Thursday, April 1, 2010

prime number theorem and postulates

1. the maximum periodicity of the reciprocal of a number n is n-1. (Gauss)
2. if the periodicity of 1/n is n-1 then n must be prime. the converse is not true.
3. if the periodicity of 1/p is always a factor of p-1.
4. if the periodicity of 1/p is p-1 then it must exhibit the property of revolution. hence a revolution characteristic can be associated to the prime number p.
5. on the basis of periodicity of reciprocals of prime numbers, they may be classified as true primes and pseudo-primes. (total-reciprocal-periodic prime (trp) or partial-reciprocal-periodic prime (prp))
6. the periodicity of 1/p2 is p*(periodicity of 1/p). In fact the periodicity of 1/n2 is either n*(periodicity of 1/n) or LCM of n and periodicity of 1/n.
7. the perodicity of 1/(p1*p2) is the LCM of the perodicities of 1/p1 and 1/p2.
8. if the decimal part (including left zeroes) of 1/n is denoted by d, then the decimal part of 1/d is n preceded by [d]-[n] zeroes. ([ ] signifies number of digits)
9. the location of nulls in a discontinuous revolution characteristic may be symmetrical or asymmetrical. (Symmetrically discontinuous or Asymmetrically discontinuous)
10. the diameter of circular revolution of a number the length of whose reciprocal is d is always d/2.