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.

It was discovered that any whole number can be written as a product of powers of prime numbers &ldquo;in an unique way&rdquo;. 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.

There are many theorems in Number Theory which involve Prime numbers. We will now see a few interesting facts about them.

Goldbach&rsquo;s Conjecture -

In 1742, Christian Goldbach made the conjecture that &ldquo;any even number can be written as the sum of 2 prime numbers&rdquo;. Though the above statement has not been proved mathematically to be correct, until today, even after analysing millions of numbers, no exception to the rule has been found.

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

Infinite number of primes

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.

Use of prime numbers in cryptography

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

With the development of computers &amp; telecommunications vast amounts of financial and confidential data were 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 &lsquo;code&rsquo; the data using various techniques so that only those who knew the code could read them.

One simple technique could be to change a word like &ldquo;Money&rdquo; into another word &ldquo;Nlmvb&rdquo;. Here each letter has been replaced by its counterpart reading the alphabet in the reverse direction. &ldquo;y&rdquo; is the 25thletter (2ndfrom last) and it is replaced by &ldquo;b&rdquo; (2ndfrom the first). This is called the &ldquo;key&rdquo; to the coding. Unless some one knows the key, she cannot decipher the original word.

These techniques were needed badly during World War II and developed into the mathematical science of Cryptography. Invention of computer software accelerated the development.

Many sophisticated systems of coding were invented. One of them was the use of 2 very large prime numbers. The keys were the product of these prime numbers and one of the primes. The safety of this method rested on the fact that finding 2 prime factors of very large 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. Faster computers were used for this. Even with a computer, the larger the number, more difficult is the process to check if it is a prime. Hence faster algorithms and computer programs were also developed to check if a number is prime.

The largest known prime number discovered, as of July 2018, is 277,232,917 &minus; 1, a number with23,249,425 digits.

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