Primes Definition and 291 Threads

From the fact that \sum_{\mathbb{P}}\frac{1}{p} diverges, how do I conclude that the sequence \frac{n^{1+e}}{p_n} diverges for all e>0? (p=prime, P_n=nth prime)

Conjecture, For p>2, the 2^(p-1) th square triangular number is divisible by M_{p} if and only if M_{p} is prime. I checked this for 2<p<27. For instance the first four square triangular numbers are 0,1,36 and 1225 and the fourth is divisible by . PS In fact it appears that if is prime then...

It,s proven that there can not be any Polynomial that gives all the primes..but could exist a function to its roots are precisely the primes 8or related to them) if we write: f(x)=\prod_{p}(1-xp^{-s})=\sum_{n=0}^{\infty}\frac{\mu(n)}{n^{s}}x^{n} wher for x=1 you get the classical...

Other "primes" If, instead a defining a prime as an integer divisable only by 1 or itself, we define it as an integer not divisable by a number other than 1, 2, half of itself, or itself, (numbers not being divisable by 2 still "able" to be primes) then we generate a new "prime" sequence: 1...

I'm working on understanding the following relation which was referenced in the Number Theory Forum some time ago: x-\text{ln}(2\pi)-\sum_{\rho} \frac{x^{\rho}}{\rho}-\frac{1}{2}\text{ln}(1-\frac{1}{x^2})= -\frac{1}{2\pi...

Every number can be considered a bit string. For a bit string one can define some algorithmic complexity (the shortest algorithm/program that reproduces the desired bit string). Can something be said, in general, about difference in complexity for primes as compared to composites?

Hi folks. I'm back to make a fool of myself again. It seems that I have a proof that the 'music' of the primes is caused by the interaction of a collection of sine waves. However, I have no idea whether this is old hat, trivial or interesting. Can somebody here tell me? PS. I don't know why...

I've been able to prove that the set {8n+7} has infinite primes by manipulating Fermat's Theorem, but I'm trying to reprove it using quadratic residue and Legendre Polynomials. I've been able to show that for p=8n+7, (2/p)=1 and (-1,p)=-1 So it follows that (-2/p)=-1. And that (-2/p)=1 iff...

I somewhere read that Euler proved that primes are infinite by proving that the series 1/2 +1/3 + 1/5 +... diverges. Can anybody tell the proof? Aditya

Bear with me. I'm new to forum and don't yet know all protocol. My question concerns twin primes. The previous thread on this topic seems to be closed. My question is this: When considering the Twin Primes Conjecture, has anyone researched the idea that (heuristically speaking) there is...

1.) Show that (n+1)! = 2(n-1)! mod n+2 I finished this one. Actually very easy. 2.) Let n > 2 be odd. Prove that if 4[(n-1)! + 1] + n = 0 mod n(n+2) holds, then n, n+2 are twin primes. Hint says to use the previous problem. I don't even know what to do for this problem. 3.) Prove the...

here's one more: pairs of primes separated by a single number are called prime pairs. Example: 17 and 19 are a pair. Prove that the number between prime pair is always divisible by 6 (assuming both numbers are greater than 6).

Can somebody please give me a hint as to how to do the following proof please? I have no idea what to use or where to start :( Suppose that p is a prime and a is an integer which is not divisible by p. Prove that there exists an integer b such that ba is congruent to 1 mod p^2. Thank you. T

Sieving for Primes For a while now I've been messing about in a spreadsheet trying to make sense of the pattern of the primes. I've at last managed to construct a simple programme that will sieve a range of numbers leaving only the primes. But I'm wondering just what it is I've done, whether it...

Hi, When I hear about the Riemann hypothesis, it seems like the first thing I hear about it is its importance to the distribution of prime numbers. However, looking online this seems to be a very difficult thing to explain. I understand that the Riemann Hypothesis asserts that the zeroes of...

"A way of conceptualizing the nature of primes..." We know Eratosthenes observed that the primes occur at 6n+-1. We also know that Ulam's spiral is considered interesting because it visually displays a 'striking non-random appearance' in the distribution of primes. What strikes me...

I have proved the primes of the order 4n+3 are infiinite but can not prove that primes of the form 4n+1 are infinite.please help prove that primes of the form 4n+1 are infinite.

prove that primes of the form 4n+1 are infinite . send the proof at tamalkuila@gmail.com

Is it possible to express all natural numbers greater than 2 as the sum of N unique prime numbers? For example, 6 = 2 + 3 and 18 = 13 + 5.

This is a particularly fun problem! Its on a homework that I already turned in. I used the proof by contradiction method. I just need a clarification point. I started by assuming finite number of primes of form 12k-1. suppose N = (6*P1*P2...*Pn)^2 - 3 and set the congruence...

It was really close, perhaps the ways you can wright n on is >= the n-1:th prime. But how could i ever prove it?

prove that there are infinetely many primes of the form 3n+1 we used : Assume there is a finitely # of primes of the form 3n+1 let P = product of those primes.. which is also of the form 3A+1 for some A. Let N = (2p)^2 + 3. Now we need to show that N has a prime divisor of the form...

Hi, what would be the best estimate in the # of primes between 10^{100} and 10^{101} thanks

Question from a mathematics dunce. I recently read that Merseinne primes are searched for using 2^x -1 where x is a number. Is this correct? If so why not use 6n+/-1 instead?

Primes in ring of Gauss integers - help! I'm having a very difficult time solving this question, please help! So I'm dealing with the ring R=\field{Z}[\zeta] where \zeta=\frac{1}{2}(-1+\sqrt{-3}) is a cube root of 1. Then the question is: Show the polynomial x^2+x+1 has a root in F_p if...

I have a conjecture that the equation X^2 - 2Y^2 = P has solutions in odd integers if P is a prime of the form 8*N+1. I know of a paper that requires one to find Q such that Q^2 = 2 mod P inorder to solve these equations using continued fractions. To get to first base in proving my conjecture...

Hi hi, I've been messing about with primes for the last couple of weeks now to try and generate very strong primes for my RSA project. I've made an algorithm which generates a 100 digit prime within about 3 seconds. For my own amusement I thought I'd add a little check to see if any prime I...

I'm working on encrypting a small message using RSA. Are there any types of primes or primes of particular property that would make RSA decryption particularly difficult? Furthermore are there any better ways that just randomly trying to factorise to break RSA encryption or is there some...

Hey everyone, I need help on this problem: Find all prime numbers x such that x^2 = v^3 + 1 for some integer v. Thanks a lot for your help, i appreciated.

I've been asked to research primes of the form 2^m + 1, I've found all the known primes but now I've been asked to find out why m must be of the form 2^x for some natrual number x for 2^m + 1 to be prime. I've found quite a lot on this but nothing that clearly proves it, can anyone give me a...

Hi, I need help solving this problem. The question asks me to find all prime numbers p such that p^2 = n^3 + 1 for some integer n.

Given the probability of flipping a heads with a fair coin is \frac{1}{2}, what is the probability that the first heads occurs on a prime number?

Know why the cicada have 13 and 17 year life cycles ? Is there a reason why evolution picked prime numbers for their life-cycles ? Check out this neat article in the Post : http://www.washingtonpost.com/wp-dyn/articles/A61426-2004May2.html

Hi I've been wondering...the conjecture which states that the number of twin primes is infinite has neither been proved nor disproved so far. We know that the number of primes is infinite and I have come across two methods of proving this. My question is: why can't we actually prove that...

How can I prove that the sum of two odd primes will never result in a prime? Would this be proof?: Proof by contradiction: The sum of two odd primes will sometimes result in a prime. This is true because 2 + 3 = 5, which is a prime. So since this is true, does this proof the...

If a partition P(n) gives the number of ways of writing the integer n as a sum of positive integers, comparatively how many ways does the partition P'(n) give for writing n as a sum of primes?

Maybe most of you have seen this before, but I find it cool and that's why I thought I share it with you. :smile: Lets look at the sum \sum_{p\leq N}\frac{1}{p} where p represents only prime numbers. If I calculate the value for this sum when taking into account all prime numbers < 105...

There are many occurrences where real primes are composites when including complex factors with integral magnitude components, e. g. 2=(1+i)(1-i); 1 X 2 5=(2+i)(2-i); 1 x 5 . . . Using complex numbers gives no insight, though, into a formula for prime distribution. Both sets of...

A prime number is a positive ineger greater than 2 whose only integer divisors are itself and 1. A Mersenne prime in of the form 2^(n) - 1 where p is a prime. For example 2^(5) - 1 = 31 is a Mersenne prime. One of the larger Mersenne prime is 2^(216091) - 1. Estimate the number of decimal digits...

Hello everyone, My first post on these forums and I was wondering if I could have some assistance/direction with a problem: Prove that if p is a prime number and a and b are any positive integers strictly less than p then a x b is not divisible by p. The first thing I thought to myself...

1/7 = .142857... (repeated) 2/7 = .285714... 3/7 = .428571... 4/7 = .571428... 5/7 = .714285... 6/7 = .857142... So, you get all n/7 from the same 'cycle' of 6 digits. Let's call 7 a 'cyclic' integer. The next cyclic integers are 17, 19, 23,... They are all prime. But 11, 13, or 37 are...