This is delivery number 416 science gamesWhich means it’s been appearing in the front pages, week after week, for 8 consecutive years. Subject, This is a good opportunity to thank once again the readers who, with their numerous and rich comments, have made this volume more than just a popularization column and mathematical pastimes.
The number 416 is not particularly interesting, but 8 is not pointless: it is a perfect cube (smallest after the trivial case of 1), it is the only positive power that differs from one unit to another positive power, it is a It’s a Fibonacci number, it’s a Leyland number, it’s a Keck number, it’s a tau number, it’s a panarithmic number… and, lying down, it represents infinity.
In 1884, the Belgian mathematician Eugène Catalan (among those numbers named after him we’ve dealt with on more than one occasion) conjectured that 8 and 9 (2³ and 3²) were the only powers of consecutive natural numbers. This conjecture was proved in 2002 by the Romanian mathematician Preda Mihalescu, so Catalan’s earlier conjecture is now called Mihalescu’s theorem.
8 is the sixth Fibonacci number: 1, 1, 2, 3, 5, 8… Is any other term of the sequence such that 8—and not counting the trivial case of 1—is a perfect cube?
Leyland numbers (after British mathematician Paul Leyland) are 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 pi number of order n is the maximum number of sectors into which a cube can be divided by n faces. The name comes from a famous riddle (which we tackled at some point) that asks to divide a cake into 8 equal parts with only 3 cuts. And 8 is therefore a pi number of order 3. The first Pi numbers are:
1, 2, 4, 8, 15, 26, 42, 64, 93…
The first term, 1, corresponds to the 0 planes, which stands for zero division. Can you find a general formula for the number pi?
A tau number or refactorable number is one that is divisible by any number of its divisors (including 1 and the number itself). Since 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…
With regard to the use of the lying 8 as a symbol for infinity (), dates back to the 17th century. It was the English mathematician John Wallis, a pioneer of trivial calculus, who first used it, apparently inspired by the Greek symbol for it. Ouroboros, the snake that bites its tail as a representation of an endless cycle,
On the other hand, we should not forget the recurring appearance of 8 in geometry (and especially with regard to 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, the number of cells of a hypercube, are 8 convex deltahedrons (among them the regular tetrahedron, octahedron and icosahedron) … which is enough for a few more articles.
And surely my astute readers will discover other remarkable features of the unbreakable number 8.
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