Question about "primes"... Hello..i've got a question that will seem "strange" or perhaps trivial... why are primes so important in Number theory or in maths?..there're many primality tests but my question is ..do real primes have any importance in real life?.... in fact if we knew the generating sequence of primes so [tex] a(n)=p_{n} [/tex] n-th prime...we could perform every sum over primes and similar..but do primes have any "secret" interest to mathematician, or are they involved in code-breaking ?..thanks.
There is a such an a(n) that generates the primes, but it is infeasible, from a computational perspective, for very large n. For example, a brute-force algorithm for outputting the first n primes would take about 2^[n(n+1)/2] operations. Code-breaking is more concerned with factoring than identifying the primes.
If you are interested in applications that affect the everyday person, look into cryptography. Otherwise, Hardy's A Mathematician's Apology is a good read.
-We, Physicist should also write ' A Physicist's Apology'... we aren't better than Him (Hardy), some friends of mine and an Ex-girlfriend always questioned that my thesis or any subject had any realistic application.. I'm downloading the e-book you pointed. - I have proposed my teachers papers about "Riemann Hypothesis" (Hilbert-polya operator version) "renormalization" (involving Abel-Plana formula, and Zeta regularization), "Quantization of NOn-Polynomial Hamiltonians" (involving continous Taylor series and Poisson summation) or "Riemann Gas work" (get the log of the primes by getting the band-structure or the Phonon dispersion relation)..as you can see they have no real life application. - But i think it's beatiful to think about "Number theory" and primes as something similar to the"Passtime" (pasatiempos in spanish) some people makes "sudokus" and "crosswords" you make analytic number theory....
Hmm, I did not consider that it is now in the public domain: http://www.math.ualberta.ca/mss/books/ in Canada at least, I don't know about the legality elsewhere. If you want to learn more about Hardy, a later edition has a lengthy intro by Snow that was interesting.
" Hallsoftivy"..with the "term" real life i meant that...had some appliaction to daily problems you find for example to construct bridges, or to study gases, solids or computers... i also agree that theoretical investigation is more beatiful than a practice one...
Many subjects in the sciences attempt to find and understand the "fundamental unit" for their specific area. Biologists examine cells and proteins. Chemists deal with atoms and molecules. Quantum physicists search for subatomic particles. Understanding these basic units leads to a greater understanding of the everyday plants, animals, and physical objects that are made up of them. In mathematics, arithmetic begins with an understanding of the natural numbers. The Fundamental Theorem of Arithmetic states that each natural number (other than 1) is a unique product of primes. That means that the prime numbers serve as a fundamental unit for the natural numbers. That is, we can break the natural numbers into their prime factorizations, and it is guaranteed that each one is unique. By better understanding how primes work, we gain a better sense of the way other number sets, such as the Integers, Rationals and Irrationals are constructed. This is like saying a nutritionist can give good diet advice because they have a good understanding of the way the body's various systems respond to certain vitamins, etc. Maybe I'm off, but it's the way I've always seen it. The primes are important because they are fundamental. That's good enough for me.
As long as numbers become an increasingly essential part of life, through the growth of computation and information industry, it follows that number theory would be more important to life. However, we are seriously just in the early embryonic stage of "number life," if I could coin a term. :)
As a curiosity..if we define the "Partition function" (see Statistical mechanics at wikipedia) in the form: [tex] Z(s)=\sum_{n}e^{-sp_{n}}=s\int_{0}^{\infty} dt \pi (t) e^{-st} \sim \frac{\sqrt \pi }{\sqrt s}\int_{-\infty}^{\infty}dxe^{-sV(x)} [/tex] Where V is a potential so [tex] -D^{2} \Phi (x)+ V(x)\Phi (x)= p_{n} \Phi (x) [/tex]