Search results for query: *

(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 [Broken] 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" [Broken], 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...

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" [Broken] 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...

Iterating Euler's totient function \varphi(n), it eventually arrives at 1. Let h(n) denote the least number of iterations to arrive at 1. For instance, h(18)=3, ie \varphi^{3}(18)=1. There is a conjecture (or is there a proof?) that the largest n for which h(n)=k is 2\times 3^{k-1}. The only...

Are you looking for a number, an approximation, a formula, or an algorithm? A062311, in the Encyclopedia of Integer Sequences, lists the first 13 "number of ways writing n! as a sum of two primes", that is it goes as far as 13!, it does not have a formula for a "reasonable" calculation by hand...

I think there is a lot of information about this topic on the internet. As so often, a useful starting point is Wikipedia (see "Pascal's triangle"). You can also search Amazon.com; for instance, a review on 'Pascal's Arithmetical Triangle: The Story of a Mathematical Idea' (Edwards, A.W.F...

Using the form -(4^k)*(8n-1), I can count 333 residues in [-2007,-1]. With 0 also a residue, I get 670.

Is it at this point that quadratic reciprocity should be applied? But how if it applies to odd primes? Something else (in my previous post): m any natural number > 2.

Wikipedia, under "quadratic residue", says that a number is a residue mod 2^m (m any natural number) if and only if it is of the form (4^k)*(8n+1). I counted 336 integers in [1,2007] that are of that form. I thought an equal number might be in [-2007,-1], but that would give me 672, not 670.

Is it possible to extend this theorem by saying that n divides the sum of the squares of the quadratic residues of n if and only if n can be written as p^m where p is a prime > 5 and m is any natural number?

Is it possible to show that if p divides the sum of the quadratic residues of p then p also divides the sum of the squares of the quadratic residues of p? Then it would follow that, as p divides the sum of the squares of integers 1 to p-1, p also divides the sum of the squares of the quadratic...

Hi (1) I wonder if it is right to say that, for a prime p > 5 (actually, p >=5), p divides the sum of the quadratic residues of p (if right, it should not be difficult to prove but I haven't done it). Further, a^2 = (p-a)^2 mod p, if 1 <= a <= (p-1)/2, and so the first (p-1)/2 integers, as...

Is Goldbach's conjecture difficult to prove because there isn't a tight enough upper bound for prime gaps? Is it possible to construct a sequence a(n) of odd numbers, with a gap, depending on, and increasing with, n, between a(n+1) and a(n), such that all even integers greater than a given...

Thanks. Why are the two prime gap lengths, 803 and 4247, quoted from Wolfram MathWorld (under "Prime Gaps"), odd numbers?

I hope I am correct in saying that Bertrand's postulate can be rephrased this way: the maximum prime gap following prime p is p-3, if p > 3. Is this the closest proven result for the maximal prime gap? Wolfram MathWorld mentions 803 as a large known prime gap following 90874329411493, and...

Sorry, I shouldn't have mentioned chaos theory as I know too little about it.

The Sieve of Eratosthenes is a simple algorithm to find primes, and what possibly looks so strange is that the application of simple rules can lead to randomness or unpredictability. As I understand, chaos theory and complexity theory deal with such issues: the iterative application of simple...

There is a generalised Binet formula for the recurrence relation G(n) = aG(n-1) + bG(n-2), which explains much more than the problem raised. The system does not allow me to post URLs to other sites as I have made less than 15 posts(why ?), but googling "generalised fibonacci" helps, and I also...

Julian Havil (2003), in "Gamma", p.163, quotes Euler: "Mathematicians have tried in vain to this day to discover some order in the sequence of prime numbers, and we have reason to believe that it is a mystery into which the mind will never penetrate." And then Havil goes on to quote Erdős...