Number theory Definition and 475 Threads

Homework Statement Show that for every odd positive integer n the following is correct xn + yn = (x+y)(xn-1 - xn-2y + xn-3y2 - ... - xyn-2 + yn-1) Homework Equations The one above. The Attempt at a Solution I have an idea about using induction to prove this. My idea is to...

Homework Statement Prove that for two integers m,n: all the common divisors divides the g.c.d.(m,n). Homework Equations The Attempt at a Solution g.c.d = aA +bB ; where a, b are the integers and let d be a common divisor, then: d|a and d|b. After this I have no clue where...

I am on the http://cow.temple.edu/~cow/cgi-bin/manager website working some congruence problems, here you can plug in answers over and over until you get them right. Three problems still baffle me: 1) With Mod24, find the solution of 3-15-21=. Here I just pretended that none of the...

Homework Statement When a is odd, show \frac{a^2-1}{8} is an integer. Then prove by induction n \geq 2 that for all odd numbers a_1,a_2,...,a_n, \frac{(a_1a_2...a_n)^2 - 1}{8} \equiv \frac{a^2_1 - 1}{8} + \frac{a^2_2 - 1}{8} + ... + \frac{a^2_n - 1}{8} \ mod \ 2 Homework Equations The Attempt...

{SOLVED}Number theory/ divisibility Show that m^2 is divisible by 3 if and only if m is divisible by 3. MY attempt: I assumed that 3k=m for some integers k and m. squared both sides and now get. 3n=m where n=3*(3k^2). Thus 3|m^2 Now the problem is when i assume: 3k=m^2 and need...

so I have 2^{1990}=(199k+2)^{10} expanding I have. 2^{1990}=2^{10}+10.2^9. (199k)+\frac{10.9}{1.2} 2^8.(199k)^2+...+10.2. (199k)^9+(199K)^{10}-(1) now its clear 199|2^{1990}-2^{10} since I can take 199 out of the RHS. but the book seems to imply that the above equation(1) says...

Does anyone have any comments or reviews to share on Gareth Jones' Elementary Number Theory? Is it suitable for an introduction to the subject? If not, what is the recommended book?

Hi guys, I want to know about some information of books, I recently finished a Number Theory course by Pommersheim and you know it was pretty easy, and it took for 3 days. Please suggest advanced and serious Number Theory book. + set theory. Thank you in advanced.

Dear all, I am attending a taught postgrad programme starting next October. I can not decide whether to take the Algebraic Number Theory or the (additive/arithmetic) Combinatorics modules. My choice will determine my PhD route, so it is a choice of career rather than just a choice of...

Homework Statement Let X=10000000099 represent an eleven digit no. and let Y be a four digit no. which divides X.Find the sum of the digits of the four digit no. Y. Homework Equations None The Attempt at a Solution I guess I have to factorize X. But it is really difficult to...

I signed up for the coaching service for the GRE and when looked through the questions I struggled with elementary number theory. What's the most efficient way to deal deal with the following kind of questions. 1. Positive integer Z_1 divided by 7 gives a remainder of 5 and Z_2 divided by 4...

Homework Statement If k is a prime number find all k that satisfy k²=n³+1 n is not a prime number Homework Equations I really have no idea, use any suitable one The Attempt at a Solution all prime numbers are odd except 2. n must be positive natural number n³ = k² -1 =...

Hello! I am a sophmore physics/math major who will probably be going into Mathematical or Theoretical Physics. My question is should I take Number Theory at some point during my undergrad years? On the one hand, it looks like an interesting/fun class (and I love math :) ) , but I've heard it's...

Hi, I am a computer science student that has found an interest in mathematics. I am currently exploring number theory, among other fields such as abstract algebra, and have gathered an interest in it after glancing at HAKMEM and Hacker's Delight, as well as learning of its importance in fields...

I have wrestled with the following two problems for a couple of hours each and have been successful. Now I am interested in how a more experienced mathematician would go about solving these. I encourage you to look past my lack of latex skills (lol), and do your best with the attachment I...

Homework Statement Find the least positive integer N such that every integer n \geq N can be written in the form 4a + 7b, where a,b are non-negative integers. Prove your N has this property Homework Equations The Attempt at a Solution Well, I kind of went about doing trial and error. I...

hi, I'm entering my 3rd year of PMAT degree and need to make a choice between differential geometry and number theory. These are both undergrad courses. I am trying to decide which would be more interesting/useful to take. I am planning on going into grad school, so it would be nice to choose a...

Positive integers 30, 72, and N have the property that the product of any two of them is divisible by the third. What is the smallest possible value of N? Note I have not yet taken a Number Theory course. I think I have found the solution using a bit of reasoning and some luck. N=60? I...

Show that if x, y ∈ Z and 3|x2+z2 then 3|x and 3|z Solution: 3|(x-z)(x+z) => 3|x+z or 3|x-z if 3|x+z then 3|(x+z)2 = x2+z2 +2xz => 3|x2+z2 +2xz - (x2+z2) => 3|2xz => 3|xz so 3|x or 3|z where to go from here? just the argument that if 3|x then 3|x2 and therefore 3 must divide z2 =>...

Homework Statement i'm sure everyone has seen this: Solve the following ancient Indian problem: If eggs are removed from a basket 2, 3, 4, 5, and 6 at a time, there remain, respectively, 1, 2, 3, 4, and 5 eggs. But if the eggs are removed 7 at a time, no eggs remain. What is the least...

I'm trying to do the homework for a course I found online. A problem on the first homework goes as follows: Suppose A is an integral domain which is integrally closed in its fraction field K. Suppose q in A is not a square, so that K(sqrt(q)) is a quadratic extension of K. Describe the...

Wouldn't mind a hint on how to start part iii), thanks. edit: in my notes i have for a similar question: 'L=Q(20.5, 30.5) F=Q(60.5) degree of the min polynomial = 2, because L=F(a) and [L:F]=2' (a = alpha) Could someone clarify what L=F(a) means so I can understand the example...

P.S. I'm not sure where to post this question, in particular I can't find a number theory forum on the coursework section for textbook problems. Please move this thread to the appropriate forum if this is not where it should belong to. Thanks!

Alright, having problems with this question too. It seems to be the same type of number theory problem, which is the problem. Homework Statement Prove "The square of any integer has the form 4k or 4k+1 for some integer k. Homework Equations definition of even= 2k definition of...

Homework Statement Use direct proof to prove "The product of any two even integers is a multiple of 4." Homework Equations definition of even is n=2k The Attempt at a Solution My proof is going in circles/getting nowhere. So far I have (shortened): By definition even n=2k...

I have a few questions I am having troubles with. If someone can push me in the right direction that would be awesome. Here are the questions: 1. Prove that the prime divisors, p cannot equal 3, of the integer n2-n+1 have the form 6k+1. (Hint: turn this into a statement about (-3/p) ) 2...

I was just working on some problems from a textbook I own (for fun). I am not sure how to start this problem at all. Here's the question: Show that 3 is a quadratic non-residue of all Mersenne primes greater than 3. I honestly don't know how to start. If I could get some help to push me...

1. For any positive integer n, if 7n+4 is even, then n is even. 2.Sum of any two positive irrational numbers is irrational. 3. If m, d, and k are nonnegative integers with d=/=0 then (m+dk) mod d = m mod 4. For all real x, if x^2=x and x=/=1 then x=0 5. If n is an integer not divisible by 3...

For i) I said for a in [0,1], then the group of units are = {f in R | f(a) =/= 0} i.e a continuous function f on [0,1] would have a continuous function g on [0,1] such that f.g=1 but the function would have to be g = 1/f, but this wouldn't be continuous if f(a) = 0 for ii) I have to show for...

Homework Statement Prove that the ring R of polynomials with real coefficients (i.e. f(x) = a0 + a1x + ... + anxn, ai real, are elements of R) has only the constant term a0 as the group of units, providing the constant term isn't zero.Homework Equations u is a unit if there exists a v such...

Homework Statement Prove that for any n \in Z+, the integer (n(n+1)(n+2) + 21) is divisible by 3 Homework Equations A previously proved lemma (see below) The Attempt at a Solution I sort of just need a nudge here. I have a previously proven lemma which states: If d|a and d|b...

Let a,b>1 be integers such that for all n>0 we have a^n-1|b^n-1. Then b is a natural power of a. I can't find a solution.

Prove this converse of Wilson’s Theorem: if m > 4 is a composite number then (m − 1)! ≡ 0 (mod m). (Note: This isn’t true for m = 4, so make sure that this fact is reflected in your proof.) My train of thought...: If m is composite, which has a prime factors r and s such that r does not equal...

I am aiming to prove that p is the smallest prime that divides (p-1)!+1. I got the first part of the proof. It pretty much follows from Fermat's Little Theorem/ Wilson's Theorem, but I am stuck on how to prove that p is the smallest prime that divides (p-1)! +1. I am assuming that every...

I'm having some trouble addressing the following two questions in a text I am going through: 1. Show that n is a prime number iff whenever a,b ∈ Zn with ab=0, we must have that a=0 or b=0. 2. Show that n is a prime number iff for every a,b,c ∈ Zn satisfying a not =0, and ab=ac, we have...

Homework Statement Show 7 divides 3^(2n+1) + 2^(n+2) The Attempt at a Solution Have proved base case K=1 and for the case k+1 I have got ot the point of trying to show 7 divides 9.3^(2k+1) + 2.2^(k+2). Any pointers would be much appreciated. Thanks in advance

The first part of the problem is as follows: Any nonempty set of integers J that fulfills the following two conditions is called an integral ideal: i) if n and m are in J, then n+m and n-m are in J; and ii) if n is in J and r is an integer, then rn is in J. Let Jm be the set of all integers...

Hello dear forum member I wanted to know how about the research on this branch of science.Are many people working on or is it a very very small area because it is too difficult . Are researchers in this area seeking new maths in the short what is the general policy direction of their research

Hi, I was wondering what introductory book to Number Theory for self study would you recommend? I don't need one that is too streched, just one that will give me a flavor of number theory. Thanks in advance!

Homework Statement Prove if there exists an integer whose decimal notation contains only 0s and 1s, and which is divisible by 2009. Homework Equations Dirichlet's box principle :confused: The Attempt at a Solution I'm new to number theory, and I'm aware that I do not have the...

I'm looking for a good number theory book which doesn't hesitate to talk about the underlying algebra of some of the subject (e.g. using group theory to prove Fermat's Little Theorem and using ring theory to explain the ideas behind the Chinese Remainder Theorem). I'm still an undergraduate, so...

I am trying to understand (I've already seen the rigorous proof in a real analysis class) why exactly rational numbers have periodic decimal expansions. I have basically boiled it down to proving a seemingly simple statement of number theory (I say seemingly because I don't know any number...

Find a primitive root modulo 101. What integers mod 101 are 5th powers? 7th powers? -I tested 2. -2 and 5 are the prime factors dividing phi(101)=100 so i calculated 2^50 is not congruent to 1 mod 101 and 2^20 is not congruent to 1 mod 101. -Therefore 2 is a primitive root modulo 101 I guess...

This is not a homework , i got this question from the internet. Prove:

Homework Statement Why is R\Q not countably infinate or denumerable? Given R (Real Number) is not countably infinate or denumerable and Q (rational number) is denumerable. Homework Equations A set is said to be denumberable or countably infinate if there exists a bijestion of N...

Homework Statement If M and N are positive integers >2, prove that ((2^m)-1) is not a divisor of ((2^n)+1) Homework Equations The Attempt at a Solution Is this correct? I use the well-ordering principle. Let T be the set of all M,N positive integers greater than 2 such...

Hello, I wonder if somebody could point me to a book (preferably), or paper, link, etc. which explores the relations between number theory and group theory. For example, I am (more or less) following Burton's "Elementary Number Theory" and there is no mention of groups. I also have the...

Homework Statement compute 59x +15 \equiv 6 mod 811 Homework Equations The Attempt at a Solution 59x \equiv -9 mod 811 I really don't know hoow to do from here.

Homework Statement consider the number m=111...1 with n digits, all ones. Prove that if m is Prime, then n is prime Homework Equations def of congruence. fermat's and euler's theorem. can also use σ(n): the sum of all the positive divisors of n, d(n): the number of positive divisors of n...

Homework Statement x cong 1(mod m^k) implies x^m cong 1(mod m^(k+1)) Homework Equations x cong 1(mod m^k) <=> m^k|x-1 <=> ym^k=x-1 The Attempt at a Solution starting with ym^k=x-1 add one to both sides ym^k+1=x now rise to the power m. (ym^k+1)^m=x^m <=> subtract the 1^m from...