Number theory Definition and 475 Threads

Homework Statement Prove that 2^15-2^3 divides a^15-a^3 for any integer a. Hint: 2^15-2^3 = 5*7*8*9*13Homework Equations fermats theorem eulers theoremThe Attempt at a Solution I think that the problem is equal to show that 4080 divides any number a^13-a^3, that is a^15-a^3 = k * 5*7*8*9*13...

Scope of this thread is to supply [when possible...] an answer to unsolved question in other sites in the field of analysis [real or complex...] and number theory, avoiding to make dispersion in different threads... The first unsolved question is 'easy enough' and was posted on...

Homework Statement There are exactly 33 postage amounts that cannot be made up using these stamps, including 46 cents. What are the values of the remaining stamps? Homework Equations stamp 1 = x stamp 2 = y Im assuming postage amounts range from 1 to 100 cents The Attempt at a...

Homework Statement p is an odd prime (a) show that x^2+y^2+1=0 (mod p) is soluble (b) show that x^2+y^2+1=0 (mod p) is soluble for any squarefree odd m Homework Equations For (a) hint given : count the integers in {0,1,2,...,p-1} of the form x^2 modulo p and those of the form...

Homework Statement Solve the following equations positive integers: (i) a!+b!+c!=d! (ii) a!+b!=25*c! (iii)a!=b^2 Homework Equations For the first two one , i have no idea how to begin . But the third one I may use Bertrand's Postulate some where. Could anyone give me some ideas??

What the best way to introduce congruences in a number theory course? I am looking for something which will have an impact. What are the really interesting applications of congruent mathematics?

What are the most interesting examples of unsolved problems in number theory which an 18 year can understand?

What are the most interesting examples of unsolved problems in number theory which an 18 year can understand?

This new challenge was suggested by jostpuur. It is rather number theoretic. Assume that q\in \mathbb{Q} is an arbitrary positive rational number. Does there exist a natural number L\in \mathbb{N} such that Lq=99…9900…00 with some amounts of nines and zeros? Prove or find a counterexample.

In the late 80’s and early 90’s there was the idea of ‘calculus reform’ and some emphasis and syllabus changed. The order of doing things in calculus also changed with the advantage of technology. Similarly in linear algebra there was a linear algebra curriculum study group which produced some...

In the late 80’s and early 90’s there was the idea of ‘calculus reform’ and some emphasis and syllabus changed. The order of doing things also changed in calculus with the advantage of technology. Similarly in linear algebra there was a linear algebra curriculum study group which produced some...

I have to teach a course in elementary number theory next academic year. What topics should be included on a first course in this area? What is best order of doing things? The students have a minimum background in proof but this is a second year undergraduate module. I am looking for...

I need to teach a course in elementary number theory next academic year. What topics should be included on a first course in this area? What is best order of doing things? The students have a minimum background in proof but this is a second year undergraduate module. I am looking for...

Homework Statement Hi, sorry to be a pain, if anyone could help me understand this I'd be very grateful (exams next week, no more revision classes and no tutors I can easily ask...) Let K be a number field, OK its ring of integers, and Δ(W)2 be the discriminant. Write Z for set of...

I'm sorry if this has been posted already, but here's the article. I don't know much about number theory, but it seems like many of the biggest problems in number theory are quite simple to state, like this one, even a school child could understand it. Sounds like some really exciting...

Homework Statement Prove that if gcd(a, 133) = 1, then 133 divides (a^18 - 1). The Attempt at a Solution This is an old homework question as I'm going over the homeworks to review for the test, but can't seem to get this right. Which is annoying because I remember I did it fine back in the...

1)Prove that x,y are positive integers such that $x^2=y^2-9y$, then x=6 or 20. 2) Let p and q be distinct primes. Show that $p^{q-1}+q^{p-1}=1$ (modpq) Hint for 2) Use Fermats little theorem.

If a and b are relatively prime natural numbers, how many numbers cannot be written on the form xa+yb where x and y are nonnegative integers? My thoughts: Let n be a fixed integer such thatn=ax_{0}+by_{0}. Assume that we want to minimize x_{0} but keep it nonnegative. Then the following...

I'm going to be at an REU for discrete math and combinatorics over the summer. Do you have suggestions for combinatorics books to look at? An introduction to graph theory would especially help (I'm taking an algorithms class but I doubt that it would go into much depth regarding graph theory)? I...

Does anyone have suggestions for number theory references? I'm already familiar with elementary number theory and have some algebra background, but I'm not sure what kind of number theory I'm interested in yet. Thanks!

Homework Statement For all non zero natural numbers n prove that: 1- 24\mid n(n+1)(n+2)(n+3)(n+4) and that : 2- 120\mid n(n+1)(n+2)(n+3)(n+4)(n+5) The Attempt at a Solution 1- For n=1 we get that 24 divides 120 so we assume that 24 divides n(n+1)(n+2)(n+3)(n+4) and we...

As I've read and been told, you can never know too much mathematics when you study physics, and I think I read it somewhere here. I have also read that cutting-edge theories like M-Theory need most likely a new the invention of a new type of mathematics to be developed. But my question is...

Homework Statement Does there exist an integer n, such that 1+2+3+...+n, ends with the last two digits 13? Homework Equations 1+2+3+...+n = n(n+1)/2 The Attempt at a Solution I reached a conclusion that 1+2+3+...+n \equiv 13 (mod 100). Also the sum has to be greater than 100, but...

Homework Statement Prove or disprove: If n is a positive integer, then n=p+a^2 where a\in\mathbb{Z} p is prime or p=1 Note that the interpretation of "prime" used here includes negative primes. So, an exhaustive list of possibilities for p is p=1,\pm2,\pm3,\pm5,\pm7,\pm11,\cdots...

Let p be a prime number and 1 <= a < p be an integer. Prove that a divides p + 1 if and only if there exist integers m and n such that a/p = 1/m + 1/n My solution: a|p+1 then there exists an integer m such that am = p+1 Dividing by mp a/p = 1/m + 1/mp So if I choose n = mp(which is...

Let $n = pq$ such that $p$ and $q$ are distinct primes. Let $a$ be coprime to $n$. Show that the following holds: $$a^{p^k + q^k} \equiv a^{n^k + 1} \pmod{n} ~ ~ ~ ~ ~ \text{for all} ~ ~ k \in \mathbb{Z}$$

As the title suggests, I am looking for a topic to do my senior thesis on. It doesn't have to be original work, but it does have to be rather math intensive. Also, I have to do it on a topic in Number Theory for reasons beyond my control. I have the typical senior level background(e.g. Modern...

Homework Statement How many four-digit numbers formed of only odd digits are divisible by five? Homework Equations Permutations The Attempt at a Solution Here is what I think should be done : Ans : 4P3 * 1 = 24 Is that right ?

Can anyone suggest a good book on basic number theory or an introduction text to it. I have been looking but don't have much of an idea on what to use. Thank you.

Homework Statement see attachment Homework Equations Combinations and Arithmetic progression formulae Nth term = 4+(N-1)(3) The Attempt at a Solution After recognizing that this is an arithmetic progression, I calculated the number of terms as : 15 Then , as we have to...

Physics Kiddy ask for suggestions of what books to read on number theory, so I did a search and came up with this link. Others may have their own favorite sites or titles. www.freebookcentre.net/Mathematics/Number-Theory-Books.html. Some may be downloaded

Homework Statement a, b, c, d and e are distinct integers such that (5-a)(5-b)(5-c)(5-d)(5-e) = 28. What is the value of a+b+c+d+e? Homework Equations N/A The Attempt at a Solution I tried to solve this in the following manner:- On factorizing 28, we get 28 = 2x2x7. So...

I am about to choose my classes for my up and coming semester and was debating between taking Optics or Number Theory. I know they are very different courses but for those of you who have had one or both which one would you say "in general" is the more interesting course?

Prove that ordda | ordma, when d|m. Some conditions are 1 ≤ d, 1 ≤ m, and gcd(a,d)=1. What I have so far: let x=ordma, which gives us ax\equiv 1 (mod m) \Rightarrow ax=mk+1 for some k\inZ Let m=m'd. Then ax=mk+1=d(m'k)+1

Homework Statement If k is an integer, explain why 5k +2 cannot be a perfect square. Homework Equations n/a The Attempt at a Solution I'm in way over my head and not really sure what type of proof I should be using. In my course, we just went over some number theory and modular algebra so...

Homework Statement Suppose that u, v ∈ Z and (u,v) = 1. If u | n and v | n, show that uv | n. Show that this is false if (u,v) ≠ 1. Homework Equations a | b if b=ac [b]3. The Attempt at a Solution I understand this putting in numbers for u,v, and n but I don't know how to...

Homework Statement Show there exist infinitely many (p, q) pairs, (p ≠ q), s.t. p | 2^{q - 1} - 1 and q | 2^{p - 1} - 1 Homework Equations We are allowed to assume that 2^{β} - 1 is not a prime number or the power of a prime if β is prime. The Attempt at a Solution Using fermat's little...

Homework Statement I've got two questions out of my textbook. I'll list both of them and my attempts below. (1) Suppose : a, b, c\in Z, a|c \space \wedge \space b|c.\spaceIf a and b are relatively prime, show ab|c. Show by example that if a and b are not relatively prime then ab does not...

is there any kind of relationships between number theory and physics?i would also like to know if there is any kind of applications of euclidean geometry in theoritical physics.

Homework Statement Several of us claimed that if d=gcd(a,b,c) then d is a linear combination of a,b and c, i.e. that d=sa+tb+uc for some integers s,t, and u. That is true, but we only proved the analogous claim for the greatest common divisor of two numbers, i.e. when d=gcd(a,b). We need...

Homework Statement Argue that (17^4)*(5^10)*(3^5) is not the square of an integer. Homework Equations N/A? The Attempt at a Solution Do I break these up, and show that each is not a square? I'm not sure if that would be correct, but sqrt(17^4)=289 * sqrt(5^10)=3125 *...

Homework Statement If d=gcd(a,b) show that gcd((a/d),(b/d))=1 Homework Equations N/A? The Attempt at a Solution Basically, I know that I need to show that 1 is a linear combination of a/d and b/d. I'm not exactly sure how to go about this. Dividing by d gives...

Professional Help Needed--Elementary Number Theory I will preface this by saying that I have no formal training in Mathematics. I've taken Calc 1 and a couple of Symbolic Logic classes. Forgive me if I butchered any terminology. However, this has been bugging me for a while, and I would like...

This semester I decided to take elementary number theory instead of intro to philosophy. While I so far am enjoying the class, I'm a physics major, and am looking to pursue research in gravity later down the road (only a freshman, so that's far away). The description for the course: This...

Homework Statement Prove that the equation x^k \equiv 1 (mod p) has exactly k solutions if k|p-1. I'm also curious to know if it's possible to generalize this theorem this way: Prove that the equation x^k \equiv 1 (mod n) has exactly k solutions if k|\varphi(n) where \varphi(n) indicates the...

Let a, b and c be positive integers such that a^(b+c) = b^c x c Prove that b is a divisor of c, and that c is of the form d^b for some positive integer d. I'm not sure how to solve this question at all, I need some help.

Hello, The following problem appears in my number theory text: The answer: I have tried to trace the reasoning in reverse. I understand how we get to the finish (by showing that the number is divisible by all of the relatively prime factors of n, but I don't understand how we...

Hey guys. Does anyone know of a good undergraduate level textbook on number theory? I have a pretty solid undergraduate level math background but have never had the chance to take a course on this particular topic. If anyone could recommend a textbook that he/she likes, or is widely used at the...

Does anybody of a good book in number theory for deep understanding of concepts like kiselev in geometry and apostol in calculus ? I have 'introduction to theory of numbers' by I. Niven and H. S. Zuckerman but I feel it is not suitable for my purpose , there is no description of 'why we did...

Hello, I was browsing a set of number theory problems, and I came across this one: "Prove that the equation a2+b2=c2+3 has infinitely many solutions in integers." Now, I found out that c must be odd and a and b must be even. So, for some integer n, c=2n+1, so c2+3=4n2+4n+4=4[n2+n+1]...