# Importance of prime numbers

1. Oct 15, 2014

### preceptor1919

Just want to know if there are applications in the derivation of prime numbers. My instructor and the textbook that we are using seems to be obsessed with it, there is at least one problem about deriving prime numbers in each chapter. And also different versions like palindromic prime, emirp, mersenne prime, twin primes etc. I am starting to be fascinated myself.

Is it just because solving primes(and variations of it) can be tough or is there a real world application?

2. Oct 15, 2014

### phinds

Prime numbers are used in cryptography so I suppose there is a practical application there for their generation

3. Oct 16, 2014

### preceptor1919

Thanks for the info, at least now I know why they're so interesting

4. Oct 16, 2014

### mathman

Prime numbers have been interesting to mathematicians long before cryptography. Important names are Fermat and Ramanujan.

5. Oct 16, 2014

### rcgldr

In addition to encryption, prime numbers are also used for error correction, such as 929 which is used to create a finite field (numbers modulo 929) used for the error correction on PDF417 bar codes. However, most error correction schemes use finite fields based on "prime" polynomials that use 1 bit coefficients (so add and subtract effectively become xor). AES encryption uses Rijndael S-box on 8 bit bytes, finding the multiplicative inverse of that byte modulo x^8 + x^4 + x^3 + x + 1 (hex 11B) (division by a 9 bit polynomial produces an 8 bit remainder). For a software implementation, typically a 256 byte lookup table is used. However in hardware, which may include 10 or more S-box'es in parallel, there's been a lot of effort made to reduce the gate count well below the hardware equivalent of a lookup table, using some interesting properties of fields based on 1 bit coefficients, in this case being able to map an 8 bit field into two 4 bit fields and then into four 2 bit fields. There are a lot (but not anastronomically large number) of possible mappings, and a brute force approach to simply try them all and select the one that needs the fewest number of gates has been done.

The point here is that prime numbers and finite field math at one time were just exercises in higher level mathematics, but once there was a commercial application for this stuff, a lot more people and more effort became involved, and the was significant advancement in the commercial aspect for this branch of mathematics.

Last edited: Oct 17, 2014
6. Oct 16, 2014

### preceptor1919

Wow it seems that people can actually write a book about prime numbers

7. Oct 17, 2014

### phinds

Well, there IS a book called "The Even Primes". All the pages are blank except somewhere around the middle, one page has a big "2" on it.

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