This is delivery number 416 from The Game of Sciencewhich means that it has appeared in the pages of week in and week out for the past 8 years THEME. A good opportunity to once again thank the readers who, with their numerous and enriching comments, have helped make this section more than just a popularization column and mathematical pastime.
Dragon (1952) directed by MC Escher.
The number 416 isn’t particularly interesting, but 8 isn’t a waste: it’s a perfect cube (the smallest after the trivial case of 1), it’s the only positive power that differs by one unit from any other positive power, it is a Fibonacci number, it’s a Leyland number, it’s a pie number, it’s a Tau number, it’s a panarithmic number… And lying flat, it represents infinity.
In 1884, the Belgian mathematician Eugène Catalan (the one of the numbers bearing his name, which we have considered more than once) conjectured that 8 and 9 (2³ and 3²) were the only powers of consecutive natural numbers. The conjecture was proved by the Romanian mathematician Preda Mihailescu in 2002, which is why Catalan’s earlier conjecture is now called Mihailescu’s theorem.
8 is the sixth Fibonacci number: 1, 1, 2, 3, 5, 8… Is there another term in the sequence that, like 8—and not counting the trivial case of 1—is a perfect cube?
Leyland numbers (named after British mathematician Paul Leyland) are those of the form xʸ + yᵡ, where x and y are integers greater than 1 that are not necessarily distinct. The first of these 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 circle number of order n is the maximum number of regions into which a cube can be divided by n levels. The name comes from a well-known riddle (which we dabbled in at some point) involving dividing a cake into 8 equal parts with just 3 cuts. And 8 is the circle number of order 3. The first circle numbers are:
1, 2, 4, 8, 15, 26, 42, 64, 93…
The first term, 1, corresponds to 0 levels, i.e. the zero partition. Can you find a general formula for circle numbers?
A tau number, or refactorable number, is a number that is divisible by the number of its divisors (including 1 and the number itself). Because 8 has four factors (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…
An 8 is in the tarot.
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 calculus, who first used it, apparently taking inspiration from the Greek symbol of uroboros, the serpent biting its tail, representing 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 The number of faces of an octahedron, the number of cells in a hypercube, there are 8 convex deltahedrons (including the regular tetrahedron, octahedron, and icosahedron)… All of that is enough for a few more articles.
And surely my astute readers will discover other remarkable features of the inexhaustible number 8.
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