2024: a new year with many properties

With climate change in mind, will 2024 be apocalyptic? No one knows, but we all hope that the world depicted by Jean Dutourd in his dystopian novel of the same name (2024, Gallimard, 1975) will not become reality. The integer 2024, however, is apocalyptic! The decimal expansion of 2²⁰²⁴, which has 610 digits, contains the sequence 666 starting at the 471^st digit after the decimal point. A number n is called apocalyptic if the decimal expansion of 2*ⁿ contains the sequence 666. The number 666, which appears in the Book of Revelation, the last book of the New Testament, is called the number of the Beast* (see box), because it symbolizes the Beast, though there is no clear explanation of what the Beast represents: the Roman emperor Nero, who persecuted the early Christians, the Antichrist, Satan… All this is a very long way from mathematics.

2024 with oranges ---------------------

Stack spheres in a pyramid with a triangular base, as you would stack oranges. A pyramid with twenty-two layers requires exactly 2,024 spheres.

Better still, 2024 is a figurate number (to learn more about this class of integers, visit the page "Figurate numbers", where an entire feature is devoted to them!). It is the eighth dodecahedral number, given by the following formula:

2024 = \dfrac{n(3n-1)(3n-2)}{2} = \begin{pmatrix} 3n\\\ 3\[0,5mm] \end{pmatrix}

This is the number of regularly spaced points in a regular dodecahedron when eight points are placed on each edge. The regular dodecahedron is one of the five Platonic solids, and its faces are regular pentagons: it has twenty vertices, thirty edges and twelve faces. In Plato's symbolism, it represented the Whole because it "came closest to the sphere." Aristotle later named it aether, maintaining that the universe was made from this element and that it contained all the others: water, earth, fire and air, each represented by another Platonic solid.

Untouchable but practical! -------------------------------------

Let's return to apocalyptic numbers, which should not be confused with another class of integers known as Apocalypse numbers: these must themselves contain the sequence 666 in their decimal representation. There are infinitely many apocalyptic numbers, and it is conjectured that every integer n satisfying n ≥ 29,785 is apocalyptic. Clearly, then, this property is not rare. The nearest neighbors of 2024 with this property are 2015, 2019 and 2021 below it, and 2025, 2032 and 2033 above it.

The number of the Beast ---------------------

The number 666 has played a part in many religious conflicts throughout history. In the 16^th century, for example, Catholics and Protestants each used it to accuse the other of embodying the Antichrist. Christians later used it against Muslims. Each time, the name of the "enemy"—Luther, the pope or Muhammad—was converted into numbers, producing 666.

Today, many properties involving this number are explored by "amateur" mathematicians, doubtless more for fun and as a challenge than out of any particular belief… For example:

• it can be written as a sum of numbers that use every digit: 666 = 1 + 2 + 3 + 4 + 567 + 89 = 123 + 456 + 78 + 9;

• in Roman numerals, it uses each symbol below M exactly once, in descending order: 666 = DCLXVI;

• it is the sum of the squares of the first seven prime numbers: 666 = 2² + 3² + 5² + 7² + 11² + 13² + 17²;

• it is a striking sum of cubes: 666 = 1³ + 2³ + 3³ + 4³ + 5³ + 6³ + 5³ + 4³ + 3³ + 2³ + 1³;

• it is equal to the sum of its digits and the cubes of its digits: 666 = 6 + 6 + 6 + 6³ + 6³ + 6³;

• it is related to the golden ratio:

\sin(666^\circ) = -\dfrac{\phi}{2}

where

\phi = \dfrac{1+ \sqrt{5}}{2}

denotes the golden ratio.

2024: a new year with many properties

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