My view on Number Theory

[Steven G]

As a student I chose not to study number theory as I though that it would be boring compared to say algebra classes. Having been studying number theory for a while now I can say that this subject is not exciting at all! However, I am still enjoying studying this subject as I am learning about numbers. Numbers is the most basic part of math and I am realizing, and of course this is all my opinion, that exciting or not one really needs to know about numbers. I have some learned some interesting facts about numbers which I never knew. Number theory, to me is not exciting but very interesting! I have never studied such a math subject like this before.

What are your opinions of this subject?

 

[blamocur]

I too find number theory generally unexciting most of the time. But I'm fascinated by a number of pretty simple theorems with very difficult proofs, like Great Fermat Theorem, the greatest of them all. Also, modular arithmetic leads one to finite fields and commutative rings theories, both very elegant parts of math, at least to my under-educated eye.

 

[khansaheb]

Some other great proof is:

number of "irrational number" is greater than the number of rational number.

For number of rational "prime number" , there is no upper bound (another name for infinity).

Another very simple statement (I do not know any proof of this statement)

12³ + 1³ = 10³ + 9³ [=1729 is the smallest integer, that can be expressed as sum of cubes of two sets of integers (10,9) & (12,1)] ​

 

[blamocur]

number of "irrational number" is greater than the number of rational number.

I would not consider this as a number theory result: it does not use or refer to any arithmetic properties of rational/integer numbers (https://en.wikipedia.org/wiki/Cantor's_diagonal_argument).