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An invitation

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An invitation

This is an invitation to solve a problem occupying my mind:

All odd prime numbers can be divided into two mutually exclusive classes: a)Mangammal primes (for definition see OEIS-a123239) and b) non-Mangammal primes.

For testing Mangammal primality I use the following in pari:

Let M be the prime to be tested for Mangammal primality.

{p(n)=if(Mod(3,M)^n == 2, print(" integer"))}; for (n=1,M-1,print(p(n))).

Q: This is ok in the case of comparitively small M-prime suspects. What should we do when we want to test a very large prime for Mangammal primality?
A.K. Devaraj

Thsnk you very much; am surprised to see that a member has responded. Generaly there is no response -the reason being that few are, nowadays, interested in number theory. Wishing you success and am looking forward to your formal approach.
A.K. Devaraj

Incidentally you may look up, if interested, my blogs of 19th March and ist May 2011 ( These have something to do with the construction of Carmichael numbers made up exclusively of Mangammal primes.

A.K. Devaraj

u r right. Let N = number of digits of the suspected Mp. Run the following: {p(n)=if(Mod(3,Mp)^n==1,print("integer"))}. for(n=N^1/3 to N/2,print(p(n))). There is consdierable saving in running time of this programme.
A.K. Devaraj

I have not considered throughly but in few examples that I have tried I see that if it is a mangammal prime, it satisfies 3^n == 1 mod (M) for some n <= M/2. I guess it has something to do with pigeonhole principle.

Later, I will try to make up my "formal" mind.

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