number 8

This is delivery number 416 of The game of sciencewhich means that it has been continuously appearing for 8 years, week after week, in the pages of SUBJECT. A good opportunity to thank, once again, the readers who, with their numerous and enriching comments, have made this section more than just a popularization column and mathematical pastimes.

The number 416 is not particularly interesting, but 8 is not wasted: it is a perfect cube (the smallest after the trivial case of 1), it is the only positive power that differs by one unit from another positive power, it is a Fibonacci number , it is a Leyland number, it is a cake number, it is a tau number, it is a panarithmic number… And, lying down, it represents infinity.

In 1884, the Belgian mathematician Eugène Catalan (he of the numbers that bear his name, which we have dealt with on more than one occasion) conjectured that 8 and 9 (2³ and 3²) were the only powers of consecutive natural numbers. The conjecture was proved in 2002 by the Romanian mathematician Preda Mihailescu, so Catalan’s former conjecture is now called Mihailescu’s theorem.

8 is the sixth Fibonacci number: 1, 1, 2, 3, 5, 8… Is there another term of the sequence that, like 8 —and not counting the trivial case of 1— is a perfect cube?

Leyland numbers (after British mathematician Paul Leyland) are those of the form xʸ + yᵡ, where x and y are integers greater than 1, not necessarily distinct. The first of them is, therefore, 2² + 2² = 8. The first Leyland numbers are:

8, 17, 32, 54, 57, 100, 145, 177, 320…

Why do you think 1 is excluded for the values ​​of x and y?

A pie number of order n is the maximum number of regions into which a cube can be divided by n planes. The name comes from a well-known riddle (which we have dealt with at some point) that asks to divide a cake into 8 equal parts with only 3 cuts. And 8 is therefore the pie number of order 3. The first pie numbers are:

1, 2, 4, 8, 15, 26, 42, 64, 93…

The first term, 1, corresponds to 0 planes, that is, to the null partition. Can you find a general formula for the pie numbers?

A tau number or refactorable number is one that is divisible by the number of divisors it has (including 1 and the number itself). Since 8 has four divisors (1, 2, 4, and 8) and is divisible by 4, 8 is a refactorable number. The first tau numbers are:

1, 2, 8, 9, 12, 18, 24, 36, 40…

Regarding the use of a lying 8 as a symbol of infinity (∞), dates back to the 17th century. It was the English mathematician John Wallis, precursor of the infinitesimal calculus, who used it for the first time, apparently inspired by the Greek symbol for ouroboros, the snake that bites its tail as a representation of an endless cycle.

On the other hand, we must not forget the recurring presence of 8 in geometry (and very specifically in relation to the Platonic solids and hypersolids): 8 is the number of symmetries of a square, the number of vertices of a cube, the number of faces of an octahedron, the number of cells of a hypercube, there are 8 convex deltahedrons (among them the regular tetrahedron, octahedron and icosahedron)… All of which is enough for a few more articles.

And surely my shrewd readers will discover other notable features of the inexhaurable number 8.

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