Number theory Definition and 475 Threads

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...

hey guys been staring at this question for a few days and frustratingly nothing springs to mind. If any1 could offer some direction that would be awsome :) let n be a positive integer, and let p be a prime. Show that if e is a positive integer and p^e|n then p-1|∅(n) and p^(e-1)|∅(n)

Proving "simple" things (elementary number theory) Hey, I was wondering if I could ask for some help proving simple things in number theory, like divisibility things etc. The kind of stuff that you stare at and say "Duh, properties of real numbers...next?" but maybe don't know how to start as a...

Hi guys I am really interested in number theory and I want to start studying it,because I am interested in logic of numbers. Do you recommend any book to start with ?

Homework Statement if an integer n >= 2 and if n divides ((n-1)! +1) prove that n is prime. Homework Equations a divides b iff b = ma for integers a, b, m. The Attempt at a Solution by contrapositive: Assume n is not prime. Then we have by definition of divisibility...

It's been some time that I've been studying abstract algebra and now I'm trying to solve baby Herstein's problems, the thing I have noticed is that many of the exercises are related to number theory in someway and solving them needs a previous knowledge or a background of elementary number...

Homework Statement Suppose n is a natural number and A is a subset of natural number with n elements. Prove that a subset of A exists that the sum of its elements is dividable by n. The Attempt at a Solution well, This problem is harder than I can solve it. I first tried to use the...

Homework Statement Prove the following proposition: For any positive integers x and y, (x^2 - y^2) is not equal to 6.Homework EquationsThe Attempt at a Solution I'll try to prove using contradition. Assume x^2 - y^2 = 6. (x+y)(x-y) = 6 (x+y)=6 and (x-y)=1 (OR) (x+y)=1 and (x-y)=6 (OR)...

Homework Statement If 3 | m^2 for some integer m, then 3 | m. Homework Equations a | b means there exists an integer c such that b = ca. The Attempt at a Solution I realize that this is a corollary to Euclid's first theorem, and that there are plenty of ways to solve this. However, I...

Here is my situation, In short I want a number theory book that doesn't assume knowledge of previous number theory but assumes all knowledge of mathematics. I have been knowing multivariable calculus since about 4 years(learned it from Thomas 4th edition(this book was not like the later...

How to prove that if a-c | ab+cd then a-c | ad+cb is correct?? And how to prove the gcd(a^2+b^2, a+b) is 1 or 2. where gcd(a,b)=1.

First note that the operation * denotes the Dirichlet product, and µ denotes the Möbius function. Ok so here is the problem: Let f(x) be defined for all rational x in 0≤x≤1 and let F(n)=\sum_{k=1}^nf(\frac{k}{n})\;\;\;\;\; and \;\;\;\;\;\; G(n)=\sum_{k=1,\;(k,n)=1}^nf(\frac{k}{n}). Prove that...

I'm interested in analytic number theory and from what little I understand of it complex analysis will be more important than real analysis(measure theory). Thus I will be taking a year of graduate complex analysis this fall, however, I do also have the option of taking a year of graduate real...

Are there any basic prerequisites before learning about these branches of mathematics?

Homework Statement Use the principle of mathematical induction to show that the value at each positive integer of a function defined recursively is uniquely determined. I understand the problem and its related concepts. However, I feel that my attempt at a proof doesn't use the principle...

If Riemann's Hypothesis is proved as true, would number theory collapse?

Homework Statement Let p\ge 2. Prove if 2^p-1 is prime, then p must be prime. Homework Equations The Attempt at a Solution I am a physics student. I need help from those studying mathematics. Thank you very much!

Hello. I am looking for learning materials for the field of number theory. I took a class this semester in number theory and the topic fascinated me, but I don't feel like I learned that much. I am a mathematics minor so I have a very strong background in basic math, abstract/linear algebra, and...

NUMBER THEORY: Show that 8^900 - 7 is divisable by 29 ... Help Homework Statement Show that 8^900 - 7 is divisable by 29 Homework Equations The Attempt at a Solution By Fermats little theorem (8^28)^32 x 8^4 - 7 =1^32 x 8^4 - 7 =8^4 - 7 =(8^2)^2 - 7 =64^2 - 7 NB: 64...

Prove that the last three digits of n^100 can be only: 000, 001, 376, or 625. I can easily show that the last digit is either 0, 1, 6 or 5 because n^100=((n^25)^2)^2, so if our last three digits are 100a+10b+c, with a, b, c belonging to {0, 1, 2, 3, 4, 5, 6, 7, 8, 9}, any digit for c...

I'm struggling with how to even begin with this problem. Find the remainder of the division of 75!*130! by 211. 211 is prime, so I know the remainder is not 0. I'm not sure where to start though. Thanks!

So basically I am trying to prove that the sum ∑1/2k from k=1 to n is a fraction of the form odd/even, that is to say that the denominator will contain more 2's than the numerator. Now I'm almost positive this is true, and I suppose it might be more tractable to consider the stronger...

I'm interested in taking a course in number theory as the material excites me very much, however, I'm not sure how such material would be taught. What can I expect from lectures, homework, exams, etc? (on a somewhat related note, anyone here have any information regarding any special...

Number Theory -- Find Remainder .. when dividing by 17 Homework Statement Find the remainder when 3^24*5^13 is divided by 17. Homework Equations I know that 3^24 = 16 (mod 17) and calculated that 5^13 mod 17 = 3 (mod 17) The Attempt at a Solution BUT, I'm completely unsure...

Here's the question. Starting with an integer a≥2, we write on its left, below it, the number a+1, and on its right, below it, the number a^2, and obtain four numbers, to which we continue the process. We thus obtain a binary tree, whose root is a. Prove that the numbers in every line of the...

Homework Statement I am trying to prove that lim as x->infinity of pi(x)/x =0 Homework Equations pi(x) is the counting function that describes the number of prime numbers equal to or less than x and greater than 1. The Attempt at a Solution I'm really stuck about where to start...

1. For n ≥2, n^(1/n) is irrational. Hint provided: Use the fact that 2^n > n2. This is probably familiar to many. By contradiction, n = a^n/b^n --> a^n = n(b^n) --> n|a^n --> n|a Am I trying to force the same contradiction as with 2^1/2 is rational, that is, that a/b are not in lowest terms? Or...

I was wondering if one wanted to pursue learning more about cryptography which of these classes would be the most important? Number theory of abstract algebra?

Homework Statement If I have a matrix M, say 30 5 20 16 How do I calculate M^{1870} mod 101 using Euler's Theorem. Homework Equations I have so far worked out M ^{2} mod 101 to be 91 28 11 53 and thought I could use this as 2x935=1870 The Attempt at a Solution I...

Would the field of the number theory collapse or flourish if the Riemann Hypothesis is proved as true?

Homework Statement Decipher the following text KQEREJEBCPPCJCRKIEACUZBKRVPKRBCIBQCARBJCVFCUPKRIOF KPACUZQEPBKRXPEIIEABDKPBCPFCDCCAFIEABDKPBCPFEQPKAZ BKRHAIBKAPCCIBURCCDKDCCJCIDFUIXPAFFERBICZDFKABICBB ENEFCUPJCVKABPCYDCCDPKBCOCPERKIVKSCPICBRKIJPKABI Homework Equations I know that...

Homework Statement Let b1 through b_phi(m) be integers between 1 and m that are coprime to m. Let B be the product of these integers. Show that B must be congruent to 1 or -1 (mod m) Homework Equations None. The Attempt at a Solution Well, I know that the quantity B appears during the...

Given: N is a four digit number. S is the sum of N's digits. Prove: N minus S is a multiple of 9.

hi, i have only basic knowledge of number theory, but would like to know a hell lot, like maths major level or something(especially about fractals). is there any good site where i could? and please don't suggest wikipedia.

I need help. I'm trying to prove that if (2^n)+1 is prime, then there exists an integer k>=0 such that n=2^k. If n is odd, then (2^n)+1=(2^(2k+1))-(-1)^(2k+1)=(2+1)(stuff...)=(3)(stuff) so it's not prime, a contradiction. So I've knocked out half of the possible n's. What I'm struggling...

Homework Statement A base b repunit is an integer with base b expansion containing all 1's. a) Determine which base b repunits are divisible by factors b-1 b) Determine which base b repunits are divisible by factors b+1 Homework Equations R_{n}=\frac{b^{n}-1}{b-1} The Attempt...

Number theory: confused about the phrase "an integer of the form" Homework Statement Prove that any prime of the form 3k+1 is of the form 6k+1 Homework Equations The Attempt at a Solution I'm not sure where to start at all. I tried rewriting 3k+1 as 6k+2=6k+(6-4)=6(k+1)-4. But...

Homework Statement If ax+by=1, then (a,b)=1. Homework Equations The Attempt at a Solution I am just wondering if this is true. Because I know it is not true if ax+by=c, then (a,b)=c. Here is a proof I came up with: Suppose (a,b)=c, c>1.Then c|a and c|b, but then from our...

Homework Statement The actual problem is: "Does x^2+y^2=3 have any rational points? If so, find a way to describe all of them. If not, prove it."Homework Equations NoneThe Attempt at a Solution I found a book on Google Books (can't find it again) that said that this circle has no rational...

Homework Statement Prove that if N=abc+1, then (N,a)=(N,b)=(N,c)=1. Homework Equations The Attempt at a Solution Assume N=abc+1. We must prove (N,a)=(N,b)=(N,c)=1. Proceeding by contradiction, suppose (N,a)=(N,b)=(N,c)=d such that d\not=1 . Then we know, d | N and d | abc. Thus...

Homework Statement 1) What are the possible values of m^{2} + n^{2} modulo 4? 2) Let d_{1}(n) denote the last digit of n (the units digit) a) What are the possible values of d_{1}(n^{2})? b) If d_{1}(n^{2})=d_{1}(m^{2}), how are d_{1}(n) and d_{1}(m) related? 3) a)...

Homework Statement My domain i numbers of form 4k+1. n divides m is this domain if n=mk for some k in the domain. A number is prime in this domain if its only divisors are 1 and itself. My problem is to find a number in the domain with multiple prime factorizations. Homework Equations...

Homework Statement prove using induction: for any n =1,2,3... the product of the divisors of n = n^(number of divisors of n (counting 1 and n)/2) Homework Equations The Attempt at a Solution I understand why this is the case, but I'm having trouble with the induction step. if...

Homework Statement Does the sum of the reciprocals of natural numbers starting with nine converge? In other words, does Sigma 1/n with n being numbers starting with nine, converge? Homework Equations The Attempt at a Solution I know that subsets of the natural numbers with...

Homework Statement The problem is that I have to prove that there aren't three or more primitive pythagorean triples with the same value of c. A primitive pythagorean triple has has no values, a, b, or c that have common factors. The actual question is if this is possible, and if not prove it...

Let q be a prime, k= q-2 and X be an Integer, I have found solutions for q=5,7,17 for the equation (2^k) - 7 = X^2 . I have checked q for up to 1000000 but was not able to find any other solutions. Please prove if there can be more solutions or none for this equation. Thanks in...

So I'm stil deciding whether or not I want to do a math/physics major (as opposed to just a physics major), and I was wondering if Number Theory is at all useful to physicists. I ask this because it's the easiest of the three classes I have left for my math major, which would make it perfect...

Homework Statement Let p = a prime. Show {x}^{2} ≡ a (mod {p}^{2}[/tex]) has 0 solutions if {x}^{2} ≡ a (mod p) has 0 solutions, or 2 solutions if {x}^{2} ≡ a (mod p) has 2. The Attempt at a Solution OK, my mistake, I don't think this has anything to do with the phi function. But I don't...

What is a good introductory textbook for beginners?

For instance, let's say that you want to study fermat x^n+y^n=z^n for n=3; do not mind that we already know the answer :-) We could consider the densities of exact cubes, d(n), and then to calculate joint probabilities for d(Z), d(X) and d(Y). The mechanism can be applied, for instance, to...