While asleep, I had an unusual experience. There was a red screen formed by flowing blood, as it were. I was observing it. Suddenly a hand began to write on the screen. I became all attention. That hand wrote a number of elliptic integrals. They stuck to my mind. As soon as I woke up, I committed them to writing.

S. Ramanujan

Welcome to the blog Math1089 – Mathematics for All.

2022 is about to pass, and 2023 is coming. This is the moment to leave the old and welcome the new. Let’s hope that 2023 will

add the joys;

subtract the sorrows;

multiply the happiness; and

divide the love

among your loved ones. Happy New Year!

2023

In this blog post, consider the number 2023 and let us discuss the mathematical beauties of this number.

The factors of 2023 are 1, 7, 17, 119, 289 and 2023. The prime factors are 7 and 17, whereas the prime factorisation of 2023 is 7 × 17².

The corresponding representation of 2023 in Roman numeral is MMXXIII and in binary is (11111100111)₂.

Various Representations of 2023

Using the digits 1 to 9 and basic mathematical operations

Below are representations of 2023 using the digits 1, 2, . . . , 9 and basic mathematical operations like addition, subtraction, multiplication etc.

2023 = 12 × 3 × (4 + 5) × 6 + 7 + 8 × 9

2023 = 9 × 8 + 7 + 6 × 54 × 3 × 2 × 1

Using the numbers 1 to 10 and basic mathematical operations

Below are representations of 2023 using the numbers 1, 2, . . . , 10 and basic mathematical operations like addition, subtraction, multiplication, exponentiation etc.

2023 = 12 × (3 × 4 + 5) × 6 + 789 + 10

2023 = 10 + 98 + (7 + 6) × 5 + 43² + 1

Using the digits 1 to 9 and various powers

We can represent 2023 using the digits 1 to 9, where the bases are in increasing order and powers in decreasing order.

2023 = 1⁹ – 2⁸ + 3⁷ + 4⁶ – 5⁵ – 6⁴ + 7³ + 8² + 9¹

As the sum of digits

If we divide 2023 by 289, we get 7 as the answer which is same as 2 + 0 + 2 + 3, the sum of the digits of the number.

Using all the digits of the number

The digits of the number are 0, 2 and 3. Using these, we can write

2023 = (2 + 0 + 2 + 3)(2² + 0² + 2² + 3²)²

Using algebraic identities

Applying the difference of squares identity, we have

As a sum of squares

We can write 2023 as the sum of squares in the following ways:

2023² = 952² + 1785²

2023² = 1127² + 1680²

As a sum of cubes

We can write 2023 as the sum of cubes in the following way:

2023 = 2³ + 5³ + 6³ + 7³ + 11³

As various powers of 2

It’s easy to represent 2023 as the sum/ difference of various powers of 2. Here are few examples.

2023 = 2¹¹ ‒ 2⁵ + 2² + 2¹ + 2⁰

2023 = 2¹⁰ + 2⁹ + 2⁸ + 2⁷ + 2⁶ + 2⁵ + 2² + 2¹ + 2⁰

2023 = 2¹⁰ + 2⁹ + 2⁸ + 2⁷ + 2⁶ + 2⁵ + 2⁴ ‒ 2³ ‒ 2² + 2¹ + 2⁰

Using factorials

Allowing for factorial representation, another way to represent is 

As a trigonometric equation

If cos x₁ + cos x₂ + ···+ cos x₂₀₂₃ = 0, then

sin x₁ + sin x₂ + ···+ sin x₂₀₂₃ = 2023.

As a limit

We can write 2023 as a limit in the following way:

As a definite integration

Allowing for definite integral representation, another way to represent is 

Using matrices and determinants

If we consider a square matrix M, its determinant value is 2023.

As a continued fraction

We have the following identity:

Repeated application of this identity yields

As a result, we can write

As an infinite series

We can also represent 2023 as the sum of an infinite series. For example

In fact, a simple computation shows that

The cube of 2023 is 8279186167. We can write the same number by rearranging the digits in their order and using various mathematical operations. As a result, we have the following number relationship:

2023³ = 8279186167 = (82 + 79 + 1861 − 6 + 7)³

Single Digit Representation

A single-digit representation of any number is always a challenge. The same goes when we represent 2023 using only the digits 1 or 2 or 3 . . .  or 9 etc. Here, we are presenting four types of representations, but the reader can explore more. During your explorations, if you find something not listed here, feel free to mail us or add yours using the comments.

Representation of 2023 using the digit 1

A few representations using only 1’s are given below:

Representation of 2023 using the digit 2

Similar representations using only 2’s are given below:

Representation of 2023 using the digit 3

Using only 3’s, a few representations are given below:

Representation of 2023 using the digit 4

A few representations using only 4’s are given below:

Representation of 2023 using the digit 5

Similar representations using only 5’s are given below:

Representation of 2023 using the digit 6

Using only 6’s, a few representations are given below:

Representation of 2023 using the digit 7

A few representations using only 7’s are given below:

Representation of 2023 using the digit 8

Similar representations using only 8’s are given below:

Representation of 2023 using the digit 9

Using only 9’s, a few representations are given below:

During your explorations, if you find something not listed here, feel free to mail us or add yours using the comments. Your suggestions are eagerly and respectfully welcome! See you soon with a new mathematics blog that you and I call “Math1089 – Mathematics for All!“.

References:

Inder J. Taneja: 23 and 2023 in Numbers and Patterns; https://inderjtaneja.com