Prime Definition and 756 Threads

greetings . i have a couple of questions about the prime counting function . when \pi _{0}(x) changes by 1, then it's logical to assume that it should happen at a prime argument . meaning : \lim_{\xi\rightarrow 0}\pi _{0}(x+\xi )-\pi _{0}(x-\xi )=1 implies that x is a prime . is this a true...

When I teach GCF to students, I show them how to find via the prime factorization and explain to them how the PF can get you all the factors of a number by multiplying different combinations of the Prime Factors and then proceed to explain why they are supposed to multiply the common Prime...

I am trying to find nonzero prime ideals of \mathbb{Z} \oplus \mathbb {Z}, specifically those which are not also maximal. If I try to do direct sums of prime ideals, the resulting set is not a prime ideal. (e.g., 2 \mathbb{Z} \oplus 3 \mathbb{Z} is not prime since (3,3) \cdot (2,2) = (6,6)\in...

I would really like to get some constructive feed back on this prime-seeking algorithm. Computationally it's no better than the rest. However, it does offer some unique insight. I have partitioned the set of naturals between prime and composites using a rigorous structural schema that I prove...

Hi, Can anyone help me how to prove that the two numbers 2^((n-2)/2) -1, 2^(2n) + 1 are relatively prime? Thanks.

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 if 1 + 2^n + 4^n is a prime number then n is a power of 3 Homework Equations The Attempt at a Solution If n is not a power of 3, i need to show that we can factorize 1 + 2^n + 4^n But I am not really sure what should n be if it is not a power of 3 ? is...

What is the probability that the stability of strings depends on prime quantities in order to be unique? Without prime numbers, would the strings break into composite states.

Hello all! I realize I am new to the community of online math forums, so I'm probably breaking a few etiquette rules (and possibly more important rules too - if so, please let me know and I'll fix what I can.) However, I am working on a math problem, and I am stuck on bounding a particular...

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...

The prime numbers are the multiplicative building blocks of the integers. Even so, their distribution escapes all methods of rationalization. As with building a pyramid, the primes are most densely distributed near zero, the point of origin, and as we move towards larger numbers the primes are...

Homework Statement Determine the index [G:H], where H is a subgroup of the group G and; G = GL(2,Z_p) H = SL(2,Z_p) p is a prime Where GL(2,Z_p) is the general linear group of 2x2 invertible matrices with entries in Z_p, SL(2,Z_p) is the general linear group of 2x2 invertible matrices...

Help with Mobius Inversion in "Riemann's Zeta Function" by Edwards (J to Prime Pi) Can someone please add more detail or give references to help explain the lines of math in "Riemann's Zeta Function" by Edwards. At the bottom of page 34 where it says "Very simply this inversion is effected...

Here is a good question for maths enthusiasts. I really find this sum very tough. Any help, advice or guidance for this question will be greatly appreciated. find the product upto n terms (1+(1/2^2)) . (1+(1/3^2)) . (1+(1/5^2)) . (1+(1/7^2)) . (1+(1/11^2))...upto n terms Where the nth...

I'm trying to write a program that will find and then display the prime factor of a number. I have written this in C# but it works for some numbers but doesn't work for others? using System; class primefactors { public static void Main() { int number; bool...

Homework Statement Show that the ring of rational numbers whose reduced form denominator is not divisble by a prime, p, mod an ideal the set of elements of the above set whose numerators are divisible by p is isomorphic to Z_pHomework Equations The Attempt at a Solution It seems very trivial...

Hi. I need to: prove the group (Zp*,x) has exactly one element of order 2. Here, p is prime and (Zp*,x) is the set {1, 2,..., p-1} under multiplication modulo p. Any help would be much appreciated!

Homework Statement Prove that there are infinitely many prime of the form 6k+5, where k is nonnegative integer. Homework Equations The Attempt at a Solution Prove by contradiction. Suppose there are finitely many prime of the form 6k+5. Then i get stucked. Anyone can help me ??

Definition: Proth number is a number of the form : k\cdot 2^n+1 where k is an odd positive integer and n is a positive integer such that : 2^n>k My question : If Proth number is prime number are there some other known relations in addition to 2^n>k , between exponent n and coefficient k ?

I'm trying to find a general formula for an algebraic equation, I'm studying the behavior of ∏_{i=2}^n(1-\frac{1}{i^m}) for m=3 and so far I've seen that I can find a general formula if n^2 + n + 1 produces only prime numbers. if not, it would get way harder to find a general formula for it by...

Having worked on prime ideals recently and finding them for Z_n I was wondering how you can to find all the prime ideals of a multiplication of Z/3Z X Z/9Z or Z/2Z X Z/4Z for example. I'm mostly having trouble starting this problem and feel that if I could get an idea where to start that I...

Is there a way within reasonable errors to say what part of the positive integers are prime and what part is factored greater than one? Oh course one is a factor of all numbers greater than zero. Yeats ago playing around a floating constant became known to me. to the tenth decimal place is...

Homework Statement I am attempting to allow a user to enter a number and check to see if it is in a list of already generated prime numbers Here's my attempt Where can I put the code to allow the number to be checked, the prime list generator works fine alonedef buildPrimeList ()...

So the problem has to deal with Mersenne numbers 2^p-1 where p is a odd prime, and Mp may or may not end up being prime. The theorem given is: If p is an odd prime, but Mp=2^p-1 is not a Mersenne prime, then every divisor of the Mersenne number 2^p-1 is of the form 2*c*p+1 where c is a...

Homework Statement (2) P is a prime number and a and b are positive integers . We Know... p | a^6 \ and p | a^3 + b^7. how do i find out how to prove that p | b? Homework Equations if a | b, then a | bx for every x in Z if a | b, and a | c, then a | bx + cy for any...

Prime Division HELP! Homework Statement I need to be able to understand and likely prove that for any positive odd integer n, 8 | (n^2 -1 ) Homework Equations The Attempt at a Solution odd can be said to be n = 2k +1 so 8 | (2k + 1)^2 -1

Hello friends, The problem I am trying to solve sounds simple, but I still haven't been able to find the solution: Find a prime divisor of 1111111111111 (13 ones), also known as a repunit. I know the answer(53, 79 and some big prime), but I have no idea how Mathematica calculated those values...

Homework Statement In my Number Theory class, we learned how to calculate the value of large exponents modulo primes using Euler's Theorem. I understand how to do this with exponents larger than the value of the totient function of the prime, which is p-1, but what about when the exponent is...

I am stuck on this proof. Zp[x]/(x^2 + 1) is a field iff p is a prime p = 3 (mod 4) We're assuming p is odd, so p is either 4m + 1 or 4m + 3. ==>/ let Zp[x]/(x^2 + 1) be a field I need to find that x^2 + 1 is reducible if p =4m+1 I can see it for Z5, Z13, Z17 for instance but I...

Homework Statement Write a program that takes a command line argument N and a sequence of N positive integers and prints the numbers that are prime only, followed by their sum. Homework Equations for loop The Attempt at a Solution This has been a vexing program to think of. We...

Hi guys, I always wanted to know if one could generate prime numbers according to an equation,so I wanted to go and study prime numbers little bit and know how exactly its gets formed and its logic. So as we all know that prime number is any natural number that is divisable by itself and 1...

Homework Statement Let p be an odd prime and n be an integer. Show that -1 and +1 and the only solutions to x^2 \equiv 1\ mod\ p^n. Hint: What does a \equiv b\ mod\ m mean, then think a bit.Homework Equations x^2 \equiv 1\ mod\ p^n \rightarrow x^2 = 1 + p^nk for k an integer. The...

Homework Statement Define the numbers G_n = \prod_{k=1}^n (\prod_{j=1}^{k-1}\frac{k}{j}). (a) Show that G_n is an integer, n>1; (b) Show that for each prime p, there are infinitely many n>1 such that p does not divide G_n. Homework Equations The Attempt at a Solution I can see that the...

Is there any standard (or reasonably standard) notation for the following function? Basically, it's ∏(n) - ∏(n-1) or, which is the same thing, \frac{\Lambda(n)}{\log n} (where ∏(n) is the Riemann prime counting function and \Lambda(n) is the Von Mangoldt function) Basically...

Hi folks, The CRT says there's a unique solution to the system of congruences x = a (mod m) x = b (mod n) x = c (mod p) in (mod mnp) when m, n, p are pairwise relatively prime. But what if m, n, p are NOT pairwise relatively prime. Is there a systematic way to solve...

Show that if a,b\in\mathbb{N}^+,\ \gcd(a,b) = 1 and p is an odd prime, then \gcd\left(a+b,\frac{a^p+b^p}{a+b}\right)\in \{1,p\}

According to the prime number theorem, the number of prime numbers that are less than N is approximately N\ln(N) for a sufficiently large N. But can we find the product of prime numbers that are less than N? (For example, N=20 then 2x3x5x7x11x13x17x19 although I think 20 isn't large enough haha)

I need to factorize large numbers (some of them have about 200 decimal digits). Wolfram alpha is a dead end and programming with python is not working for me too. Any suggestions?

Here is the process: For example, take 28. Its prime factors (taken with multiplicity) are 2, 2, and 7. The sum of these is 11. Double it and you get 22. So, the sum of prime factors of 28 multiplied by 2 is 22. Repeat this process until you reach a loop or a stable number. For 28, if you...

A positive integer is called an additive prime number if it is prime and the sum of its digits is also prime. For example, 11 and 83 are additive prime numbers. OEIS gives the sequence of additive primes the number http://oeis.org/A046704" for that info). I've done many Google and...

Prove that if "a" is prime, then "b" is prime Homework Statement Suppose that "a" and "b" are associates. Prove that if "a" is prime, then "b" is prime. Homework Equations The Attempt at a Solution By the definition of an associate a=ub where u is a unit.

Why is it not possible for a filed to have have two prime fileds one isomorphic to Zp and the other isomorphic to Zq for p and q primes. Thank you

One of my professors told us about a prime number equation he came up with back in his own college days in Russia. It was published in a Russian journal in the problems section years and years ago and has not been translated to English. The thing is, his professor thought this equation must have...

Homework Statement prime mover = 220 hp crane = 2000 tonne velocity crane = 10m/min = 0.17m/sec acceleration crane = 0.017m/sec every 100 tonne = 6 tonne of friction 1hp = 726 W 1 tonne = 1000 kg Homework Equations how many prime mover are needed to pull the crane?prime...

I recently got interested in number theory and have been fiddling around with Scilab trying to find interesting things. I came across the following mildly interesting property, which I couldn't find much about on Google. Form the difference d_n between the sum of the squares of the prime...

I know that any prime p = 1 mod 4 can be expressed as sum of 2 squares. But how many different pairs of integers a,b such that p = a^2+b^2? (with a>b!) It seems there is only one pair. How to prove it? I try in this way: assume p = a^2+b^2 = c^2+d^2 (with a>c>d>b) and try to show it has...

Homework Statement Why are prime ideals so important? Homework Equations The Attempt at a Solution

Homework Statement Let D = Z[sqrt(10)], and let P be the ideal (2,sqrt(10)) 10). Prove that P is a prime ideal of D. Homework Equations The Attempt at a Solution Not sure where to start. I think elements are of the for a+b*sqrt(10), a,b integers. Any hints as to what to do next?

Is there any method to show that their are infinitley many prime numbers?

how do you prove, if p is prime, then a derived equation of p is prime, if true?