Happy New Year! The properties of 2023

When a mathematician studies a number, one of the first questions is how it factors. Is 2,023 a prime number?

Well, no: it factors as 2,023 = 7 × 17².

We can also note that 2,023 = 2,048 – 25 = 2¹¹ – 5².

Raphaël Douady takes the opportunity to express 2,023 using only the two digits 2 and 5:

2,023 = 2^5×2 × 2 – 5²

or alternatively: 2,023 = 2^{(52 – 2(5 + 2))} – 5²

or even: 2,023 = 2^{((5 – 2)2 + 2)} – 5².

Using only the two digits 3 and 1, we can also write:

2,023 = (33 – 1)⁽¹⁺¹⁾ + ((3+3)⁽¹⁺¹⁾ + 1) × 3³.

Now it is your turn to be inventive, like Dominique Souder, whose magic square appears on the following page.

"Kitano" for 2023 ------------------------

For several years, Tangente has been exploring ways of calculating the year number subject to rules devised by Japanese filmmaker Takeshi Kitano.

Kitano is not a mathematician but a film director. At the "Mathematics: A Beautiful Elsewhere" exhibition held in Paris in 2011 at the Fondation Cartier pour l’art contemporain (see Tangente 143, 2011), he set visitors a challenge. They had to find the shortest expression for the current year using the integers starting at 1 in order, together with:

• the usual operations chosen from +, ‒, ×, /,

• arbitrary powers or square roots,

• and, if desired, factorials and digit concatenations.

Any combination is allowed, but the main aim is to succeed using as few digits as possible. Here are some proposals for 2,023 from two Tangente contributors.

Alain Zalmanski (the record holder, with only six digits):

2,023 = 1×2 + (3!)⁴ + 5 + 6!,

2,023 = (1+ 2 + 3)⁴ + (5!) × 6 + 7,

2,023 = (((1+ 2)³ + 4) × 5) × (6 + 7) + 8,

2,023 = ((1 + 2 + 3 – 4) × 56 + 7) × (8 + 9).

François Lavallou has some proposals of his own:

2,023 = (‒1 + 234 + 56) × 7,

2,023 = (1 + 2³) × 4 × 56 + 7,

2,023 = 1234 × (‒5 + 6) +789.

2023 and the Josephus sieve ------------------------------------------

Beyond these playful considerations, one of the most remarkable properties of the integer 2,023 is that it is a lucky number: it is generated by a special "sieve" resembling the sieve of Eratosthenes (see, for example, Tangente 149, 2012), which generates the prime numbers.

Flavius Josephus (c. 37–c. 100), who was Jewish and born in Judea under Roman rule, commanded the military forces in Galilee during the war against the Romans before later settling in Rome. He is regarded as one of the foremost historical chroniclers of Greco-Roman antiquity. We owe to him a Greek account of the events and conflicts of his time between Rome and Jerusalem. His name is associated with a combinatorial problem illustrated by the tragic tale told in the box.

Inspired by this story, the Josephus sieve is defined as follows.

• Write down the sequence of integers: 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49…

• In the first step, eliminate every second number from the list. This leaves: 1, 3, 5, 7, 9, 11, 13, 15, 17, 19, 21, 23, 25, 27, 29, 31, 33, 35, 37, 39, 41, 43, 45, 47, 49…

• In the second step, eliminate every third number from this new list: 1, 3, 7, 9, 13, 15, 19, 21, 25, 27, 31, 33, 37, 39, 43, 45, 49…

• In the third step, eliminate every fourth number from the list: 1, 3, 7, 13, 15, 19, 25, 27, 31, 37, 39, 43, 49…

• Repeat this process indefinitely, eliminating every (n + 1)-th number at the n^th step.

The numbers that survive this sieve are, by definition, lucky numbers. Those below 100 are 1, 3, 7, 13, 19, 27, 39, 49, 63, 79 and 91.

Many more integers follow, including… 2,023. It is the fifty-first lucky number!

Although infinitely many lucky numbers exist, they seem much sparser than the primes. There are thirty-five lucky numbers below 1,000, compared with 168 primes. And the gap keeps widening: below 10,000, there are 112 lucky numbers and 1,229 primes; below 100,000, there are 357 lucky numbers and 9,592 primes.

The asymptotic distribution of lucky numbers differs from that of the prime numbers. As n tends to infinity, the number of lucky numbers below n is asymptotic to $2\sqrt{{\displaystyle \frac{n}{\pi }}}2\; \backslash sqrt\{\backslash dfrac\{n\}\{\backslash pi\}\}$, whereas the prime number theorem tells us that the number of primes below n is asymptotic to ${\displaystyle \frac{n}{\ln n}}\backslash dfrac\{n\}\{\backslash ln\; n\}$ (where ln denotes the natural logarithm): the latter grows faster than the former as n increases.

It may seem surprising that such numbers have been studied. They were introduced by the Polish-born American mathematician Stanislaw Ulam (1909–1984), in particular to distinguish which properties of prime numbers result from their being generated by a sieve and which are unique to them. Ulam is also known for having been part of the team that developed the H-bomb during the Second World War. Thanks to him, we can chalk up a delightful property to 2023—perhaps that will make the year ahead all the better!