2003 divides k^143+1

[Matthew Vandermast]

Ed,

2002 = 2 * 7 * 11 * 13 = 143 * 14 = 286 * 7

Since 2003 is a prime, it will always divide evenly into n^2002 - 1
(Fermat's Little Theorem).  In some cases it will divide evenly into n^d - 1
where d is a divisor of 2002.  In all the cases listed below where 2003 is a
divisor of n^143 + 1, it is also a divisor of n^286 - 1; 286 is a divisor of
2002, as shown above.

Best,
Matthew Vandermast


----- Original Message -----
From: "Ed Pegg" <edp at wolfram.com>
To: <seqfan at ext.jussieu.fr>
Sent: Thursday, April 17, 2003 1:31 PM
Subject: 2003 divides k^143+1


>
>   2003 divides evenly into 2^143+1
>
>   2003 divides evenly into 11^143+1
>
> k for which 2003 is a factor of k^143+1 are as follows:
>
> {2, 8, 11, 21, 23, 32, 44, 45, 67, 71, 84, 87, 91, 92, 95, 128, 149, 176,
> 178, 180, 195, 226, 239, 242, 249, 268, 282, 284, 310, 313, 336, 348, 364,
> 368, 370, 377, 380, 381, 443, 462, 506, 512, 538, 563, 579, 595, 596, 704,
> 712, 720, 780, 811, 813, 845, 877, 882, 904, 927, 954, 956, 958, 966, 968,
> 979, 990, 991, 996, 1002, 1046, 1058, 1062, 1072, 1079, 1094, 1117, 1128,
> 1136, 1222, 1240, 1241, 1243, 1249, 1252, 1263, 1266, 1267, 1275, 1307,
> 1318, 1331, 1344, 1370, 1377, 1383, 1392, 1435, 1439, 1456, 1467, 1472,
> 1474, 1480, 1502, 1505, 1508, 1519, 1520, 1524, 1525, 1526, 1551, 1562,
> 1613, 1643, 1647, 1651, 1705, 1734, 1747, 1750, 1772, 1813, 1818, 1819,
> 1821, 1829, 1835, 1848, 1861, 1862, 1869, 1882, 1890, 1913, 1914, 1915,
> 1939, 1957, 1961, 1981, 1987, 1999}
>
>   I feel I should know the connection between 143 and 2003, but I'm not
> seeing it.  Can someone enlighten me?
>
>   --Ed Pegg Jr
>
>