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.
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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.
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!!
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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.