In the ring
\[\Z[\sqrt{2}]=\{a+\sqrt{2}b \mid a, b \in \Z\},\] show that $5$ is a prime element but $7$ is not a prime element.

Hint.

An element $p$ in a ring $R$ is prime if $p$ is non zero, non unit element and whenever $p$ divide $ab$ for $a, b \in R$, then $p$ divides $a$ or $b$.

Equivalently, an element $p$ in the ring $R$ is prime if the principal ideal $(p)$ generated by $p$ is a nonzero prime ideal of $R$.

Proof.

5 is a prime element in the ring $\Z[\sqrt{2}]$.

We first show that $5$ is prime in the ring $\Z[\sqrt{2}]$.
Suppose that
\[5|(a+\sqrt{2}b)(c+\sqrt{2}d)\] for $a+\sqrt{2}b, c+\sqrt{2}d \in \Z[\sqrt{2}]$.
By taking the norm, we obtain
\[25| (a^2-2b^2)(c^2-2d^2)\] in $\Z$.
From this, we may assume that $5|a^2-2b^2$.
Now look at the following table.

\begin{array}{ |c|c|c|c| }
\hline
a, b & a^2, b^2 \pmod{5} & 2b^2 \pmod{5} \\
\hline
0 & 0 & 0 \\
1& 1 & 2 \\
2& 4 & 3 \\
3 & 4 & 3\\
4 & 1 & 2\\
\hline
\end{array}

From this table, we see that $a^2-2b^2=0 \pmod{5}$ if and only if $a, b$ are both divisible by $5$.
Therefore $5|a+\sqrt{2}b$, and $5$ is a prime element in $\Z[\sqrt{2}]$.

7 is not a prime element in the ring $\Z[\sqrt{2}]$.

Next, we show that $7$ is not a prime element in $\Z[\sqrt{2}]$.
To see this, note that we have
\[7=(3+\sqrt{2})(3-\sqrt{2})\] and $7$ does not divide $3+\sqrt{2}$ and $3-\sqrt{2}$.
Hence $7$ is not a prime element in the ring $\Z[\sqrt{2}]$.

Related Question.

For a proof of this fact, see that post “The Ring $\Z[\sqrt{2}]$ is a Euclidean Domain“.

The post 5 is Prime But 7 is Not Prime in the Ring $\Z[\sqrt{2}]$ first appeared on Problems in Mathematics.

Find the largest prime number less than one million.

What is a prime number?

A natural number is called a “prime number” if it is only divisible by $1$ and itself.
For example, $2, 3, 5, 7$ are prime numbers, although the numbers $4,6,9$ are not.

The prime numbers have always fascinated mathematicians.
There are a lot of unsolved problems related to prime numbers.

There are many special types of prime numbers named after famous mathematicians.
My favorites are Mersenne primes, Fermat primes, and Wagstaff primes.

• A natural number of the form
  \[2^n-1\] is called a Mersenne number.
• A Mersenne prime is a prime number of the form
  \[2^p-1.\]
• A natural number of the form
  \[2^{2^n}+1 \] is called a Fermat number.
• A Fermat prime is a prime number of the form
  \[2^{2^n}+1.\]
• A Wagstaff prime is a prime number of the form
  \[\frac{2^p+1}{3}.\]

Unsolved problems

For these prime numbers the followings are still unknown.

• Are there infinitely many Mersenne/Fermat/Wagstaff prime numbers?
• Are there infinitely many nonprime Fermat numbers?
• Are there infinitely many composite Mersenne number $2^p-1$ for a prime $p$?

What is the largest prime number less than one million.

It is known for a long time (Euclid’s Elements (circa 300 BC)) that there are infinitely many primes.

Here are the first $95$ prime numbers. These are all prime numbers less than $500$.

2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61,
67, 71, 73, 79, 83, 89, 97, 101, 103, 107, 109, 113, 127, 131, 137,
139, 149, 151, 157, 163, 167, 173, 179, 181, 191, 193, 197, 199, 211,
223, 227, 229, 233, 239, 241, 251, 257, 263, 269, 271, 277, 281, 283,
293, 307, 311, 313, 317, 331, 337, 347, 349, 353, 359, 367, 373, 379,
383, 389, 397, 401, 409, 419, 421, 431, 433, 439, 443, 449, 457, 461,
463, 467, 479, 487, 491, 499.

List of prime numbers less than one million.

In fact, there are $78,498$ prime numbers less than $1,000,000$=one million.
To list them here takes a lot of space, so I created a PDF file of the list of primes less than one million.

It takes $95$ pages just to list $78498$ prime numbers less than one million.

Prime numbers less than one million

From this list, we see that

the largest prime numbers less than one million is $999983$.
(The last number in the PDF file.)

Other Facts

Here are several facts that we can find from the list (with time and energy)

• The largest twin prime pair less than one million is $999959$ and $999961$.
• The second largest twin prime pair less than one million is $999611$ and $999613$.
• The third largest twin prime pair less than one million is $999431$ and $999433 $.
• There are 7 Mersenne primes less than one million. These Mersenne primes are
  \[3, 7, 31, 127,8191, 131071, 524287.\]
• The know Fermat prime numbers are all less than one million.These are
  \[ 3, 5, 17, 257, 65537.\]
• $11$ is the only prime number containing only the decimal digit 1 and less than one million. (The second largest such prime is $1111111111111111111$.)
• Wagstaff prime numbers less than one million are \[3, 11, 43, 683, 2731, 43691, 174763.\]

Try to find an interesting property of prime numbers from the list of primes <100000.

The post Find the Largest Prime Number Less than One Million. first appeared on Problems in Mathematics.