# Practical applications of number theory?

## Main Question or Discussion Point

I am currently a college student who just started taking my first number theory course this week, and with all the stuff I am learning my only question is... are there practical applications to number theory?

I mean there are many theories in physics yet many of those have been put to practical use, but does the same apply to number theory? If so, then what situations could it apply too in the real world? I'm an engineering major so I'm really curious in knowing how this will help me in future engineering maths or the real world.

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alu's

mathman
I am currently a college student who just started taking my first number theory course this week, and with all the stuff I am learning my only question is... are there practical applications to number theory?

I mean there are many theories in physics yet many of those have been put to practical use, but does the same apply to number theory? If so, then what situations could it apply too in the real world? I'm an engineering major so I'm really curious in knowing how this will help me in future engineering maths or the real world.
Much of modern day cryptography uses number theory.

I've heard some of the divisibility algorithms are used in computer science...

Otherwise, though, number theory is a field of math which is very "pure".

Rather unfortunately, parts of number theory (even those once held as the pinnacle of esoteric) has succumbed to trivial applications.

The most immediate is in cryptography and cryptanalysis, then there are computer algorithms as was said and there are even, surprisingly enough, connections to physics! (mainly in the area of analytic regularizations in quantum theories)

I think data compression as well. From data storage to data transmission, that is billions of dollars of a big deal.

Rather unfortunately, parts of number theory (even those once held as the pinnacle of esoteric) has succumbed to trivial applications.

The most immediate is in cryptography and cryptanalysis, then there are computer algorithms as was said and there are even, surprisingly enough, connections to physics! (mainly in the area of analytic regularizations in quantum theories)
quoting Yasiru, Number theory has lots of applications in physics

- Selberg Trace formula, is very similar to Gutzwiller formula for the trace of resolvent
$$(E-H+i\epsilon)^{-1}$$

- Negative values of Zeta function $$\zeta (-m)$$ or Bernoulli Numbers (related to negative value of Zeta) can be used to compute divergent series and integrals

- Combinatorial problems of Number theory has application to Bose-Einstein or Fermi-Dirac statistics

to know more: http://secamlocal.ex.ac.uk/people/staff/mrwatkin/zeta/physics.htm

Wow, mhill I have to say the link you gave definitely answered all my questions.

I really appreciate everyones responses. I'm an engineering major taking number theory because I wanted to strengthen my math skills and I always think that's a point that all engineers need to work on. However, being as theoretical as the course was I got more than what I expected and kind of got lost in it all. So by finding out all the practical applications everyone has given me I have definitely gained a better appreciation for the course.

Yeah the link mhill gave has it all; as an engineer explicit number theory knowledge will only come in useful in telecommunication and perhaps computer science. The uses in physics are largely theoretical, so unless you get into a physics technology sector chances of coming across applications of number theory is still pretty low.