Busy Beaver Number 5 is Alive!

In theoretical computer science, the objective of the busy beaver is to find a terminating program of a given size that (depending on definition) either produces the most output possible, denoted by BB(n), or runs for the longest number of steps.

The latest value for n = 5 was recently discovered.

The known values are BB(1) = 1, BB(2) = 6, BB(3) = 21, BB(4) = 107, BB(5) = 47,176,870.

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An Amazing Approximation to e

An amazing pandigital approximation to e that is correct to 18,457,734,525,360,901,453,873,570 decimal places is given by:

`e\approx(1+9^(−4^(6⋅7)))^(3^(2^85))`

It was discovered by Richard Sabey in 2004.

Proof:

`(1+9^(−4^(6⋅7)))^(3^(2^85))=(1+9^(−4^42))^(3^(2^85))`

`=(1+9^(−4^42))^(3^(2*2^84))`

`=(1+9^(−4^42))^(3^(2*2^84))`

`=(1+9^(−4^42))^(9^(2^(84)))`

`=(1+9^(−4^42))^(9^(4^42))`

`=(1+\frac{1}{9^(4^42)})^(9^(4^42))`

`=(1+\frac{1}{n})^n`.

Permutation Rotations

In this paper, we discuss certain properties of permutation rotations on each other.

Click here to read my paper.

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.

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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.

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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.

Mathematicians Discover the Perfect Way to Multiply

Four thousand years ago, the Babylonians invented multiplication. Last month, mathematicians perfected it.

On March 18, two researchers described the fastest method ever discovered for multiplying two very large numbers. The paper marks the culmination of a long-running search to find the most efficient procedure for performing one of the most basic operations in math.

“Everybody thinks basically that the method you learn in school is the best one, but in fact it’s an active area of research,” said Joris van der Hoeven, a mathematician at the French National Center for Scientific Research and one of the co-authors.

Click here and here for more information.

A 53-Year-Old Network Coloring Conjecture Is Disproved

A paper posted online last month has disproved a 53-year-old conjecture about the best way to assign colors to the nodes of a network. The paper shows, in a mere three pages, that there are better ways to color certain networks than many mathematicians had supposed possible.

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Andrew Booker, a Mathematics Professor at the University of Bristol, Just Solved a Deceptively Simple Puzzle That Has Boggled Minds for 64 Years

A mathematician in England has cracked a math puzzle that's stumped computers and humans alike for 64 years: How can the number 33 be expressed as the sum of three cubed numbers?

While it might seem simple on its face, this question is part of an enduring number-theory conundrum that goes back to at least 1955 and may have been mulled over by Greek thinkers as early as the third century. The underlying equation to solve looks like this:

`x^3 + y^3 + z^3 = k`

That answer is:

`(8,866,128,975,287,528)^3 + (–8,778,405,442,862,239)^3 + (–2,736,111,468,807,040)^3 = 33`.

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Karen Uhlenbeck is first woman to win prestigious maths Abel prize

Mathematician Karen Uhlenbeck has become the first woman to win the Abel prize, sometimes called the Nobel prize of mathematics. She has been awarded the 6 million Norwegian kroner ($700,000) prize for her work in the fields of gauge theory and geometric analysis, which have been credited with far-reaching impact in both mathematics and physics.

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Mathematician Sir Michael Atiyah dies aged 89

One of the world's foremost mathematicians, Prof Sir Michael Atiyah, has died at the age of 89.

Sir Michael, who worked at Cambridge University before he retired, made outstanding contributions to geometry and topology.

Sir Michael was a recipient of the highest honour in mathematics, a Fields Medal. He died on Friday.

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51st Known Mersenne Prime Found!

The Great Internet Mersenne Prime Search (GIMPS) has discovered the largest known prime number, 2^82,589,933-1, having 24,862,048 digits. A computer volunteered by Patrick Laroche from Ocala, Florida made the find on December 7, 2018. The new prime number, also known as M82589933, is calculated by multiplying together 82,589,933 twos and then subtracting one. It is more than one and a half million digits larger than the previous record prime number.

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Riemann Hypothesis Solved By Sir Michael Atiyah After 160 Years, He Says

One of the world's most renowned mathematicians showed how he solved the 160-year-old Riemann hypothesis at a lecture on Monday — and he will be awarded $1 million if his solution is confirmed.

Sir Michael Atiyah, who has won the two biggest prizes in mathematics — the Fields Medal and Abel Prize — took the stage at the Heidelberg Laureate Forum in Germany on Monday to present his work.

To solve the hypothesis you need to find a way to predict the occurrence of every prime number, even though primes have historically been regarded as randomly distributed.

Aityah's solution will need to be checked by other mathematicians and then published before it is fully accepted and he can claim the prize from the Clay Mathematics Institute of Cambridge.

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Every Positive Integer Is A Sum Of Three Palindromes

For integer g ≥ 5, we prove that any positive integer can be written as a sum of three palindromes in base g.

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"Nobel of Mathematics" Stolen Minutes After Awarded

One of the winners of the award known as the Nobel Prize for mathematics had his gold medal stolen minutes after it was given to him. Caucher Birkar, a Kurdish refugee turned Cambridge University math professor, was among four winners of the prestigious Fields Medal on Wednesday in Rio de Janeiro.

It was an embarrassing debut for crime-ridden Rio, the first Latin American city ever to host the Fields ceremony, which takes place every four years. Less than an hour had passed since Birkar, a 40-year-old specialist in algebraic geometry, had been handed his 14-karat gold medal when his briefcase went missing.

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