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, ", "); ))