My nickname is miket and have many other nickname, research on natural science.I am interested in number theory.I put forward a conjecture on cycle length and primes : prime abc conjecture PAC.
prime abc conjecture PAC:
Suppose a>9 is odd and b is the cycle length of a as defined below. Then I conjecture that if
(a - 1) / b = 2 ^ c
for some positive integer c then a is prime.
Cycle length:
a is an odd number > 9 and Od is the odd part function, Sloane's A000265.
Let a(0) = 1 and a(n) = Od(a(n-1) + a) for n > 0. If a(n) = 1 for 0 < n < N/2 - 1 then the cycle length is the smallest such n, otherwise it is a/2 - 1.
For more information see A179382.
Example:
11 = 5*2^1+1
11 (1,3, 7, 9, 5)
97 = 24*2^2+1
97 (1,49, 73, 85, 91, 47, 9, 53, 75, 43, 35, 33, 65, 81, 89, 93, 95, 3, 25, 61, 79, 11, 27, 31)
if c = 1 that's OEIS A001122
if c = 2 that's OEIS A155072
if c = 3 that's OEIS A001134
if c = 4 that's 1217, 1249, 1553, 4049, 4273, 4481, 4993..., not in the OEIS
PARI/GP code:
/*** Prime abc conjecture PAC latest version 2013-05-11 by Mike T Rewrited by Charles 2013-05-10 ***/
is(a,c) = {
k=(a+1)>>valuation(a+1,2);
v = List();
listput(v,k);
for(i=0,(a-5)/(2*c),
k=(k+a)>>valuation(k+a,2);
listput(v,k);
if(k==1,break)
);
/*** Here's the Prime ABC Conjecture PAC ***/
#v == (a-1)/(2^c)
}
forstep(i=11,100000,2,if(is(i,4),print1(i, ", "); ))