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A182090
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Carmichael numbers divisible by 11.
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1
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561, 41041, 75361, 101101, 656601, 658801, 852841, 2100901, 2704801, 4903921, 6049681, 6313681, 8341201, 9890881, 10837321, 10877581, 11205601, 18162001, 26932081, 27062101, 29020321, 33302401, 37167361, 37280881, 40430401, 43286881, 67994641, 72108421, 76595761, 77826001, 88689601, 93614521, 100427041, 104404861, 105869401
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OFFSET
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1,1
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COMMENTS
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Conjecture: Any Carmichael number C divisible by 11 and not divisible by 3 can be written in one of the following four forms: C = 990k+11; C = 990k+11^2; C = 990k+11*41 or C = 990k+11*71, where k is a natural number.
In other words, C mod 990 can be, for C divisible by 11 and not divisible by 3, just 11; 11^2; 11*41 or 11*71.
For comparison, we issue the following related conjectures:
(1) Any Carmichael number C of the form 10n+1 divisible by 7 and not divisible by 3 and 5 can be written in one of the following three forms: C = 630k+7*13; C = 630k+7*43; C = 7*73.
(2) Any Carmichael number C of the form 10n+3, 10n+7 or 10n+9 divisible by 7 and not divisible by 3 and 5 can be written as C = 630k+7^3, C = 630k+7*31 or C = 630k+7*67.
(3) Any Carmichael number C divisible same time with 7 and 11 can be written as C = 7*11*90*k + 7*11*n, where n can be 23, 53 or 83.
Note: there exist Carmichael numbers that can be written as C = p*(90k+1) for every p prime divisor of C.
Example:
C = 197531244744661; C mod 90*1531 = 1531; C mod 90*3061 = 3061; C mod 90*4591 = 4591; C mod 90*9181 = 9181.
The first counterexample to the conjecture is 6049681 which is 781 mod 990. 341 and 671 mod 990 also appear. Conjecture 1 is true; the only excluded possibility is 161 mod 210 which is never a 2-pseudoprime (check mod 7). All three parts of conjecture 2 are false; counterexamples include 461502097, 1378483393, and 7044493729. Conjecture 3 is true, consider Korselt's criterion mod 30. - Charles R Greathouse IV, Dec 07 2014
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LINKS
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MATHEMATICA
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CarmichaelNbrQ[n_] := ! PrimeQ[n] && Mod[n, CarmichaelLambda@ n] == 1; Select[ Range[10000000], CarmichaelNbrQ[ 11#] &] (* Robert G. Wilson v, Aug 24 2012 *)
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PROG
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(PARI) Korselt(n)=my(f=factor(n)); for(i=1, #f~, if(f[i, 2]>1||(n-1)%(f[i, 1]-1), return(0))); 1
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CROSSREFS
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KEYWORD
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nonn
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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