# Find the Largest Prime Number Less than One Million.

## Problem 90

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.

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1. 01/01/2017

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