# Mathematics About the Number 2017

Happy New Year 2017!!

Here is the list of mathematical facts about **the number 2017** that you can brag about to your friends or family as a math geek.

Contents

## 2017 is a prime number

Of course, I start with the fact that **the number 2017 is a prime number**.

The previous prime year was **2011**.

The next prime year is **2027** and it is actually a twin prime year (2027 and 2029 are both primes).

- 2017th prime number is 17539.
- Combined number 201717539 is also prime.
- Yet combined number 175392017 is composite.
- 2017 is 306th prime number. $306=2\cdot 3^2\cdot 17$ contains a prime factor 17.
- 2017+2+0+1+7=2027 is the next prime year.

You may find more prime years from the list of one million primes that I made.

### 2017 is not a Gaussian prime

The number 2017 is congruent to 1 mod 4. (When we divide 2017 by 4, the remainder is 1.)

Such a number can be factored in the ring of **Gaussian integers** $\Z[i]$, where $i=\sqrt{-1}$. Explicitly we have

\[2017=(44+9i)(44-9i).\]

### 2017 is not an Eisenstein prime

The number 2017 can be factored in the ring of **Eisenstein integers** $\Z[\omega]$, where $\omega=e^{2\pi i/3}$ is a primitive third root of unity, as

\[2017=(-7-48\omega^2)(41+48\omega^2).\]

## 2017 is a sum of squares

We can write 2017 as a **sum of two squares**:

\[2017=44^2+9^2.\]

## 2017 is a part of Pythagorean triple

A triple $(a, b, c)$ of integers is called a **Pythagorean triple** if we have

\[a^2+b^2=c^2.\]
The triple

\[(1855, 792, 2017)\]
is a Pythagorean triple because we have

\[1855^2+792^2=2017^2.\]

(To obtain these numbers note that in general for any integers $m>n>0$, the triple $(a, b, c)$, where

\[a=m^2-n^2, b=2mn, c=m^2+n^2\]
is a Pythagorean triple by Euclid’s formula.

Since we know $2017=44^2+9^2$, apply this formula with $m=44, n=9$.)

A Pythagorean triple $(a, b, c)$ is said to be **primitive** if the integers $a, b, c$ are coprime. A Pythagorean triple obtained from Euclid’s formula is primitive if and only if $m$ and $n$ are coprime. In our case, $m=44$ and $n=9$ are coprime, the Pythagorean triple $(1855, 792, 2017)$ is primitive.

By the way, **Carl Friedrich Gauss** passed away on February 23rd 1855.

(Reference: Wikipedia Carl Friedrich Gauss.)

## 2017 is a sum of three cubes

The number 2017 can be expressed as **a sum of three cubes of primes**:

\[2017=7^3+7^3+11^3.\]

## 2017 appears in $\pi$

The number 2017 appear in the decimal expansion of $\pi=3.1415…$.

Look at the last four numbers of $\pi=3.1415…2017$ truncated to $8900$ decimal places.

The number 2017 does not appear in the decimal expansion of $2017^{2017}$.

## Exam problem using 2017

Let

\[A=\begin{bmatrix}

-1 & 2 \\

0 & -1

\end{bmatrix} \text{ and } \mathbf{u}=\begin{bmatrix}

1\\

0

\end{bmatrix}.\]
Compute $A^{2017}\mathbf{u}$.

This is one of the exam problems at the Ohio State University.

Check out the solutions of this problem here.

## How many prime numbers are there?

2017 is a prime number. How many prime numbers exist?

In fact, there are infinitely many prime numbers.

Please check out the post

A One-Line Proof that there are Infinitely Many Prime Numbers.

As the title suggests, the proof is only in one-line.

## More fun with 2017?

If you know or come up with more interesting properties of the number 2017, please let me know.

I hope 2017 will be a wonderful year for everyone!!

Add to solve later

2019 is not prime. I presume this is a misprint for 2027 and 2029.

Dear George Jelliss,

Yes, that’s right. Thank you for pointing out the misprint. I modified the post.