20081203, 08:41  #1 
May 2004
13C_{16} Posts 
3factor Carmichael numbers
NECESSARY & SUFFICIENT CONDITIONS FOR A THREEFACTOR COMPOSITE
NUMBER WITH FOLLOWING SHAPE TO BE A CARMICHAEL NUMBER Let N, the composite number, have the shape (2m+1)(10m+1)(16m+1). Here m belongs to N. The necessary and sufficient conditions: a) (80m^2 + 53m + 7)/20 should be an integer b) The values of m which render the above an integer should also render 2m + 1, 10m + 1 and 16m + 1 prime. This is a corollary of the DevarajPomeranceMaxal theorem (ref:: www.crorepatibaniye.com/failurefunctions 
20081203, 13:52  #2 
"Lucan"
Dec 2006
England
2×3×13×83 Posts 

20081203, 15:26  #3 
Aug 2002
Buenos Aires, Argentina
2×17×43 Posts 
I tested the first five million values of m and it appears to work. This is the program in UBASIC, noticing that the point "a" is equivalent to m=1 (mod 20). Nothing is printed, so it is OK.
Code:
10 for M=1 to 5000000 20 if 2*M+1<>nxtprm(2*M) or 10*M+1<>nxtprm(10*M) or 16*M+1<>nxtprm(16*M) then 60 30 A=2*M+1:B=10*M+1:C=16*M+1:D=A*B*C 40 if (D1)@(A1)<>0 or (D1)@(B1)<>0 or (D1)@(C1)<>0 then K=0 else K=1 50 if (M@20=1 and K=0) or (M@20<>1 and K=1) then print M,(2*M+1)*(10*M+1)*(16*M+1) 60 next M Last fiddled with by alpertron on 20081203 at 15:44 Reason: Added program in UBASIC 
20081203, 16:02  #4  
"Robert Gerbicz"
Oct 2005
Hungary
1577_{10} Posts 
Quote:
Why you don't write: (80m^2 + 53m + 7)%20==13m+7==13*(m1)==0 mod 20 so the simple condition is that m1 is divisble by 20. Last fiddled with by R. Gerbicz on 20081203 at 16:03 

20081206, 04:16  #5 
May 2004
2^{2}·79 Posts 
3factor C.Ns
I wonder whether 561 is the only 3factor C.N. with 3 as a factor . The reason is that if we were to keep 3 fixed and increase p_2 and p_3 indefinitely, k becomes asymptotic to 6.
A.K.Devaraj 
Thread Tools  
Similar Threads  
Thread  Thread Starter  Forum  Replies  Last Post 
Carmichael numbers  devarajkandadai  Number Theory Discussion Group  14  20171115 15:00 
Carmichael numbers and Devaraj numbers  devarajkandadai  Number Theory Discussion Group  0  20170709 05:07 
Carmichael Numbers  Stan  Miscellaneous Math  19  20140102 21:43 
Carmichael Numbers  devarajkandadai  Miscellaneous Math  0  20060804 03:06 
Carmichael Numbers II  devarajkandadai  Math  1  20040916 06:06 