The Golden Ratio's Siblings

On real numbers whose fractional part remain constant after squaring.

Click here to read my paper.

52nd Known Mersenne Prime Found!

The Great Internet Mersenne Prime Search (GIMPS) has discovered the largest known prime number, 2^136,279,841-1, having 41,024,320 decimal digits. Luke Durant, from San Jose, California, found the prime on October 12th.

Click here for more information.

A Chiral Aperiodic Monotile

An update to the aperiodic tile. This update shows that a shape that tiles the plane aperiodically without reflections is possible. The original one did not.

Click here for the article.

Mathematicians have finally discovered an elusive ‘einstein’ tile

A 13-sided shape known as “the hat” has mathematicians tipping their caps.

It’s the first true example of an “einstein,” a single shape that forms a special tiling of a plane: Like bathroom floor tile, it can cover an entire surface with no gaps or overlaps but only with a pattern that never repeats.

Click here for more information.

Value-Counting Up to N

Some interesting properties arise when value-counting the integers sequentially up to N using N digits or fingers and comparing the number of values to the prime-exact equation; with a simple method for testing primes and prime powers (particularly Mersenne and Fermat primes).

Click here to read my paper.

Primality Testing and Factoring Using Pascal's Triangle

An interesting if not impractical way of primality testing and factoring a number using Pascal’s Triangle.

Click here to read my paper.

Collatz Conjecture

I've previously linked to Jason Davies website for another article. He has another JavaScript program for the Collatz Conjecture.

To pretty it up, remove the circle fill in collatz.css and modify the circle append (line 83) in collatz.js as follows:

      nodeEnter.append("circle")
          .attr("fill", function(d) {
            var cc;
            var i = parseInt(d.data);
            if ((i && (i & (i - 1)) === 0) && (i <= 16)) {
              cc = "#0000ff";
            }
            else if ((i % 3) === 0) {
              cc = "#c8c8c8";
            }
            else if ((i % 6) === 1) {
              cc = "#ffff00";
            }
            else if (((i % 2) === 1) || (((i % 3) === 2) && (((i / 2) % 2) === 0))) {
              cc = "#ffa500";
            }
            else {
              cc = "#000000";
            }
            return cc;
          })
         .attr("r", 5);

I was only interested in the initial node for those in orange and yellow.

Minimal Set for Powers of 2

The minimal set for powers of 2 is currently nondeterministic and can be shown to be more complex than previously proposed.

Click here for my analysis on it.