Unless you understand the beauty of Prime numbers, you won’t run behind
them. Well, I leave it as an exercise for the reader to find their beauty. To get a
feel of how important Primes are, try writing a simple code to find if the given
number is Prime [Wisely choose your algorithm!]. Now, put a massive Prime
number (Yes, you can use the web to find these massive primes) having 20
digits and see what your code returns. If at all you perform this exercise
[though imaginary], I assume that you have got some insight into what I’m
trying to convince.
So, the principle which we use in our welfare is that Prime numbers do
not have any factors other than themselves! Thus, if you have a pair of 20-30
digit Primes then you may multiply them and use it to encrypt your data, and let
the intruder take years to find your data. I won’t go into details of encryption,
for it will make us deviate from the title.
If you have used the Euclid’s algorithm (Which reduces the complexity from
O (n) to O () for the primality search and if you have an inquisitive mind then
you may ask: Are there any more such algorithms? I’m glad you asked! And
I’m blushing here to explain them. You may refer, as I did, to the ‘so-called’
Black Bible of the Primes by my favourite author, Richard Crandall which bears
the name same as the title of this very article. Years before, Dr. Mahindra
Agarwal, a professor at IIT, Kanpur invented an algorithm which is proved to
have minimum complexity until now! With his two Ph. D students who
completed their Ph. D within 2 months!! What I want to highlight here is, as the
community over which this article aims is having some computational
background, I encourage them to work out these algorithms. One can use the
Parallel Programming approach (in some tests like the famous Lucas-Lehmer
test for Merssene Primes). This field is so much into work these days, see the
Bitcoin, the mining of these is rests its basement on Primes!.
If you are a student of Mathematics or not, it is a truth that Mathematics is a
tool, I say, in analyzing every physical phenomenon. If you know to use it well,
force is with you! If you don’t know to use it, say hello!!
The above article is just a crude description of what computational mathematics
aims at. To get a hands-on experience into it, I suggest referring following

1. https://www.youtube.com/watch?v=lEvXcTYqtKU