A Counter example, anyone?

[R.D. Silverman]

Originally Posted by maxal

I implicitly assumed that the base is always less than the number being tested (there is no point to consider larger bases). And for prime n, any positive integer less than n is co-prime to n.

That's what you may have *assumed*, but what you WROTE was:

"In particular, 2 is psp(3) (as well as psp(4), psp(5) etc.)."

The last time I looked, 3,4,5 etc. are all bigger than 2.
You are certainly suggesting that 3,4,5, etc. may be used as a base for the prime 2 in what you WROTE.

[maxal]

Originally Posted by R.D. Silverman

That's what you may have *assumed*, but what you WROTE was:

"In particular, 2 is psp(3) (as well as psp(4), psp(5) etc.)."

The last time I looked, 3,4,5 etc. are all bigger than 2.
You are certainly suggesting that 3,4,5, etc. may be used as a base for the prime 2 in what you WROTE.

Yes, that's my fault. I should have excluded all multiples of 2 stating pseudoprimeness of 2 this way.

[mfgoode]

[QUOTE=R.D. Silverman]NO! NO! NO!

Here are the standard definitions.

A number N is a probable prime if, for some a != 0, 1 mod N and (a,N) = 1,
then a^(N-1) = 1 mod N

A number N is a pseudoprime if it is a probable prime and it is composite.

Primes are not pseudoprimes.[UNQUOTE/]

Thank you Bob (if I can take the liberty)!

I accept your definition of a pseudo prime as very clear and precise and unambiguous.
However the number two of all numbers sticks out like a sore thumb. I was wondering how you would categorise it.

As you well know it is the only even prime, even ( not composite) and a Fermat pseudoprime. It seems to be an exception to the rule.
How do we accommodate it to make the definition universal?

I ask to clean up the half baked and erroneous definitions we find from the various websites.

Also to eliminate all doubts in future discussions of this thread. : Thank you.
Mally

[mfgoode]

As a sequel to my previous post in the Thread ‘An invitation to Dr, Silvernman’ I give more details in this thread as its more relevant here. These are more guidelines for those interested in further study on Devraj numbers (abbrv. Dev. No.s) as a ready reference from Mathworld and from my texts.

Dev’ cyclic formula : ( P1 -1) (N-1)/(P2 -1)(P3-1) where N is a square free composite number and P1,P2,P3 are its prime factors leads to a sequence of numbers which give strange composites.
Besides these it covers all the Carmichael numbers (abbr:Car. No.s) which is truly amazing and which are closely linked to (Dev. No.s).

1) If a prime p divides the Car. No. N then it should satisfy the Korselts Criterion viz: N = = p(mod p(p-1)
1 (a) A Dev. No. should satisfy at least one of the three cyclic expressions where the formula gives at least one integer out of the three possible ones for the primes p1,p2,p3 which will be the Dev Criterion.. The other two expressions are (P2-1)(N-1)/(P3-1)(P1-1) and (P3-1) (N-1)/P1-1)(P2-1).

2)Every Car.No. is square free and so is a Dev. No

3)Car.No.s (also called absolute pseudoprimes) have at least 3 non repeating primes so also Dev No.s.
3(a) If any of the primes p1,p2,p3, is put equal to 1 leaving only two primes balance, Dev’s formula breaks down proving that at least 3 non repeating primes are required. So does Korselts Criterion

4) Any solution to Lehmer’s totient problem must be a Car.No.*

5)The first few Car.Nos are 561,1105,1729,2465,2831,6601,8911…(OEIS seq. A 104016)..
5)(a) Besides the Car.No.s the Dev No.s. are 11305,39865,401401, 464185,786961 (OEIS seq A104017) 6)The smallest Car. No.s having 3 or
4 factors are
561=3*11*17
41041=7*11*13*41
7)Numbers of the form (6k+1)(12k+1)(18k+1) are Car.No.s if each of the factors are primes. *
Any odd square free composite N is a Car.No. iff N divides the denominator of the Bernoulli number Bn-1. *
Hope that helps.

* Kind Attn: Maxal pl. check this out for Dev No.s
Mally

[devarajkandadai]

Originally Posted by mfgoode

As a sequel to my previous post in the Thread ‘An invitation to Dr, Silvernman’ I give more details in this thread as its more relevant here. These are more guidelines for those interested in further study on Devraj numbers (abbrv. Dev. No.s) as a ready reference from Mathworld and from my texts.

Dev’ cyclic formula : ( P1 -1) (N-1)/(P2 -1)(P3-1) where N is a square free composite number and P1,P2,P3 are its prime factors leads to a sequence of numbers which give strange composites.
Besides these it covers all the Carmichael numbers (abbr:Car. No.s) which is truly amazing and which are closely linked to (Dev. No.s).

1) If a prime p divides the Car. No. N then it should satisfy the Korselts Criterion viz: N = = p(mod p(p-1)
1 (a) A Dev. No. should satisfy at least one of the three cyclic expressions where the formula gives at least one integer out of the three possible ones for the primes p1,p2,p3 which will be the Dev Criterion.. The other two expressions are (P2-1)(N-1)/(P3-1)(P1-1) and (P3-1) (N-1)/P1-1)(P2-1).

2)Every Car.No. is square free and so is a Dev. No

3)Car.No.s (also called absolute pseudoprimes) have at least 3 non repeating primes so also Dev No.s.
3(a) If any of the primes p1,p2,p3, is put equal to 1 leaving only two primes balance, Dev’s formula breaks down proving that at least 3 non repeating primes are required. So does Korselts Criterion

4) Any solution to Lehmer’s totient problem must be a Car.No.*

5)The first few Car.Nos are 561,1105,1729,2465,2831,6601,8911…(OEIS seq. A 104016)..
5)(a) Besides the Car.No.s the Dev No.s. are 11305,39865,401401, 464185,786961 (OEIS seq A104017) 6)The smallest Car. No.s having 3 or
4 factors are
561=3*11*17
41041=7*11*13*41
7)Numbers of the form (6k+1)(12k+1)(18k+1) are Car.No.s if each of the factors are primes. *
Any odd square free composite N is a Car.No. iff N divides the denominator of the Bernoulli number Bn-1. *
Hope that helps.

* Kind Attn: Maxal pl. check this out for Dev No.s
Mally

Dear Malcolm,
Tks;yes I think your study will help in the arriving at the
final answer .
A.K.Devaraj

[devarajkandadai]

Originally Posted by mfgoode

As a sequel to my previous post in the Thread ‘An invitation to Dr, Silvernman’ I give more details in this thread as its more relevant here. These are more guidelines for those interested in further study on Devraj numbers (abbrv. Dev. No.s) as a ready reference from Mathworld and from my texts.

Dev’ cyclic formula : ( P1 -1) (N-1)/(P2 -1)(P3-1) where N is a square free composite number and P1,P2,P3 are its prime factors leads to a sequence of numbers which give strange composites.
Besides these it covers all the Carmichael numbers (abbr:Car. No.s) which is truly amazing and which are closely linked to (Dev. No.s).

1) If a prime p divides the Car. No. N then it should satisfy the Korselts Criterion viz: N = = p(mod p(p-1)
1 (a) A Dev. No. should satisfy at least one of the three cyclic expressions where the formula gives at least one integer out of the three possible ones for the primes p1,p2,p3 which will be the Dev Criterion.. The other two expressions are (P2-1)(N-1)/(P3-1)(P1-1) and (P3-1) (N-1)/P1-1)(P2-1).

2)Every Car.No. is square free and so is a Dev. No

3)Car.No.s (also called absolute pseudoprimes) have at least 3 non repeating primes so also Dev No.s.
3(a) If any of the primes p1,p2,p3, is put equal to 1 leaving only two primes balance, Dev’s formula breaks down proving that at least 3 non repeating primes are required. So does Korselts Criterion

4) Any solution to Lehmer’s totient problem must be a Car.No.*

5)The first few Car.Nos are 561,1105,1729,2465,2831,6601,8911…(OEIS seq. A 104016)..
5)(a) Besides the Car.No.s the Dev No.s. are 11305,39865,401401, 464185,786961 (OEIS seq A104017) 6)The smallest Car. No.s having 3 or
4 factors are
561=3*11*17
41041=7*11*13*41
7)Numbers of the form (6k+1)(12k+1)(18k+1) are Car.No.s if each of the factors are primes. *
Any odd square free composite N is a Car.No. iff N divides the denominator of the Bernoulli number Bn-1. *
Hope that helps.

* Kind Attn: Maxal pl. check this out for Dev No.s
Mally

Further to Mr.Malcom G.'s study perhaps the following will help in arriving at the final soln.:

Members of A 104017 have the following properties:

a) The no. of factors are even

b)(p2 -1) is a multiple of (p1-1)
or
(p3-1) is a multiple of (p1-1) and (p4-1) is a multiple of (p2-1)
or
The rest are multiples of (p1-1).

This produces the integer effect for k= (p1-1)(N-1)^2/ (p2-1)(p3-1)(p3-1)

resulting in members being pseudo-Carmichael (w.r.t. Devaraj's conjecture).

A.K. Devaraj