Prime Numbers

< 8.5 Number Theory | Topic Index | 8.7 Fundamental Theorem of Arithmetic >

We have seen that whole numbers which cannot be divided by any number, other than 1 and the number itself, are called Prime Numbers. For example, 7 is a prime as its factors are only 1 &amp; 7. 6 is not prime since its factors are 1, 2, 3 &amp; 6.

Greeks discovered that any whole number can be written as a product of powers of prime numbers "in an unique way". This is called the Fundamental Theorem of Arithmetic which we will see in detail in the next chapter. For this reason, prime Numbers are considered the building blocks of all numbers.

Sieve of Eratosthenes

Prime numbers have been studied right from early days of mathematics. Eratosthenes, a Greek mathematician who lived in the 3rd century BCE, evolved a systematic way of locating all prime numbers up to any desired number.

You start with a grid of numbers from 1 to the desired number. The user systematically crosses out numbers which can be divided by prime numbers starting with 2, 3, 5 and so on. Ultimately only the prime numbers would be left uncrossed.

For obvious reasons, it was called the Sieve of Eratosthenes.

A lot of higher mathematics is based on primes. There are many theorems in Number Theory which involve Prime numbers. We will now see a few interesting facts about them.

1 is not prime

There are reasons in mathematics why 1 is not considered a prime. This is due to the basic theorem of arithmetic (refer to Chapter 8.7) which states that every number can be represented as a product of powers of prime numbers in an unique manner.

1 is a factor of every number and product of any number of 1s has the same value. So if 1 is taken as a prime, there cannot be an unique representation for any number. 6 can be written as 2 & 3 or 1, 2 & 3 or 1, 1, 2 & 3!. Hence mathematicians have decided that 1 will not be considered a prime.

2 is a prime as well as even

Every alternate whole number is an even number which has 2 as a factor. So for the sake of unique representation, 2 is considered as a prime number.

Also 2 obeys the basic condition for being a prime number. It has only 2 factors; 1 and itself which is 2!.

So 2 becomes the only even prime!

Goldbach&rsquo;s Conjecture -

In 1742, Christian Goldbach made the conjecture that "any even number can be written as the sum of 2 prime numbers". Even after analysing millions of numbers, no exception to this conjecture has been found.

At the same time, no one has been able to provide a proof as to the truth of this conjecture.

This is an example of a math idea which can be understood by a primary class student, whose proof is still eluding the best mathematical brains.

This is also an example of a proposition from Number Theory which we dealt with in the previous chapter.

Infinite number of primes

We know that whole numbers never end. There are infinite number of whole numbers. Is this true of prime numbers also?

Question like this reveal the abstract nature of math. And the fact that we can answer this question with logical thinking. This also shows the power of math.

Euclid gave a very elegant proof that there are an infinite number of primes. His proof showed that with the known primes, we can always construct another bigger prime.

Further, no detectable pattern has been seen in the sequence of primes. Finding a function which will predict how primes are distributed over the real number line is an intensely studied problem.

Use of Cryptography

Until 1970s prime numbers were studied only as exercises in mathematical logic.

With the development of computers & telecommunications vast amounts of financial and confidential data started being sent through them. Such data could easily be accessed by persons who know the technical workings of a computer network. Hence the issue of safety of the data became critical. One solution was to "code" the data using various techniques so that only those who knew the code could read them.

One technique could be to change a word like "Money" into another word "Nlmvb" using a simple rule. Each letter has been replaced by its counterpart reading the alphabet in the reverse direction. "y" is the 25th letter (2nd from last) and it is replaced by "b"(2nd from the first).

The rule is is called the "key" to the coding. Unless some one knows the key, the original word cannot be deciphered.

Such techniques were needed badly during World War II, for exchanging secret messages and developed into the mathematical science of Cryptography. Invention of computer software accelerated the development.

With the spread of digital transfers of data there rose a need to encrypt the data for secrecy, in real time. Many sophisticated systems of encrypting were invented.

Use of primes in Cryptography

When 2 large primes are multiplied to get a product, it is almost impossible, even using computers, to find the two original primes from the product.

The product of 2 very large (with hundreds of digits) prime numbers was used to generate the keys required to decrypt the data. The safety of this method rests on the fact that factorizing the product of such huge prime numbers was almost impossible even for superfast computers.

Largest identified prime

The use of primes in cryptography motivated the discovery of larger and larger primes. Even with a computer, the larger the number, more difficult the process to check if it is a prime.

Faster computers were used for this. Hence faster algorithms and computer programs were also developed to check if a number is prime.

About 30 years back mathematicians have started an ongoing project called the Great Internet Mersenne Prime Search, basically to discover larger & larger primes.

The largest known prime number discovered, as of July 2018, is 2 77,232,917  &minus; 1, a number with 23,249,425 digits.

Primes in the Search for Intelligent Life in the Universe

Scientists have made a reasonable guess that if there are other intelligent beings in the Universe, they would have discovered the basics of mathematics and would be aware of prime numbers. Hence one idea of discovering any intelligent life is to send a message containing the sequence of primes in terms of simple dots & dashes.

The first such effort was made in 1967 when the Arecibo radio telescope in Puerto Rico was inaugurated. A message of length 1679 was sent to a star cluster about 25,000 light years away. It was a pictograph involving the use of primes, designed by Frank Drake of Cornell University. 1679 was a semi-prime which was the product of 2 primes 73 X23. No reply has been received until now.

< 8.5 Number Theory | Topic Index | 8.7 Fundamental Theorem of Arithmetic >