Recent content by huba

(r+s)(r-s)

I was reading W.J. LeVeque's Elementary Theory of Numbers. Theorem 6-8 says what I said.

I am not sure I fully understand the extension of the Unique Factorization Theorem (UFT) to Gaussian Integers (GI), by saying that the representation of a GI as a product of primes is unique except for the order of factors and the presence of units. Is there a similar problem when the UFT is...

Is it possible to prove that there is no elementary proof for FLT, or is it possible that Fermat himself did find a proof (as he claimed) which nobody since has been able to repeat?

I do not agree. Greathouse has been trying to help. As far as I am concerned, I find it very valuable that anybody interested in maths, at whatever level, has the chance in this forum to ask questions and make comments, with a good chance that they will receive help or constructive criticism.

There is certainly no point in making qualifying statements about a claim that has not been presented in the necessary detail.

Thanks. My question was related to another thread about prime numbers which resulted in the observation that Wilson's theorem can be used to test if an odd number is prime, but it is a far less efficient way of testing primality than trying factors <= the square root of a number. I assume that...

Using "[URL theorem[/URL], and the fact that \varphi(p) = p - 1 when p is prime, and the fact that 2 is coprime to all odd numbers, is it right to say that 2^{m-1} \equiv 1 \ (mod \ m) iff m is prime > 2?

The formula is interesting (it has certainly attracted many responses in a short time), but it does not produce the nth prime as effectively suggested in the first entry in this thread. For a given n, we don't know for what value of m we get H(m) = the nth prime, until finding it by trial and...

H(m) = 2 if 2m+1 is composite. When m = 4 + 3k, then 2m + 1 = 3(2k + 3) which is composite for all k. In fact, 3(2k + 3) gives all odd numbers > 9 that are divisible by 3. Similarly for m = (p^2-1)/2 + pk where p is prime, because then 2m + 1 = p(2k + p), and so 2m+1 is composite, and so...

Using Hurkyl's definition for H(m), H(m)=2 if 2m+1 is composite. H(4) = 2, and for all k, H(4+3k)=2. H(12)=2, and for all k, H(12+5k)=2, H(24)=2, and for all k, H(24+7k)=2, etc. This is Eratosthenes' sieve, and I know about one exact formula for primes which utilizes it: Riemann's...

As long as (2m)! can be calculated (although it should be difficult for larger values of m), there is no need for the proposed formula, because Wilson's theorem already tests primality: if 2m+1 divides (2m)!+1 then 2m+1 is prime. When (2m)! is difficult to calculate then the formula becomes...

There are insects, http://www.abc.net.au/science/k2/moments/s421251.htm" , that spend their lives underground, before emerging, every 7th, 13th, or 17th year, to mate. There is a theory that nature applied evolutionary selection as Eratosthenes' sieve: certain predators of cicadas also come out...

On the other hand, John Derbyshire ("Prime Obsession", 2003) mentions chaos theory in the context of the distribution of primes, and briefly describes the "Montgomery-Odlyzko law", which says that "the distribution of the spacings between successive non-trivial zeros of the Riemann zeta function...

The set of the smallest numbers n for which \varphi^{k}(n) = 1 is possibly more interesting. Sloane's http://www.research.att.com/~njas/sequences/A007755" lists the first 34 such numbers, or 33 if starting with 2 (I left out 1) : 2, 3, 5, 11, 17, 41, 83, 137, 257, 641, ... I will refer to the...