Six-Decade Math Puzzle Solved by Korean Mathematician

A Korean mathematician has won international recognition for solving a geometry puzzle, the moving sofa problem, that had resisted proof for nearly six decades.

Click here for more information.

Optimal Golomb Ruler and More

Published several short papers. However, my latest, Bounds for Optimal Golomb Rulers is promising.

Permutation Rotations

The Golden Ratio's Siblings

My Mersenne Primality Testing Algorithm

I'm not sure how fast it is compared to GIMPS, but it is acurate for all Mersenne or Fermat primes. See my article Value-Counting Up to N for why.

import sys
import time

# Remove the limit
sys.set_int_max_str_digits(0)

i = 0

def signal_handler(sig, frame):
	global i
	if sig == signal.SIGINT:
		print("index = %i" % (i))
	sys.exit(0)

def int_sqrt(n):
	"""
	Calculate the integer square root of a non-negative integer n.
	This function returns the largest integer whose square is less than or equal to n.
	"""
	if n < 0:
		raise ValueError("Cannot calculate square root of a negative number.")
	if n == 0:
		return 0
	x = n
	y = 1
	i = 0
	while x > y:
		x = (x + y) // 2
		y = n // x
	return x

def prime_power(n):
	signal.signal(signal.SIGINT, signal_handler)
	total = 0
	half_n = (n + 1) // 2
#	limit = half_n
#	limit = math.ceil((half_n*(half_n + 1) // 2 + 1) / n)
	limit = (n + 13)//8
	start_time = time.perf_counter()
	end_time = start_time

	global i
	for i in range(2, limit):
		# border_value represents the number that ends on the last
		# column when all numbers up to it are added and wrapped.
		border_value = int((int_sqrt(8 * i * n + 1) - 1)/2)
		total = (border_value * (border_value + 1)//2)
		elapsed_time = end_time - start_time
		if (total == (i * n)):# or (total + (border_value + 1)//2 == (i * n)):
			print("%i" % (border_value))
			return
		end_time = time.perf_counter()
		elapsed_time = end_time - start_time
		if elapsed_time > WAIT_TIME:
			start_time = end_time
			end_time = time.perf_counter()
	print("\tprime_power")

print("p\tborder\tprime power")

prime_power(2**89-1)

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.

Click here for more information.

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.

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.

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.

Click here for more information.

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`.

Click here for more information.

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.

Click here for more information.

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.

Click here for more information.

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.

Click here for more information.