Primes Definition and 298 Threads

I'm having trouble with this proof: Prove that there are infinitely many prime numbers, using proof by contradiction. My teacher gave it to us the first day of class, but I'm a little lost: Assume there are finitely many prime number, P1, P2, P3,...,Pn. Let q=(P1*P2*P3*...*Pn)+1 Then...

I would like to show that if a prime number P mod 8 is a) 1 or 7 or b) 3 or 5 then a) \frac{(P+1)}{2}(1-sqrt{2})(3+sqrt{8})^\frac{P-1}{2}+ \frac{(P+1)}{2}(1+sqrt{2})(3-sqrt{8})^\frac{P-1}{2} = (\frac{P-3}{2} + 2) mod P b) \frac{(P+1)}{2}(1-sqrt{2})(3+sqrt{8})^\frac{P-1}{2}+...

For which primes "P" is the following true? the function f(x) = x(x - 1) + p gives you a prime number for all x < p I've tried this with 5,11, and 41, but it doesn't work for 7 since 5(5-1) + 7 is not a prime. Btw, this isn't homework or anything, just a curiosity.

1. Homework Statement R,M are Noetherian. Prove that the radical of the annihilator of an R-moduleM, Rad(ann(M)) is equal to the intersection of the prime ideals in the set of associated primes of M (that is denoted so regretfully that I am not even allowed to spell it out by the system)...

"Every prime greater than 7, P, can be written as the sum of two primes, A and B, and the subtraction of a third prime, C, in the form (A+B)-C, where A is not identical to B or C, B is not identical to C, and A, B, and C are less than P." True?

"Every prime greater than 7, P, can be written as the sum of two primes, A and B, and the subtraction of a third prime, C, in the form (A+B)-C, where A is not identical to B or C, B is not identical to C, and A, B, and C are less than P." True?

is there any formula to compute the gaps between primes which could be true to all prime numbers?..thanks..please help!

It seems rather straight forward that if you have an abelian group G with \# G = p_1 p_2 \cdots p_n (these being different primes), that it is cyclic. The reason being that you have elements g_1, g_2, \cdots g_n with the respective prime order (Cauchy's theorem) and their product will have to...

33 x 42 __________________________ = 2 x 3-3 x 16 33-(-3) = 6 16 = 42 ... 42-2 = 0 so.. 36 x 2 <-- answer I guess I am not sure

Terence Tao has submitted a paper to arxiv: [1201.6656] Every odd number greater than 1 is the sum of at most five primes Its abstract: One can turn the Goldbach conjecture and similar problems into statements about certain integrals, but those integrals are VERY hard to do, and it has only...

Hi, can you tell me which theorem they have used here: http://everything2.com/title/proof+that+the+sum+of+the+reciprocals+of+the+primes+diverges i'm thinking on part: Well, there's an elementary theorem of calculus that a product (1-a1)...(1-ak)... with ak->0 converges to a nonzero value iff...

Homework Statement Prove that their are infinitely many primes such that \left(\frac{14}{p}\right)=1 Homework Equations The bracketed symbol is the legendre symbol (i.e. there are infinitely many primes such that 14 is a square modulo p) The Attempt at a Solution Well by...

Homework Statement Show that the number of primes of the form 4n-1 and 4n +1 are infinite Homework Equations The Attempt at a Solution I am able to show this for 4n -1 but I am having trouble doing it for primes of the other form. ( I am hoping to do it without using modular...

Complex Numbers have always facinated me. But... Do complex primes exist? If so, How?

Would like to see a proof for the following question. Let p be a prime number. Define a set interesting if it has p+2 (not necessarily distinct) positive integers such than the sum of any p numbers is a multiple of each of the other two. Find all interesting sets.

I asked this question on one another forum but didn't get any answer . Consider the following array of natural numbers : \begin{array}{ccccccccc} 1 & 2 & 4 & 7 & 11 & 16 & 22 & 29 & \ldots \\ 3 & 5 & 8 & 12 & 17 & 23 & 30 & 38 & \ldots \\ 6 & 9 & 13 & 18 & 24 & 31 & 39 & 48 & \ldots...

Homework Statement Prove the following Theorem. Let n ε Z. If n ≥ 2 and n is composite, then there exists a prime p such that p divides n and p ≤ √n. After proving this Theorem show that if 757 is not a prime, then it has a prime divisor p ≤ 23. The Attempt at a Solution I...

--> Why are prime numbers so important to number theory? (Apart from speculations of being connected to energy levels of complex quantum systems.) --> Let for time being, primes that we know of, be called primes of "type-2". Here '2' comes from the definition of primes. Since we consider...

Infinite primes proof? Someone told me Euler proved that there are infinitely many prime numbers by proving that the sum of their reciprocals is infinite. I have one concern. How can you prove the infinitude of primes by this method without assuming the set to be infinite in the first place.

Hey guys, I was reading a brief article which described the logarithmic integral for approximating π(x) in two ways: ∫x/log(x) dx and ∫1/log(x) dx I am aware that the second is the actual definition of li(x) but the top is used extremely frequently and upon trying out the top it...

As Dodo noted, my prior test excluded some primes of the form 8n +/- 3, e.g. 29. I discovered a new property of the recursive series used in the prior test to allow a correction that apparently includes all primes of the form 8n +/- 3 but apparently includes no composites. The property of the...

My question is this, is there a known convergence of the sum of primes divided by the square of the sample size? I've just been looking at it, admittedly with only the first 50,000 primes, and it looks as if it is converging on a number near 6. If you plot the points below, you might see what...

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Ramsey Primes are those generated from a simple criteria that is easy to check. I checked all odd numbers from 1 to 1 million and 29455 numbers met the criteria. None were composite. The check is to do the following sequence mod P and check to see that the (P-1)/2 term is zero and no term...

If we assume that the Twin Prime Conjecture is true (and thus there are infinite number of primes that are 2 apart), how reasonable is it to assume that there will be an infinite number of Twin Primes that are preceded by a prime that is at least 10 lower than the first of the Twin Primes? (I...

Homework Statement I'm supposed to write a program which asks for an integer input, then determines if the input is a prime or not. I wrote a program but I have 2 issues: 1) It works well for primes up to around the size of 30000~, then above that (I just tried 65537, for a weak upper bound)...

Hi all, I am spending some time learning discrete mathematics on my own using the MIT OpenCourseWare materials. On page the second to last page of the Chapter 2 notes from here...

I'm curious, can anyone think of a way to prove whether or not p^x - d^y = p - d, for any odd primes p,d and natural numbers x,y where x,y are not equal to one? This would be useful for a proof I am trying to work on. So far, I have found that 3^2 - 2^3 = 3 - 2, but for this proof I am...

Homework Statement If the sum of two primes is prime, then one of the primes must be 2. The Attempt at a Solution Proof: Since all primes bigger than 2 are odd the only way to get a sum of two primes to be odd is to add an odd prime with an even prime. Let y be an odd prime such that...

Homework Statement Show that there are infinitely many primes 4k+1 using the properties of \left(\frac{-1}{p}\right). Homework Equations \left(\frac{-1}{p}\right) = \begin{cases} 1, & \text{if }p\equiv 1\ (mod\ 4), \\ -1, & \text{if }p\equiv 3\ (mod\ 4). \end{cases}...

Let's recall the Euclidean Rule for Pythagorean Triangles: Let (m,n) be co-prime natural numbers (m<n), then h := n^{2} + m^{2} e := 2 m n d := n^{2} - m^{2} = (n - m) * (n + m) form the hypothenuse, the even and the odd leg of a primitive Pythagorean triangle (PPT) If we...

Homework Statement Two Questions: 1. Prove, by contradiction, that if a and b are integers and b is odd,, then -1 is not a root of f(x)= ax^2+bx+a. 2. Prove, by contradiction, that there are infinitely many primes as follows. Assume that there only finite primes. Let P be the largest...

"box of primes" over l V X a l ^2 Homework Statement Find B, and t for these space curves (t = torsion) r(t) = (3sint)i + (3cost)j + 4tk Homework Equations Ok, so in my textbook i have two different equations to find the answer to this. the "box of primes" over l V X a l ^2...

I have seen in various locations different conventions regarding the location of a prime symbol denoting a tensor represented in a new frame. For example, if the position four-vector is x^{\mu} then this four-vector in a different frame is often written as either x'^{\mu} or...

Can anyone find an exception to this test for odd P > 3? P = Input an odd number > 3 Msg = "not Prime" A = 4 B = 32 Do C = B B = Mod(6*C - A + 8,P) A = C If A = 4 Then If B != 0 Then Exit Do Else Msg = "Prime" Exit Do End If End If If B = 0 Then Exit Do Loop...

Homework Statement Let a, b be members of a commutative ring with identity R. If a is a a prime and a, b are associates then b is also prime. True/False Homework Equations Definitions: a is prime if a|xy implies a|x or a|y a and b are associates if there exists a unit u s.t a=bu...

Homework Statement I'm working on a problem and as long as I can show that [tex] \prod_{p\leq n}p \leq \prod_{n<p\leq 2n}p [/ tex] then I'm done. But I'm having trouble with this..Can someone help? :-p Homework Equations The Attempt at a Solution I tried to use PNT but could not solve it...

Let (N, s(n), 0) be a Peano space. That is, N=\{1,2,3,\dots \} is a set in which http://en.wikipedia.org/wiki/Peano_arithmetic" can be used. We can then define: 0=\varnothing, 1=\{0\}, 2=\{0,1\},\dots \implies n=\{0,1,2,\dots ,n-2,n-1\} s(a)=a\cup \{a\}\implies s(a)=a+1 From here we...

I believe that Fibonacci primes are infinite. Currently there is no proof that there is an infinite number of Fibonacci primes. I was wondering why we couldn't compare the set of Fibonacci primes to the set of Natural Numbers and demonstrate that both have cardinality aleph null? Indeed, why...

I have read that we do not know if there are an infinite number of Fibonacci primes. So far, no one has produced a proof to show if they are infinite. I wanted to know why this seems to be so challenging? I'm sure it is, and maybe there is a subtle mathematical principle I am missing. I...

Circular meaning all of its digit rotations are also prime.PrimeLimit = 1000000; NoRotate = Select[Range[PrimeLimit], PrimeQ]; Rotate1 = Select[FromDigits[#] & /@ (RotateLeft[IntegerDigits[#]] & /@ Select[Range[PrimeLimit], PrimeQ]), PrimeQ]; Rotate2 = Select[FromDigits[#] & /@...

I posted this to Dr. Math but I'm too excited to wait for their response. OK, so start with the following equation, http://en.wikipedia.org/wiki/1_%2B_2_%2B_3_%2B_4_%2B_%E2%80%A6#Summability" by Ramanujan and Euler: 1 + 2 + 3 + 4 + ... = -1/12 Weird, yes, but there are...

I'm not exactly sure how to prove or disprove this statement. Just by looking at the c, I'd say it's not true, but I'm not sure how to show it either way. If (a,b) = 1 and (a,c) = 1, then (ac,b) = 1. http://i111.photobucket.com/albums/n149/camarolt4z28/IMG_20110719_192439-1.jpg?t=1311127377

As we travel further up the number line, primes become more scarce. But, as n grows larger and larger, the range of numbers between n and 2n grows larger ad larger. Do these two counteract each other? Does this cause the number of primes between n and 2n to stay relatively consistant...

Divisor summatory function is a function that is a sum over the divisor function. It can be visualized as the count of the number of lattice points fenced off by a hyperbolic surface in k dimensions. My visualization is of a different conic , one of a parabola. In fact my lattice points are not...

Homework Statement prove that p is prime iff for any integer "a" either (a,p)=1 or p divides a (where (a,p) denotes the gcd of a and p) Homework Equations (a,b)= the greatest common factor of a and b The Attempt at a Solution I had no trouble with proving the forward direction...

I got this question from another forum and it's driving me crazy. Find all triples of odd primes, p,q,r such that p^2+1 is divisible by q, q^2+1 is divisible by r and r^2+1 is divisible by p. Two such triples are 5,13,17 and 17,29,421. If we assume p<q<r, then there are no other such triples...

Other than the Eucledean method (1+p1p2p3...)and so on other than this, how do we prove that there are infinite prime numbers? in algebreic way (excluding analyses)

The prime number theorem describes the distribution of the prime numbers, in a sense. Are there other prime number theorems corresponding the asymptotic distributions of primes in other arithmetic progressions containing infinitely many primes? I was just wondering.

Let N be composite number containing only two primes a and b That is, N=a*b, where a and b are primes Factorizing N[even on a computer] is an impossible task if N is very large,for example if it has 400 digits. But we can eliminate a huge number of divisors by the following rules: 1.If n...