Find the Largest Prime Number Less than One Million.

Ohio State University Mathematics 1

Problem 90

Find the largest prime number less than one million.
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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.

the list of 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 Response

  1. 01/01/2017

    […] may find more prime years from the list of one million primes that I […]

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