[seqfan] " Little Fermat theorem" for twin primes

Vladimir Shevelev shevelev at bgu.ac.il
Wed Oct 12 14:39:39 CEST 2011

Dear SeqFans,

Trivially " Little Fermat theorem" for twin primes {n,n+2} sounds as the following:  for every b prime to n*(n+2),
                     b^(n-1)==1(mod n) and b^(n+1)==1 mod(n+2).          (1)
The system of these two congruences is not equivalent to (b^(n-1)-1)*(b^(n+1)-1)==0 mod(n*(n+2)) (a simple counterexample n=7, since b^2==1(mod 3)).  However, in Little Fermat theorem there is only one congruence. I found another simple congruence
                     2*(b^(n+1)-1)==(b^2-1)*(n+2) (mod n*(n+2)).          (2)
 Now n=7 is not a solution, e.g., for b=2. A question: Is it true that (2) yields (1)?
If it is true, then the "Carmichael pseudo twin primes"  can be characterized by (2) only; otherwise, there could be exist other "Carmichael pseudo twin primes" satisfying (2) for all  b prime to n*(n+2).
Note that the sequence of Carmichael pseudo twin primes (I submitted lesser of them in A194231), i.e. numbers {n,n+2}satisfying (1) for every  b prime to n*(n+2) such that from them there exists at least one composite number,  starts with 561, 1103, 2465, 2819, 6599, 29339,...


 Shevelev Vladimir‎

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