How Are Prime Numbers Utilized in Everyday Life and Technology?

[matqkks]

Why are prime numbers important in real life? What practical use are prime numbers?

 

[chisigma]

Why are prime numbers important in real life? What practical use are prime numbers?

A nice example is illustrated here...

http://mathhelpboards.com/number-theory-27/applications-diophantine-equations-6029.html#post28283

... but is only one of the 'extraordinary' results obtained thanks to prime mumbers...

Kind regards $\chi$ $\sigma$

 

[Nono713]

Prime numbers have lots of applications in number-theoretical cryptography, such as RSA, Diffie-Hellman key exchange, etc.. well, many of them don't work *specifically* because of prime numbers, but they work on specific mathematical structures such as groups or fields, and prime numbers over the integers tend to have some interesting properties and are relatively well-understood (from a practical standpoint anyway - there's still much to learn about primes, but we know how to find out if an integer is prime efficiently, we understand many of their properties, so they are useful in real life too and not just in some abstract sense).

 

[Deveno]

One of the best examples comes from modular arithmetic.

In general, with any integer n, if you add two numbers and compute the remainder upon division by n, you get the same integer as when you compute the remainders upon division by n of each summand FIRST, and then add them together (again computing the remainder upon division by n, if this smaller sum is larger than n).

An example:

341 + 113 = 454

The remainder of 454 upon division by 6 is 4 (454 = 6*75 + 4)

The remainder of 341 upon division by 6 is 5 (341 = 6*56 + 5)

The remainder of 113 upon division by 6 is 5 (113 = 6*18 + 5)

5 + 5 = 10, when divided by 6, we get a remainder of 4.

This is usually written:

a (mod 6) + b (mod 6) = (a+b) (mod 6)

This works with multiplication, as well:

(a (mod 6))*(b (mod 6)) = (ab) (mod 6)

The trouble is, when we multiply and get 0 (mod 6), we can't "undo" the operation, in other words we can have:

ab = 0 (mod 6)

with neither a or b being 0 mod 6 (for example, a = 3 and b = 4).

If we work with a PRIME modulus, a wonderful thing occurs, we can divide, too! This means we can do "the algebra we're used to" with a much smaller number system, and things still work a lot like we expect them to.

The simplest such system, of course, is using the modulus p = 2 (also known as "parity arithmetic"). This gives us the familiar rules:

Odd + Even = Odd
Odd + Odd = Even
Even + Even = Even

Odd*Even = Even
Odd*Odd = Odd
Even*Even = Even

In this system, "Even" is the "zero", and multiplication is rather trivial, the only non-zero product is "Odd*Odd = Odd" (or, if you like, 1*1 = 1, only 1 has an inverse).

That is, we can treat the properties "even" and "odd" as if they were numbers, and do arithmetic with them. In other words addition and multiplication preserve "how far between two multiples of p" numbers are.

As a practical matter, calculations of very large numbers can then be checked by calculations of relatively small numbers, which I'm sure you can see is very useful.

 

[CellCoree]

Prime numbers are important in real life because they have numerous practical applications. One of the most significant uses of prime numbers is in cryptography. Prime numbers are used to create strong and secure encryption codes, making it difficult for hackers to decode sensitive information. This is because prime numbers have unique properties that make them difficult to factorize, making them ideal for creating secure codes.

Prime numbers are also used in computer algorithms, such as the Sieve of Eratosthenes and the Euclidean algorithm, which are used in various fields such as computer science, engineering, and finance. These algorithms rely on the properties of prime numbers to efficiently solve complex problems.

In addition, prime numbers have applications in the field of mathematics itself. They are used in number theory, which is the study of the properties of numbers. Prime numbers have also been used in the development of new mathematical concepts and theories.

Furthermore, prime numbers have practical uses in everyday life, such as in calendars and timekeeping. The number of days in a year and the number of hours in a day are determined by prime numbers. Prime numbers are also used in music, as the frequencies of musical notes are based on prime numbers.

In summary, prime numbers have a wide range of practical uses in various fields, including cryptography, computer algorithms, mathematics, and everyday life. Their unique properties make them essential in solving complex problems and creating secure systems. Therefore, understanding prime numbers is crucial for advancements in technology and our daily lives.