20050411, 06:30  #1 
May 2004
2^{2}×79 Posts 
A Counter example, anyone?
I may be wrong but I feel it may be difficult to cite a counter to the following:
Let N = P1P2P3 be a three factor composite number. The necessary & sufficient condition for N to be a pseudoprime is that atleast one of the following should be an integer: ( P1  1)(N  1)/(P2 1)(P31), (P21)(N  1)/(P1 1)(P3  1) or (P3  1)(N 1)/(P1 1)(P2 1) A.K. Devaraj 
20050411, 08:54  #2 
Aug 2004
Melbourne, Australia
10011000_{2} Posts 
Integer !=> Pseudoprime
Double check this, but if we take 42 = 2 x 3 x 7.
Then (71) x (421) / ((21) x (31)) = 6 x 41 / 2 = 123 is an integer. However there does not exist a whole number a such that a^41 = 1 (mod 42) (a != 1 mod 42) 
20050412, 02:32  #3  
May 2004
13C_{16} Posts 
Quote:
be able to cite a counter example. A.K. Devaraj Last fiddled with by devarajkandadai on 20050412 at 02:33 Reason: "am" left out 

20050412, 23:34  #4  
Feb 2005
2^{2}×3^{2}×7 Posts 
Quote:
But there are such nonpseudoprime N for which at least one of the fractions is integer. The smallest counterexamples: 105 = 3*5*7 165 = 3*5*11 195 = 3*5*13 

20050413, 03:52  #5  
Bronze Medalist
Jan 2004
Mumbai,India
2^{2}·3^{3}·19 Posts 
Quote:
There are two types of pseudoprimes viz 1) Fermat pseudoprimes and 2) Carmichael numbers. Kindly clarify which are you referring too. Mally 

20050413, 03:59  #6  
Feb 2005
2^{2}·3^{2}·7 Posts 
Quote:


20050413, 04:28  #7  
Bronze Medalist
Jan 2004
Mumbai,India
2052_{10} Posts 
A Counter example anyone?
Quote:
Thank you maxal . Could you please elaborate the distinction between them and the no.s. you have derived as Im a bit confused Mally 

20050413, 04:55  #8  
Feb 2005
252_{10} Posts 
Quote:
Quote:


20050413, 07:21  #9  
Dec 2004
The Land of Lost Content
3·7·13 Posts 
Quote:


20050413, 10:26  #10  
Bronze Medalist
Jan 2004
Mumbai,India
4004_{8} Posts 
A Counter example anyone?
Quote:
Thank you once again maxal. The no.s you have derived evidently satisfy at least one of Devraj's eqn.s ( abbr.equations). [Quote] Either one. The quoted counterexamples are neither Fermat pseudoprimes, nor Carmichael numbers.[/UNQUOTE]' maxal. If you have ruled these two sets out then what category do the numbers 105 ,165,195 etc. fall under? Thank you Mally 

20050413, 22:08  #11  
Feb 2005
FC_{16} Posts 
Quote:
(I think I've already said that). Last fiddled with by maxal on 20050413 at 22:09 

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