# Number TheoryApplications Diophantine Equations

#### matqkks

##### Member
Are there any real life applications of linear Diophantine equations? I am looking for examples which will motivate students.

#### Opalg

##### MHB Oldtimer
Staff member
Are there any real life applications of linear Diophantine equations? I am looking for examples which will motivate students.
I guess the main real life application is to public key cryptography. It might be hard to come up with realistic examples at a sufficiently elementary level, but perhaps it would be possible to go some way in that direction.

#### eddybob123

##### Active member
Other than that, I don't think number theory has any practical uses in real life. (Except maybe calculating restaurant bills, HST, etc.) The main mathematical use of number theory is to "pave a road" into more advanced studies such as differential equations and abstract algebra, which themselves have countless applications in many scientific disciplines.

#### caffeinemachine

##### Well-known member
MHB Math Scholar
Other than that, I don't think number theory has any practical uses in real life. (Except maybe calculating restaurant bills, HST, etc.) The main mathematical use of number theory is to "pave a road" into more advanced studies such as differential equations and abstract algebra, which themselves have countless applications in many scientific disciplines.
Hey eddybob.

I'd disagree with you on this. I think Number Theory has a lot of applications. The RSA encryption, for example, is a product of number theory and to understand that one doesn't even need to read very advanced stuff. There are a lot of applications of number theory in Computer Science.

EDIT: I didn't see Opalg's post when I answered this so my response can be ignored. Oops!

#### chisigma

##### Well-known member
Are there any real life applications of linear Diophantine equations? I am looking for examples which will motivate students.
I do hope that Your students will be 'motivated' by the following 'brillant' application od diophantine equations that is datec from the Middle Ages. A well known fundamental theorem of the number theory is called 'Chinese Remainder Theorem' and it extablishes that if $n_{1}$ and $n_{2}$ are coprime, then the diophantine equation...

$\displaystyle x \equiv a_{1}\ \text{mod}\ n_{1}$

$\displaystyle x \equiv a_{2}\ \text{mod}\ n_{2}\ (1)$

... has one and only one solution $\text{mod}\ n_{1}\ n_{2}$. It is easy to demonstrate in a more general case that if $n_{1},\ n_{2},\ ...\ n_{k}$ are coprime, then the diophantine equation...

$\displaystyle x \equiv a_{1}\ \text{mod}\ n_{1}$

$\displaystyle x \equiv a_{2}\ \text{mod}\ n_{2}$

$\displaystyle ...$

$\displaystyle x \equiv a_{k}\ \text{mod}\ n_{k}\ (2)$

... has one and one solution $\text{mod}\ N= n_{1}\ n_{2}\ ...\ n_{k}$. All that is well known but may be it is not as well known why this theorem is called 'chinese'. The reason seems to be in the fact that in the old China the mathematical knowledge was 'patrimony' of the highest social classes and the rest of population was able to count till twenty and no more. Taking into account that, when a chinese general wanted to know the number of soldiers of one batalion he instructed the commander to marshal the soldiers first in rows of 7, then in rows of 11 and then in rows of 13 and any time to count the soldiers in the last row. The unknown number of soldiers can be ontained solving the diphantine equation (2) where $n_{1}=7,\ n_{2}= 11,\ n_{3}=13$ so that $N=n_{1}\ n_{2}\ n_{3}=1001$. The general procedure to solve (2) is the following...

a) we define for i=1,2,...,k $\displaystyle N_{i}= \frac{N}{n_{i}}$ and $\displaystyle \lambda_{i} \equiv N_{i}^{-1}\ \text{mod}\ n_{i}$

b) we compute directly...

$\displaystyle x = a_{1}\ \lambda_{1}\ N_{1} + a_{2}\ \lambda_{2}\ N_{2} + ...+ a_{k}\ \lambda_{k}\ N_{k}\ \text{mod}\ N\ (3)$

In the case of chinese generals is $\displaystyle N_{1}= 143,\ \lambda_{1} \equiv 5\ \text{mod} 7,\ N_{2}= 91,\ \lambda_{2} \equiv 4\ \text{mod}\ 11,\ N_{3}= 77,\ \lambda_{3} \equiv 12\ \text{mod}\ 13$ so that is...

$\displaystyle x \equiv 715\ a_{1} + 364\ a_{2} + 924\ a_{3}\ \text{mod}\ 1001\ (4)$

Kind regards

$\chi$ $\sigma$

#### mathbalarka

##### Well-known member
MHB Math Helper
eddybob123 said:
I don't think number theory has any practical uses in real life.
Of course there are. For example, in molecular physics and organic chemistry.