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-   -   Elliptic Carmichael numbers (https://www.mersenneforum.org/showthread.php?t=25158)

devarajkandadai 2020-01-31 12:39

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

CRGreathouse 2020-02-01 06:04

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][/code]

devarajkandadai 2020-07-13 07:05

[QUOTE=CRGreathouse;536361]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][/code][/QUOTE]
Thanks.Are they all 2 stringed ellipses?

JeppeSN 2020-07-13 09:11

[QUOTE=devarajkandadai;550423]Thanks.Are they all 2 stringed ellipses?[/QUOTE]

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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