Here are several mathematical facts about the number 2018.

Is 2018 a Prime Number?

The number 2018 is an even number, so in particular 2018 is not a prime number.
The prime factorization of 2018 is
\[2018=2\cdot 1009.\] Here $2$ and $1009$ are prime numbers.

Identities for 2018

Here are some identities for 2018.

2018 is the sum of two squares

2018 is the sum of two squares:
\begin{align*}
2018=13^2+43^2.
\end{align*}
Here are some variants.
\begin{align*}
2018&=1^2+9^2+44^2\\
2018&=1^2+12^2+28^2+33^2\\
2018&=2^2+2^2+5^2+7^2+44^2\\
2018&=6^2+2(29)^2+3(10)^2
\end{align*}

2018 is a part of a Pythagorean triple

2018 is a part of a Pythagorean triple:
\begin{align*}
2018^2=1118^2+1680^2.
\end{align*}

This means that there is a right triangle whose hypotenuse is 2018.

Here are some variants.
\begin{align*}
2018^2&=18^2+88^2+2016^2\\
2018^2&=460^2+566^2+1172^2+1472^2\\
2018^3&=421^3+1490^3+1691^3
\end{align*}

2018 is the sum of 12 successive integers

2018 is the sum of 12 successive integers:
\[2018=7^2+8^2+9^2+10^2+11^2+12^2+13^2+14^2+15^2+16^2+17^2+18^2.\]

2018 appears in $\pi=3.14…$

2018 appears in the number $\pi=3.14…$ as in the following picture.

How about 2017?

Did you miss mathematical facts about 2017?

Check out

The post Mathematics About the Number 2018 first appeared on Problems in Mathematics.

Let $F$ be a finite field.
Prove that each element in the field $F$ is the sum of two squares in $F$.

Proof.

Let $x$ be an element in $F$. We want to show that there exists $a, b\in F$ such that
\[x=a^2+b^2.\]

Since $F$ is a finite field, the characteristic $p$ of the field $F$ is a prime number.

If $p=2$, then the map $\phi:F\to F$ defined by $\phi(a)=a^2$ is a field homomorphism, hence it is an endomorphism since $F$ is finite.( The map $\phi$ is called the Frobenius endomorphism).

Thus, for any element $x\in F$, there exists $a\in F$ such that $\phi(a)=x$.
Hence $x$ can be written as the sum of two squares $x=a^2+0^2$.

Now consider the case $p > 2$.
We consider the map $\phi:F^{\times}\to F^{\times}$ defined by $\phi(a)=a^2$. The image of $\phi$ is the subset of $F$ that can be written as $a^2$ for some $a\in F$.

If $\phi(a)=\phi(b)$, then we have
\[0=a^2-b^2=(a-b)(a+b).\] Hence we have $a=b$ or $a=-b$.
Since $b \neq 0$ and $p > 2$, we know that $b\neq -b$.
Thus the map $\phi$ is a two-to-one map.

Thus, there are $\frac{|F^{\times}|}{2}=\frac{|F|-1}{2}$ square elements in $F^{\times}$.
Since $0$ is also a square in $F$, there are
\[\frac{|F|-1}{2}+1=\frac{|F|+1}{2}\] square elements in the field $F$.

Put
\[A:=\{a^2 \mid a\in F\}.\] We just observed that $|A|=\frac{|F|+1}{2}$.

Fix an element $x\in F$ and consider the subset
\[B:=\{x-b^2 \mid b\in F\}.\] Clearly $|B|=|A|=\frac{|F|+1}{2}$.

Observe that both $A$ and $B$ are subsets in $F$ and
\[|A|+|B|=|F|+1 > |F|,\] and hence $A$ and $B$ cannot be disjoint.

Therefore, there exists $a, b \in F$ such that $a^2=x-b^2$, or equivalently,
\[x=a^2+b^2.\]

Hence each element $x\in F$ is the sum of two squares.

The post Each Element in a Finite Field is the Sum of Two Squares first appeared on Problems in Mathematics.

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.