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 2020-01-31, 12:39 #1 devarajkandadai     May 2004 22×79 Posts Elliptic Carmichael numbers I had a conjecture that the above (defined below) exist. A set of 2 or more Carmichael numbers in which the smallest and largest prime factors are common but the intervening prime factors are different. Example: 15841 = 7*31*73 126217 =7*13*19*73
 2020-02-01, 06:04 #2 CRGreathouse     Aug 2006 3·1,993 Posts About 3 minutes of brute force gave me these: Code: 6601, [7, 41] 41041, [7, 41] 15841, [7, 73] 126217, [7, 73] 29341, [13, 61] 552721, [13, 61] 10585, [5, 73] 825265, [5, 73] 670033, [7, 199] 1773289, [7, 199] 4463641, [7, 271] 9585541, [7, 271] 852841, [11, 61] 10877581, [11, 61] 16778881, [7, 181] 31146661, [7, 181] 9582145, [5, 859] 31692805, [5, 859] 18162001, [11, 241] 40430401, [11, 241] 4463641, [7, 271] 9585541, [7, 271] 41341321, [7, 271] 9890881, [7, 241] 41471521, [7, 241] 1857241, [31, 331] 42490801, [31, 331] 512461, [31, 271] 45877861, [31, 271] 13992265, [5, 397] 47006785, [5, 397] 3224065, [5, 257] 67371265, [5, 257] 37167361, [7, 193] 69331969, [7, 193] 1569457, [17, 113] 75151441, [17, 113] 67994641, [11, 181] 76595761, [11, 181] 36121345, [5, 337] 93869665, [5, 337] 17812081, [7, 1171] 94536001, [7, 1171] 4767841, [13, 199] 102090781, [13, 199] 15888313, [7, 1783] 104852881, [7, 1783] 5031181, [19, 397] 109577161, [19, 397]
2020-07-13, 07:05   #3

May 2004

22×79 Posts

Quote:
 Originally Posted by CRGreathouse About 3 minutes of brute force gave me these: Code: 6601, [7, 41] 41041, [7, 41] 15841, [7, 73] 126217, [7, 73] 29341, [13, 61] 552721, [13, 61] 10585, [5, 73] 825265, [5, 73] 670033, [7, 199] 1773289, [7, 199] 4463641, [7, 271] 9585541, [7, 271] 852841, [11, 61] 10877581, [11, 61] 16778881, [7, 181] 31146661, [7, 181] 9582145, [5, 859] 31692805, [5, 859] 18162001, [11, 241] 40430401, [11, 241] 4463641, [7, 271] 9585541, [7, 271] 41341321, [7, 271] 9890881, [7, 241] 41471521, [7, 241] 1857241, [31, 331] 42490801, [31, 331] 512461, [31, 271] 45877861, [31, 271] 13992265, [5, 397] 47006785, [5, 397] 3224065, [5, 257] 67371265, [5, 257] 37167361, [7, 193] 69331969, [7, 193] 1569457, [17, 113] 75151441, [17, 113] 67994641, [11, 181] 76595761, [11, 181] 36121345, [5, 337] 93869665, [5, 337] 17812081, [7, 1171] 94536001, [7, 1171] 4767841, [13, 199] 102090781, [13, 199] 15888313, [7, 1783] 104852881, [7, 1783] 5031181, [19, 397] 109577161, [19, 397]
Thanks.Are they all 2 stringed ellipses?

2020-07-13, 09:11   #4
JeppeSN

"Jeppe"
Jan 2016
Denmark

23·3·7 Posts

Quote:
 Originally Posted by devarajkandadai Thanks.Are they all 2 stringed ellipses?
What does that mean? I see three rows with ", [7, 271]", i.e. three different Carmichael numbers whose minimal prime is 7 and maximal prime is 271. One of them has one more prime factor than the others:

4463641 = 7*13*181*271
9585541 = 7*31*163*271
41341321 = 7*19*31*37*271

/JeppeSN

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