by lokofer
Tags: primes
 P: 108 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 $$a(n)=p_{n}$$ 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.
 Sci Advisor P: 1,132 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.
 Sci Advisor HW Helper P: 1,994 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.
 P: 994 Question about primes ... 'A Mathematician's Apology' is the most bitter book about anything I've ever read.
 P: 108 -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....
Math
Emeritus
Thanks
PF Gold
P: 39,683
 Quote by lokofer 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 $$a(n)=p_{n}$$ 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.
Define "real life".
HW Helper
P: 1,994
 P: 108 As a curiosity..if we define the "Partition function" (see Statistical mechanics at wikipedia) in the form: $$Z(s)=\sum_{n}e^{-sp_{n}}=s\int_{0}^{\infty} dt \pi (t) e^{-st} \sim \frac{\sqrt \pi }{\sqrt s}\int_{-\infty}^{\infty}dxe^{-sV(x)}$$ Where V is a potential so $$-D^{2} \Phi (x)+ V(x)\Phi (x)= p_{n} \Phi (x)$$