398,401,409,410,412,413,415,416,419,421,431,433,434,436,437,439,443,449,451,457,461,463,467,473,475,476,478,479,487,490,491,493,494,497,499,503,509,511,512,514,517,521,523,527,530,532,533,536,538,539,541,547,557,563,569,571,572,574,575,577,578,583,587,589,590,593,596,598,599,601,607,610,613,614,616,617,619,623,629,631,632,634,635,637,638,641,643,647,653,659,661,670,671,673,674,677,679,683,691,692,695,697,701,703,706,709,710,712,713,715,716,719,721,722,725,727,728,730,731,733,734,736,737,739,743,745,746,748,749,751,752,754,755,757,758,760,761,763,764,767,769,772,773,775,776,778,779,782,784,785,787,788,790,791,793,794,796,797,799,809,811,812,818,821,823,827,829,832,833,835,836,838,839,847,853,857,859,863,872,874,875,877,878,881,883,887,890,892,893,895,901,904,905,907,908,910,911,913,914,916,917,919,922,923,926,928,929,931,932,934,935,937,938,940,941,943,944,947,949,950,953,956,958,959,961,962,965,967,970,971,973,974,976,977,979,980,982,983,985,991,992,994,995,997,1003,1004,1007,1009,1012,1013,1015,1016,1018,1019,1021,1024,1028,1030,1031,1033,1034,1036,1037,1039,1040,1042,1045,1048,1049,1051,1052,1055,1057,1058,1061,1063,1067,1069,1070,1073,1075,1076,1078,1079,1081,1082,1084,1085,1087,1088,1090,1091,1093,1094,1096,1097,1099,1102,1103,1105,1106,1108,1109,1112,1114,1115,1117,1118,1120,1121,1123,1124,1126,1127,1129,1130,1132,1135,1136,1138,1139,1141,1142,1147,1150,1151,1153,1156,1157,1159,1160,1162,1163,1165,1168,1169,1171,1172,1174,1175,1178,1180,1181,1183,1186,1187,1189,1190,1192,1193,1195,1196,1198,1201,1204,1205,1207,1210,1211,1213,1216,1217,1219,1222,1223,1225,1226,1228,1229,1231,1232,1234,1235,1237,1238,1240,1241,1243,1244,1246,1247,1249,1250,1252,1253,1255,1256,1258,1259,1261,1262,1264,1265,1267,1271,1273,1274,1276,1277,1279,1282,1283,1285,1288,1289,1291,1292,1294,1295,1297,1298,1301,1303,1304,1307,1309,1310,1312,1315,1316,1318,1319,1321,1322,1324,1325,1327,1328,1330,1333,1334,1336,1337,1339,1342,1343,1345,1346,1348,1349,1351,1352,1354,1355,1357,1358,1360,1361,1363,1366,1367,1369,1370,1372,1373,1375,1376,1378,1379,1381,1382,1384,1385,1387,1388,1390,1391,1393,1394,1396,1397,1399,1403,1409,1411,1412,1415,1417,1418,1421,1423,1424,1427,1429,1430,1432,1433,1435,1436,1438,1439,1442,1444,1445,1447,1448,1451,1453,1454,1456,1457,1459,1462,1465,1468,1469,1471,1472,1474,1475,1477,1478,1481,1483,1486,1487,1489,1490,1492,1493,1495,1496,1498,1499,1501,1504,1505,1507,1510,1511,1513,1514,1516,1517,1519,1522,1523,1525,1528,1529,1531,1532,1534,1535,1537,1538,1540,1541,1543,1544,1546,1547,1549,1550,1552,1553,1556,1558,1559,1561,1562,1564,1565,1567,1568,1571,1573,1574,1576,1577,1579,1580,1582,1583,1585,1588,1589,1591,1592,1594,1595,1597,1598,1601,1603,1607,1609,1610,1612,1613,1618,1619,1621,1622,1624,1625,1627,1628,1630,1631,1633,1634,1636,1637,1639,1642,1643,1645,1648,1649,1651,1652,1654,1655,1657,1658,1663,1666,1667,1669,1670,1672,1673,1675,1676,1678,1679,1681,1682,1684,1685,1687,1688,1690,1691,1693,1694,1696,1697,1699,1702,1703,1705,1708,1709,1711,1712,1714,1715,1718,1721,1723,1724,1726,1727,1729,1730,1732,1733,1735,1736,1738,1739,1741,1742,1745,1747,1748,1750,1751,1753,1754,1756,1757,1759,1760,1762,1763,1765,1766,1768,1769,1772,1774,1775,1777,1778,1781,1783,1784,1786,1787,1789,1790,1792,1793,1795,1796,1798,1799,1801,1802,1805,1807,1808,1810,1811,1813,1814,1816,1817,1819,1820,1822,1823,1825,1826,1829,1831,1832,1834,1835,1837,1838,1840,1841,1843,1844,1846,1847,1849,1850,1852,1853,1855,1856,1858,1859,1861,1864,1865,1867,1868,1870,1871,1873,1874,1876,1877,1879,1880,1882,1883,1885,1886,1889,1891,1892,1894,1895,1897,1898,1901,1903,1904,1906,1907,1909,1910,1912,1913,1915,1918,1921,1922,1924,1925,1927,1928,1930,1931,1933,1934,1936,1937,1939,1940,1943,1945,1946,1948,1949,1951,1952,1954,1955,1957,1958,1960,1961,1963,1964,1966,1967,1969,1970,1972,1973,1975,1976,1978,1979,1981,1982,1984,1985,1987,1988,1990,1993,1994,1996,1997,1999,2003,2011,2017,2018,2027,2029,2030,2032,2035,2036,2038,2039,2053,2063,2069,2071,2072,2077,2078,2081,2083,2087,2089,2092,2093,2096,2098,2099,2101,2104,2105,2110,2111,2113,2114,2116,2117,2119,2122,2123,2126,2128,2129,2131,2132,2134,2135,2137,2138,2141,2143,2144,2146,2147,2149,2152,2153,2155,2156,2158,2159,2161,2162,2164,2165,2167,2170,2171,2173,2174,2176,2177,2179,2182,2183,2185,2188,2189,2191,2192,2197,2198,2203,2207,2209,221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SQUARES
(pour les carrés, 16% jusqu’à 100, 9,3% jusqu’à 1000, 6% jusqu’à 10000, 2,9% jusqu’à 100000). La densité diminue !
(for squares, 16% up to 100, 9.3% up to 1,000, 6% up to 10,000, 2.9% up to 100,000). The density decreases!
carrés jusqu’à 10000 :
squares up to 10,000:
{1,4,9,16,18,25,36,46,49,52,61,63,64,81,94,100,112,121,136,144,148,163,169,184,196,211,225,234,243,252,256,259,265,279,289,295,297,298,316,324,342,361,400,414,418,423,432,441,448,478,481,484,487,522,526,529,562,567,576,592,613,619,625,631,652,657,667,675,676,691,729,748,756,765,766,784,792,814,829,841,844,847,874,892,900,916,925,927,928,952,961,972,982,1006,1008,1024,1026,1042,1054,1062,1089,1098,1156,1165,1215,1216,1225,1237,1240,1251,1252,1264,1269,1273,1279,1288,1296,1297,1336,1369,1396,1405,1420,1426,1438,1444,1450,1462,1485,1498,1512,1521,1561,1566,1584,1600,1612,1618,1620,1629,1633,1647,1651,1656,1665,1674,1681,1692,1693,1732,1746,1764,1800,1816,1828,1834,1843,1849,1861,1882,1890,1894,1926,1936,1962,1963,1972,2005,2014,2016,2025,2043,2047,2052,2092,2098,2104,2116,2125,2146,2151,2161,2169,2173,2188,2196,2197,2205,2209,2250,2290,2304,2305,2340,2349,2371,2401,2410,2425,2464,2476,2500,2502,2503,2509,2521,2524,2527,2539,2565,2599,2601,2610,2619,2641,2674,2691,2704,2725,2740,2791,2809,2881,2890,2902,2905,2916,2935,2943,2961,2995,3006,3025,3052,3136,3163,3169,3196,3204,3217,3249,3250,3294,3316,3346,3361,3364,3402,3418,3448,3456,3463,3481,3492,3529,3600,3613,3643,3654,3688,3691,3697,3699,3712,3721,3796,3814,3844,3886,3925,3961,3969,3996,4006,4009,4012,4023,4027,4032,4068,4069,4072,4096,4102,4105,4126,4144,4158,4183,4189,4201,4210,4225,4246,4252,4264,4267,4276,4320,4329,4336,4356,4365,4381,4384,4414,4438,4441,4468,4477,4483,4486,4489,4498,4501,4563,4576,4600,4617,4621,4624,4633,4642,4648,4671,4675,4684,4690,4716,4720,4761,4762,4774,4806,4815,4834,4851,4860,4864,4869,4894,4900,4909,4923,4960,4968,4981,4984,5002,5014,5022,5032,5041,5092,5116,5121,5148,5166,5184,5200,5202,5203,5209,5212,5220,5221,5224,5227,5230,5242,5256,5265,5272,5290,5293,5299,5329,5392,5410,5436,5467,5476,5607,5611,5625,5634,5652,5661,5706,5764,5767,5769,5776,5841,5929,5967,5992,6001,6003,6004,6048,6049,6075,6084,6100,6102,6112,6115,6121,6129,6133,6139,6156,6174,6181,6192,6201,6214,6219,6241,6247,6291,6300,6313,6319,6331,6334,6345,6379,6388,6391,6399,6400,6409,6412,6424,6433,6448,6457,6471,6507,6511,6516,6525,6534,6543,6561,6570,6615,6651,6724,6742,6745,6754,6757,6759,6775,6840,6844,6849,6883,6889,6898,6904,6912,6913,6921,6931,6937,6940,6948,6957,6973,6993,7056,7065,7123,7164,7204,7219,7225,7252,7321,7369,7396,7402,7426,7447,7461,7546,7560,7569,7576,7596,7624,7645,7675,7695,7744,7912,7921,7963,8001,8100,8109,8128,8143,8149,8161,8209,8218,8281,8341,8344,8368,8406,8415,8443,8446,8449,8464,8496,8514,8604,8638,8644,8649,8689,8694,8812,8821,8836,8863,8901,8902,8941,8944,8986,9004,9018,9022,9025,9028,9046,9049,9052,9082,9094,9126,9127,9136,9162,9216,9220,9234,9235,9250,9253,9259,9261,9295,9316,9352,9369,9400,9406,9409,9418,9423,9432,9448,9468,9481,9486,9490,9529,9576,9604,9612,9613,9621,9631,9637,9639,9640,9657,9675,9684,9693,9721,9736,9801,9810,9814,9820,9844,9868,9886,9925,9936,9952,9963,10000}
PALINDROMES
(pour les palindromes, 1,8% jusqu’à 100, 2,6% jusqu’à 1000, 4,3% jusqu’à 10000, 5,4% jusqu’à 100 000)
(for palindromes, 1.8% up to 100, 2.6% up to 1,000, 4.3% up to 10,000, 5.4% up to 100,000)
palindromes jusqu’à 100 000 :
palindromes up to 100,000:
{1,2,3,4,5,6,7,8,9,11,22,33,44,55,66,77,88,99,110,111,112,113,114,115,116,117,118,119,121,122,131,133,141,144,151,155,161,166,171,177,181,188,191,199,202,211,212,220,221,222,223,224,225,226,227,228,229,232,233,242,244,252,255,262,266,272,277,282,288,292,299,303,311,313,322,323,330,331,332,333,334,335,336,337,338,339,343,344,353,355,363,366,373,377,383,388,393,399,404,411,414,422,424,433,434,440,441,442,443,444,445,446,447,448,449,454,455,464,466,474,477,484,488,494,499,505,511,515,522,525,533,535,544,545,550,551,552,553,554,555,556,557,558,559,565,566,575,577,585,588,595,599,606,611,616,622,626,633,636,644,646,655,656,660,661,662,663,664,665,666,667,668,669,676,677,686,688,696,699,707,711,717,722,727,733,737,744,747,755,757,766,767,770,771,772,773,774,775,776,777,778,779,787,788,797,799,808,811,818,822,828,833,838,844,848,855,858,866,868,877,878,880,881,882,883,884,885,886,887,888,889,898,899,909,911,919,922,929,933,939,944,949,955,959,966,969,977,979,988,989,990,991,992,993,994,995,996,997,998,999,1010,1111,1212,1221,1313,1331,1414,1441,1515,1551,1616,1661,1717,1771,1818,1881,1919,1991,2002,2020,2112,2121,2222,2323,2332,2424,2442,2525,2552,2626,2662,2727,2772,2828,2882,2929,2992,3003,3030,3113,3131,3223,3232,3333,3434,3443,3535,3553,3636,3663,3737,3773,3838,3883,3939,3993,4004,4040,4114,4141,4224,4242,4334,4343,4444,4545,4554,4646,4664,4747,4774,4848,4884,4949,4994,5005,5050,5115,5151,5225,5252,5335,5353,5445,5454,5555,5656,5665,5757,5775,5858,5885,5959,5995,6006,6060,6116,6161,6226,6262,6336,6363,6446,6464,6556,6565,6666,6767,6776,6868,6886,6969,6996,7007,7070,7117,7171,7227,7272,7337,7373,7447,7474,7557,7575,7667,7676,7777,7878,7887,7979,7997,8008,8080,8118,8181,8228,8282,8338,8383,8448,8484,8558,8585,8668,8686,8778,8787,8888,8989,8998,9009,9090,9119,9191,9229,9292,9339,9393,9449,9494,9559,9595,9669,9696,9779,9797,9889,9898,9999,10010,10011,10012,10013,10014,10015,10016,10017,10018,10019,10100,10101,10110,10111,10122,10133,10144,10155,10166,10177,10188,10199,10201,10210,10301,10310,10401,10410,10501,10510,10601,10610,10701,10710,10801,10810,10901,10910,11001,11011,11100,11101,11110,11111,11112,11113,11114,11115,11116,11117,11118,11119,11121,11122,11131,11133,11141,11144,11151,11155,11161,11166,11171,11177,11181,11188,11191,11199,11202,11211,11212,11221,11222,11232,11242,11252,11262,11272,11282,11292,11303,11311,11313,11323,11331,11333,11343,11353,11363,11373,11383,11393,11404,11411,11414,11424,11434,11441,11444,11454,11464,11474,11484,11494,11505,11511,11515,11525,11535,11545,11551,11555,11565,11575,11585,11595,11606,11611,11616,11626,11636,11646,11656,11661,11666,11676,11686,11696,11707,11711,11717,11727,11737,11747,11757,11767,11771,11777,11787,11797,11808,11811,11818,11828,11838,11848,11858,11868,11878,11881,11888,11898,11909,11911,11919,11929,11939,11949,11959,11969,11979,11989,11991,11999,12002,12012,12021,12100,12111,12112,12121,12122,12133,12144,12155,12166,12177,12188,12199,12210,12211,12212,12213,12214,12215,12216,12217,12218,12219,12221,12222,12312,12321,12332,12412,12421,12442,12512,12521,12552,12612,12621,12662,12712,12721,12772,12812,12821,12882,12912,12921,12992,13003,13013,13031,13100,13111,13113,13122,13131,13133,13144,13155,13166,13177,13188,13199,13213,13223,13231,13310,13311,13312,13313,13314,13315,13316,13317,13318,13319,13331,13333,13413,13431,13443,13513,13531,13553,13613,13631,13663,13713,13731,13773,13813,13831,13883,13913,13931,13993,14004,14014,14041,14100,14111,14114,14122,14133,14141,14144,14155,14166,14177,14188,14199,14214,14224,14241,14314,14334,14341,14410,14411,14412,14413,14414,14415,14416,14417,14418,14419,14441,14444,14514,14541,14554,14614,14641,14664,14714,14741,14774,14814,14841,14884,14914,14941,14994,15005,15015,15051,15100,15111,15115,15122,15133,15144,15151,15155,15166,15177,15188,15199,15215,15225,15251,15315,15335,15351,15415,15445,15451,15510,15511,15512,15513,15514,15515,15516,15517,15518,15519,15551,15555,15615,15651,15665,15715,15751,15775,15815,15851,15885,15915,15951,15995,16006,16016,16061,16100,16111,16116,16122,16133,16144,16155,16161,16166,16177,16188,16199,16216,16226,16261,16316,16336,16361,16416,16446,16461,16516,16556,16561,16610,16611,16612,16613,16614,16615,16616,16617,16618,16619,16661,16666,16716,16761,16776,16816,16861,16886,16916,16961,16996,17007,17017,17071,17100,17111,17117,17122,17133,17144,17155,17166,17171,17177,17188,17199,17217,17227,17271,17317,17337,17371,17417,17447,17471,17517,17557,17571,17617,17667,17671,17710,17711,17712,17713,17714,17715,17716,17717,17718,17719,17771,17777,17817,17871,17887,17917,17971,17997,18008,18018,18081,18100,18111,18118,18122,18133,18144,18155,18166,18177,18181,18188,18199,18218,18228,18281,18318,18338,18381,18418,18448,18481,18518,18558,18581,18618,18668,18681,18718,18778,18781,18810,18811,18812,18813,18814,18815,18816,18817,18818,18819,18881,18888,18918,18981,18998,19009,19019,19091,19100,19111,19119,19122,19133,19144,19155,19166,19177,19188,19191,19199,19219,19229,19291,19319,19339,19391,19419,1944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(those results will be submitted soon to the OEIS – many, many thanks, Jean-Marc !-)
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