A long time ago, the Chinese introduced one of the best applications of prime numbers. Since there were no electronic counters or anything like that, Chinese used an algorithm to count how many soldiers were present in the army.
To find the unknown number of soldiers, X, the entire army is organized in rows, p1 and p2 that are relatively prime, also meaning that their greatest common denominator is 1. After organizing in p1 rows, there will be a1 remainders and when organized in p2 rows there will be a2 remainders. This can also be represented with the equality:
X=a1(mod p1)
X=a2(mod p2)
X=a3(mod p3)
After this step we find M M=(p1)(p2)(p3)
M1=M/p1
M2=M/p2
M3=M/p3
These equations will give us the values of Y1, Y2 and Y3:
[p3p2](mod p1)(Y1)= 1 (mod p1)
[p1p3](mod p2)(Y2)=1 (mod p2)
[p1p2](mod p3)(Y3)=1 (mod p3)
After we know the values of a1, a2, a3, M1, M2, M3, M, Y1, Y2 and Y3. We can find the number of soldiers, X with this equality:
X=[(a1)(M1)(Y1)+(a2)(M2)(Y2)+(a3)(M3)(Y3)] (mod M)
For example: If we wanted to find the number of students in a school yard, first we would tell them to line up in 3 rows and get 1 remainder and then if we tell them to line up in 5 rows and get 4 remainders and finally we tell them to line up in 7 rows and get 5 remainders.
X= 1 (mod 3) where a1=1 and p1=3
X=4(mod 5) where a2=4 and p2=5
X= 5(mod 7) where a3=5 and p3=7
Then M will equal 105 and M1=35 M2=21 and M3=15. To find the values of Y1,Y2 and Y3 we write the equations:
[35(mod 3)] Y1=1 (mod 3)
2Y1=1 (mod 3)
Y1= 2 (mod 3)
[21(mod 5)] Y2=1 (mod 5)
Y2=1 (mod 5)
[15(mod 7)] Y3=1 (mod 7)
Y3=1 (mod 7)
Therefore X= [(1)(35)(2)]+[(4)(21)(1)]+[(5)(15)(1)] (mod 105)=229=19 (mod 105)
With all this hard work, we calculated that there were only 19 students in the yard. The Chinese Remainder Theorem might seem useless in situations involving small numbers where counting is more practical. But when it comes to big situations like counting thousands of soldiers, this algorithm may come in handy.
Conclusion
Since Euclid proved the infinitude of prime numbers there is no way that we can ever finish researching them. This is because as we find new information on prime numbers, that won�t mean that we know them. It is just like chess. You may know the rules of chess but you may never be able read you opponent�s mind to find out its next move. That is why you have to keep playing, and in the case of prime numbers, you have to keep researching and make better algorithms similar to the Sieve of Eratosthenes which can detect composite numbers easier in an array of billions of integers with multi-million digits.
In another point of view, since prime numbers aren�t all about discovering, maybe we can find use of them in our daily lives. Seeing their properties, we can make encryption and decryption systems. An ideal system might function so that when a note is transmitted in the form of numbers, the receivers only detect prime numbers and convert them into their proper alphabetical symbol.
But whatever you do about prime numbers in the future, don�t forget that each discovery will open a lock for another.