27.8.16

Collatz Factoring

Definition: A Collatz product is the product n of 2 odd primes p and q such that n+1 reaches p or q when put through the algorithm behind the Collatz conjecture.

As I already mentioned here, such products seem to appear much more frequently than single such primes, or primes that I called Collatz primes.

For example, from all products of p = 3,5,7,11,13,17 times all other primes q = from p to 53, the following are Collatz products:

p=3 for all q from 3 to 53: [0] (see conjecture at the bottom of this page)

p=5 for all q from 5 to 53: [25, 35, 55, 65, 85, 95, 115, 145, 155, 185, 205, 215, 235, 265]

p=7 for all q from 7 to 53: [ 77, 133, 161, 203, 217, 259, 287, 329, 371 ]

p=11 for all q from 11 to 53: [143, 209, 253, 341, 407, 473, 517, 583 ]

p=13 for all q from 13 to 53: [403, 481, 611, 689]

p=17 for all q from 17 to 53: [391, 493, 527, 629, 697, 799, 901

And so on. 

The following sequence emerges:

25,35,55,65,77,85,95,115,133,143,145,155,161,185,203,205,209,...


Below is an algorithm for generating all products of p given an array of odd primes.

/* -------------------------------------------------
 This content is released under the GNU License
 http://www.gnu.org/copyleft/gpl.html
 Author: Marina Ibrishimova
 Version: 1.0
 Purpose: Find Collatz Products
 ---------------------------------------------------- */
//takes an array of odd primes and a single prime cur
function genrate(primez, cur)
{
var factorable = new Array();
for(i=0; i<primez.length; i++)
{
 if(isitCollatzProduct(cur,primez[i])!=0)
 {
  factorable.push(cur*primez[i]);
 }
}
return factorable;
}
function isitCollatzProduct(p,q)
{
var n = p*q;
var cur = n+1;
while(cur != p && cur != q && cur != 2)
{
if(cur%2!=0)
{
cur = 3*cur + 1;
}else
{cur = cur/2;}
}
if(cur === p || cur === q ){return cur;}
else {return 0;}
}

Conjecture: If n is the product of two odd primes p and q and p = 3 then n is not a Collatz product.