history of prime numbers

history of prime numbers

It is not clear when humans first pondered the mysteries of

prime numbers

. The

Ishango bone

suggests humans thought about prime numbers as long ago as twenty

thousand

years ago, because it includes a

prime quadruplet

, (11, 13, 17, 19). This could just be a coincidence as this also happens to be a

partition

of 60 into distinct

odd numbers

.

The evidence is more convincing for the ancient Egyptians, with their heavy emphasis on

unit fractions

(or

Egyptian fractions

). The Rhynd mathematical papyrus, dating to four thousand years ago, concerns itself with expressing

2

n

(with

n

an odd integer in the range

4

<

n

<

102

) as a sum of unit fractions. When

n

is prime, the unit fraction expansions are considerably harder to arrive at when

n

is prime.

But the ancient Greeks of about twenty-five hundred years ago often get the credit for being the first to study prime numbers for their own sake. Eratosthenes came up with the

sieve of Eratosthenes

, and Euclid proved many important basic facts about prime numbers which today we take for granted, such as that there are infinitely many primes. Euclid also proved the relationship between the

Mersenne primes

and the even perfect numbers.

With the Roman conquest of the Greeks, much of the written Greek knowledge was translated to Latin, or at least preserved. As the Greeks taught the Romans what they knew, they preserved Greek mathematical knowledge but made no further progress in the study of pure mathematics, such as prime numbers.

The Arab mathematicians of the Middle Ages studied the work of ancient Greek mathematicians but with the added advantage of a numerical system more amenable to computational work.

Thabit ibn Qurra

, for example, proved the relationship between consecutive prime

Thabit numbers

and

amicable pairs

.

Pierre de Fermat

stated an important

theorem

(now known as Fermat’s little theorem) which states that given a prime

p

and a

coprime

base

b

, the

congruence

b

p

-

1

≡

1

mod

p

holds true. Fermat also conjectured incorrectly that

2

2

n

+

1

is always prime (the

Fermat primes

) while Marin Mersenne incorrectly conjectured that in

2

p

-

1

the primes 67 and 257 give Mersenne primes. Nevertheless these numbers were still named after them.

By this time, the most common

definition

of prime number was “a number that is

divisible

by 1 and itself.” 1 fits this definition, but some mathematicians were troubled by the ways in which 1 is different from the other prime numbers. Euler, for example, was motivated by the fact that

σ

⁢

(

p

)

=

p

+

1

for prime

p

but not for

p

=

1

to regard 1 as not a prime number. Others were no more bothered by 1’s peculiarities than those of the number 2, such as

Moritz Stern

, who did not consider 3 a

Stern prime

because he considered 1 prime.

After several attempts at a formula for the

prime counting function

, the

prime number theorem

was proven in the 19th Century. The famous

Riemann hypothesis

was also stated and to this day remains unproven despite much evidence supporting it.

In the 20th century,

computers

gradually became important in calculating data for theorists to ponder; from the 13th Mersenne prime on all the largest primes since the middle of the century have been found with the help of computers. The invention of public key cryptography in the late 1970s has precipitated the need for larger prime numbers and motivated many advances in

integer factorization

algorithms. From the 1990s onwards,

distributed computing projects

like the

Great Internet Mersenne Prime Search

and

Seventeen or Bust

have discovered some of the largest known prime numbers.

References

1

Wells, D.

The Penguin Dictionary of Curious and Interesting Numbers

, London: Penguin Group. (1986): 20

2

Derbyshire, J.

Prime Obsession:

Bernhard Riemann

and the Greatest Unsolved Problem in Mathematics

, Penguin Group. (2004)

.1

External links

PrimeFan,

http://geocities.com/primefan/Prime1ProCon.html

Arguments for and against the

primality

of 1

Title

history of prime numbers

Canonical

name

HistoryOfPrimeNumbers

Date of creation

2013-03-22 17:23:39

Last modified on

2013-03-22 17:23:39

Owner

Mathprof (13753)

Last modified by

Mathprof (13753)

Numerical id

8

Author

Mathprof (13753)

Entry type

Topic

Classification

msc 11A41

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