A Counter example, anyone?

[devarajkandadai]

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 pseudo-prime is that
atleast one of the following should be an integer:

( P1 - 1)(N - 1)/(P2 -1)(P3-1),

(P2-1)(N - 1)/(P1 -1)(P3 - 1) or

(P3 - 1)(N -1)/(P1 -1)(P2 -1)

A.K. Devaraj

[Dougy]

Double check this, but if we take 42 = 2 x 3 x 7.

Then (7-1) x (42-1) / ((2-1) x (3-1)) = 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)

[devarajkandadai]

Originally Posted by Dougy

Double check this, but if we take 42 = 2 x 3 x 7.

Then (7-1) x (42-1) / ((2-1) x (3-1)) = 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)

I should have specified the Ps as odd prime numbers;now I am not sure u will
be able to cite a counter example.
A.K. Devaraj

[maxal]

Originally Posted by devarajkandadai

Let N = P1P2P3 be a three factor composite number.

The necessary & sufficient condition for N to be a pseudo-prime is that
atleast one of the following should be an integer:

( P1 - 1)(N - 1)/(P2 -1)(P3-1),

(P2-1)(N - 1)/(P1 -1)(P3 - 1) or

(P3 - 1)(N -1)/(P1 -1)(P2 -1)

It seems to work only in one direction (which is a particular case of your previous conjecture), i.e., if N is pseudo-prime then at least one (actually, all three) of the specified fractions is integer.
But there are such non-pseudoprime 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

[mfgoode]

Originally Posted by maxal

It seems to work only in one direction (which is a particular case of your previous conjecture), i.e., if N is pseudo-prime then at least one (actually, all three) of the specified fractions is integer.
But there are such non-pseudoprime 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

There are two types of pseudo-primes viz 1) Fermat pseudo-primes and 2) Carmichael numbers. Kindly clarify which are you referring too.
Mally

[maxal]

Originally Posted by mfgoode

There are two types of pseudo-primes viz 1) Fermat pseudo-primes and 2) Carmichael numbers. Kindly clarify which are you referring too.

Either one. The quoted counterexamples are neither Fermat pseudoprimes, nor Carmichael numbers.

[mfgoode]

Originally Posted by maxal

Either one. The quoted counterexamples are neither Fermat pseudoprimes, nor Carmichael numbers.

Thank you maxal . Could you please elaborate the distinction between them and the no.s. you have derived as Im a bit confused
Mally

[maxal]

Originally Posted by mfgoode

Could you please elaborate the distinction between them

See http://mathworld.wolfram.com/FermatPseudoprime.html and http://mathworld.wolfram.com/CarmichaelNumber.html

Originally Posted by mfgoode

and the no.s. you have derived

What's "no.s." ?

[99.94]

Originally Posted by maxal

What's "no.s." ?

Numbers? (abbreviation).

[mfgoode]

Originally Posted by maxal

Either one. The quoted counterexamples are neither Fermat pseudoprimes, nor Carmichael numbers.

Yes 99.94. This abbr. is widely used in this part of the world.

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

[maxal]

Originally Posted by mfgoode

If you have ruled these two sets out then what category do the numbers 105
,165,195 etc. fall under?

These numbers are not pseudoprimes (neither "Fermat pseudoprimes", nor "Carmichael numbers") for which at least one of Devaraj's fractions is integer.
(I think I've already said that).