Maths à la De Funès…

## Life, the Universe, and Maths

For years, mathematicians have worked to demonstrate that *x*^{3}+*y*^{3}+*z*^{3} = *k*, where *k* is defined as the numbers from 1 to 100. This theory is true in all cases except for an unproven exception: 42.

By 2016 and over a million hours of computation later, researchers of the UK’s Advanced Computing Research Center had its solution for 42.

More intriguing number facts **here**.

## Prime Square

3,139,971,973,786,634,711,391,448,651,577,269,485,891,759,419,122,938,744,591,877,656,925,789,747,974,914,319,422,889,611,373,939,731 produces reversible primes in each row, column and diagonal when distributed in a 10×10 square.

Diagram by HT Jens Kruse Andersen.

## Mirror Squares

## Inverse Powers of Phi

Summation of Alternating Inverse Powers of Phi…

## The Kepler Triangle, Phi and Pi

A “Kepler triangle” is a right triangle having edge lengths in a geometric progression, in which the common ratio is √ϕ, where ϕ represents the golden ratio.

Well, let’s construct a square with side length √ϕ that inscribes a Kepler triangle, that is, a right triangle with edges 1 : √ϕ : ϕ (or approximately 1 : 1.272 : 1.618), as shown in the picture. Draw then the circumcircle of the Kepler triangle (highlighted in orange in the picture) whose diameter is the hypotenuse of the triangle.

Then, the perimeters of the square (4√ϕ≈5.0884) and the circle (πϕ≈5.083) coincide up to an error less than 0.1%. From this, we can get the approximation coincidence π≈4/√ϕ

## Fibonacci Right Triangle

The sum of the squares of consecutive Fibonacci numbers is another Fibonacci number.

## Repunit Primes

Write the digit “1” exactly 317 times, and you get a palindromic prime number. Moreover, 317 itself is a prime number!

## Infinite Pythagorean Triplets

Consider the following simple progression of whole and fractional numbers (with odd denominators):

1 1/3, 2 2/5, 3 3/7, 4 4/9, 5 5/11, 6 6/13, 7 7/15, 8 8/17, 9 9/19, …

Any term of this progression can produce a Pythagorean triplet, for instance:

4 4/9 = 40/9; the numbers 40 and 9 are the sides of a right triangle, and the hypotenuse is one greater than the largest side (40 + 1 = 41).

## A curious right triangle

The sides of a pentagon, hexagon, and decagon, inscribed in congruent circles, form a right triangle.