# Find the Largest Prime Number Less than One Million.

## Problem 90

Find the largest prime number less than one million.

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Contents

## 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.

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.

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