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.

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.

Here is the proof.

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!!

The post Mathematics About the Number 2017 first appeared on Problems in Mathematics.