Number theory Definition and 460 Threads

I've been looking for some free resources to learn a little number theory but I really can't find anything that is at a beginner level, anyone know of any sites or such of where I should start?

This isn't homework, but an interest question I found on the web. I can't solve it though... Let p be an odd prime. Determine all pairs (m,n) where m and n are positive integers and satisfy the below: (p-1)(p^n+1)=4m(m+1). I have done by some simple inspection that m must be odd, but...

Hey, I'm a little stumped on this basic number theory question. The solution is probably staring me in the face, but for some reason it's eluding me... If a^n | b^n, prove that a|b. So, say that a^n \cdot x = b^n for some integer x, there's not a lot I can go to from there. We do get that a |...

Here is the question: ----------- Suppose n has a primitive root g. For which values of a (in terms of the primitive root g) does the equations x^2 \equiv a \ \text{(mod n)} have solutions? ----------- I really don't have much of an idea of how to even begin this one. Let g be a primitive...

Here is the question from the book: ------------ Determine a primitive root modulo 19, and use it to find all the primitive roots. ------------ \varphi(19)= 18 And 18 is the order of 2 modulo 19, so 2 is a primitive root modulo 19, but I am not sure of how to use that to find all...

this is question found as part of a 25question 1hour long maths problem quiz i took today. It was just about the only one i couldn't do in the time, and even with another look after I can't do it :S I think it may involve maths I havnt come across or at least methodology i havn't (I 16 and only...

came across this in a book I am reading, it doesn't give the answer though youll know once you get there that you're right anyways. Homework Statement In a parlour game, the 'magician' asks one of the participants to think of a three-digit number abc_{10}. Then the magician asks the...

Here is the original problem: -------------- Use Euler's Theorem to compute the following: 3^{340} \ \text{mod} \ 341 --------------- So Euler's Theorem tells us that if (a,m) = 1, then a^{\phi(m)} \equiv 1 \ \text{mod} \ m \phi(341) = \phi(11\cdot 31) = 300 OK so now we need to...

Here is the question from the book: ------------ Determine all n for which \phi(n) = n -2. ------------ Now it seems that the only time this will work is for n = 4. However, I haven't any idea of how to prove (or justify) this. I have thought about working primes and composites, since we...

Homework Statement Let p be a prime number such that p \equiv 1 (mod 3) Let a be an integer not divisible by p. Show that if the congruence x^3 \equiv a (mod p) has a solution then a^{\frac{p - 1} {3}} \equiv 1 (mod p) The Attempt at a Solution Right, I'm not sure how...

I am doing a Chinese remainder theorem question and one of the equations is x \equiv 0 (mod 7). This would mean that x is a multiple of 7, but how do I use it in conjunction with the Chinese remainder theorem? Do I just ignore that equation, use the CRT on the rest of the system, and then once...

Here is the exercise from the book: -------- You are asked to design a system for numbering TV programs to facilitate the programming of VCRs. Each program should be assigned a number so that a VCR can determine the day of the week, starting time, ending time and the channel of the program...

Here is the question from our book: ------ Let F_n = 2^{2^n} + 1 be the nth Fermat numbers. Use the identity a^2-b^2 = (a-b)(a+b) to show that F_n - 2 = F_0F_1\cdots F_{n-1}. Conclude that (F_n,F_m)=1 \ \forall \ n \neq m. Show that this implies the infinitude of the primes. ------- The...

Hi guys, i m just a begineer in number theory. While solving some questions ,i came across a doubt. The expression: a+bx here, gcd(a,b)=1 There always exists a value of x(where x is a integer) such that the above expression is equal to a prime number. Can anyone prove the above...

Anna says, "We three have $100 altogether." Betty says, "Yes, and if you had six times as much and I had one-third as much, we three would still have $100." Carl says, It's not fair. I have less than $30." Who has what? (Dudley)

I have a follow-up question to the "Calculating Averages" thread. Please see the "Followup_Question" attachment. Any help is, as always, most appreciated. (See this link for the original question: https://www.physicsforums.com/showthread.php?t=143759)

I might take Number Theory in college just for fun. Is this class easy? I mean, I was told about about this class by one of my teachers, and ever since then, I became interested in it. Is this class fun? Is it easy? Should I take it? Also, where can I find an online book or something if I...

Hi all, I have been studying the Pisot-Vijayaraghavan numbers recently however I have little background in the basic theory of algebraic numbers. There are a number of standard results which the texts give without proof and which I haven't yet been able to prove myself. Here are a few that...

Ok the question is as follows: Given gcd(a,b)=d, show that Phi(ab)= (d*phi(a)phi(b))/phi(d) I know that if gcd(a,b)=1 then phi(ab)=Phi(a)phi(b) but I am just stuck here. Any help would be greatly appreciated!

I just started learning gr. 12 discrete math a few days and I’m already having trouble with two very similar questions… Using deductive proof 1) Prove that if 4 is subtracted from the square of an integer greater than 3, the result is a composite number. 2) Prove that if 25 is subtracted from...

Can anyone help me with this question? Suppose (a, b)=1 . Prove that if p is any odd prime which divides a^2 + b^2 then p ≡ 1 ( mod 4).

Here is my problem: Prove or disprove: If gcd(m, n) = d, then the gcd(a, mn) = gcd(a,m) * gcd(a.n)/d. I can seem to get it started, sort of, but it just does not seem to get anywhere. I know by definition d | m and d | n. Then arbitrary integers x and y can be used such that m = xd and...

I am new to number theory and i so far like it. I recently found this problem but there was no explanation of how they got it and i was wondering if any of you knew how they got it. It is this: 1/3+1/3^2+1/3^3... + 1/3^n=1/2(1-1/3^n) Then they used: 1/4+1/4^2+1/4^3... +...

I'm going through the book Number Theory by George E. Andrews. I'm having particular difficulty constructing proofs, which I'm sure is quite common. Problem 4 of section 2-2: Prove that \operatorname{lcm}(a,b) = \frac{{ab}}{{\operatorname{gcd}(a,b)}}.-------------------- Ok, so I guess a...

Can anyone help me with this problem? Find all solutions of the following congruence 3x^5≡1(mod 23) This is what I have so far I know 5 is a primitive root and I made a table of indices modulo 23 with respect to 5 then Φ(23)=22 Ind5(3x5)≡ind5(1)=22(mod 22) Ind5(3x5)≡ind5(3) + ind5(x5)≡16 +...

Hey all, I've got a few problems that are tripping me up tonight. 1. Let m,n be positive integers with m|n. Prove phi(mn)=m*phi(n). I know I can write n as a multiple of m, and m as a product of primes, and my best guess so far is that I can work with some basic properties of or formulas...

"Intuitive" Number theory: Now i would like to play a game called "conjecture"..we have that for asymptotic behaviour: \pi(x)=li(x) where here "li" means the Logarithmic integral.. my conjecture is that for the sum: \sum_{p}^{x}p^{n}=li(x^{n+1} i have checked it for n=-1,0,1...

I'm trying to show that 25 is a strong pseudoprime to the base 7 using millers test. Is there a better way to solve this than just brute force? Thanks

hi i want know about a good number theory (NT) book for self study. i haven't studied NT before. so what would be a good book for me? should i start with ogilvy's book? what about hardy? thanks to anyone who can help.

Hey all, I've got a few number theory exercises that are troubling me. 1. Prove a positive integer s is a square if and only if each of the exponents in its prime factorization is even. 2. Let c,d be positive, relatively prime integers. Prove that if cd is a square, c and d are squares. 3...

Could someone please give me some advice on solving these problems 1. find the last digit of the decimal expansion 7^999999? I think i have to do something like 7^999999(mod10) but I'm not really sure if that's right or how i would do that 2. Show that every positive integer...

if you have three cycles in your life that start the day you're born. the physical, emotional, and intellectual cycles, of length 23, 28, and 33 days. Each cycle follows a sine curve with period equal to the length of that cycle. Measuring time in quarter days which days of your life will you be...

1. Prove (a,b)=1 iff (a+b, ab)=1 I'm guessing the main tools I have here are xa+yb=1 and the lemma behind the Euclidean algorithm: if a=bq+r, (a,b)=(b,r). I figure I need to do lots of manipulation to build up a more complicated equality, but I can't make it quite work. Any suggestions? 2...

Q.Prove that 1+1/2+1/3+1/4+...+1/n is not an integer.n>0

Hey all, great site and I look forward to contributing more. For now, a question... For next semester I need to choose between Number Theory or Complex Variables. I am under the impression that complex variables will be the more useful class for my physics education, however I had some concerns...

I'm not saying PF is a bad forum...but *What are some good books on number theory? --I'm a HS senior, currently taking CalcIII. --I don't care about the difficulty or how the book is written-->only that it is comprehensive. (does not leave out details, whether they be significant or...

Hey, I'm interested in Number Theory and have seen a few simple proofs/concepts related to Number Theory but at moment I have no other reference. Could somebody recommend me a Number Theory which will teach it as it's taught when you start university. I've done an A-Level in Maths in...

I really really hate proofs! I've done 3 of my 5 problems, which took me 2 days and over 30-50 pieces of scrap paper. I'm serious, I didn't even eat dinner today because I spent straight hours just staring at questions, thinking I was coming close to solutions, then only to find out I've...

Hi I got a question: I have the following problem ax + by = c. Where a,b are positive integers, and c is a known integer. If I calculate the gcd(a,b) is it then possible to find the x,y which makes the above equation true ? Best Regards, Bob

Hey guys, I have a homework assignment for number theory and two of the problems I don't know how do solve. I was hoping I could get some hints or help. Thanks Problem 1 Let m,n be elements of the natural numbers where n > m. and F_n = 2^2^n + 1 a.) Show that (F_n - 2)/F_m is an...

let be the Dirichlet series in the form: g(s)=\sum_{n=0}^{\infty}a(n)n^{-s} my question is if there is a relationship between g(1-s) and g(s) for any L-Dirichlet series. another question...where could i find Vinogradov,s work on Goldbach conjecture?..thanks.

The question is Use Fermat's Theorem to compute the following: 99^{99} \equiv (mod 47) I spent over an hour on it but could only reduce it to 99^7 \equiv (mod 47) I think I am missing something. So if you can help, it would be great. Please provide an explanation as well...

I am doing a course called 'Number Theory'. It is an introductory course to the subject and some if not most of it is based on the book 'Number theory with computer applications' by Ramanjuachary Kumandari and Christina Romero (1998). Anyone have any experince with this book? If so do...

Here you are the best formula to calculate Pi(x) by means of a 4-dimensional integral: \pi(t)=\frac{1}{4\pi^2}\int_0^t\int_{c-i\infty}^{c+i\infty}\int_{d-i\infty}^{d+i\infty}\int_0^{\infty}dxdsdqdn\frac{n^{-s+2}x^{q-1}LnR(qn}}{R(4-s)} Where R(s is the Riemann Zeta function...

I saw someone say this in another thread. Why is it so important? My best guess is that it has something to do with prime factorization, but that's a pretty wild guess.

For all those starting on number theory, here are some notes from the Cambridge first year syllabus. They cover diaphatine equations etc. Regards, M

1. Find seven different unit fractions whose sum is 1. So \frac{1}{a} + \frac{1}{b} + \frac{1}{c} + \frac{1}{d} + \frac{1}{e} + \frac{1}{f} + \frac{1}{g} = 1 Would this just be purely guess and check? 2. How many different 6 digit numbers can you make using 1,2,5,6,7,9 . Would it just...

Hello all I need help with the following proofs 1. If x and y are arbitrary real numbers, prove that there is at least one real z satisfying x < z < y. (Do I just use the Archimedian Property?) Thanks

Hi, I'm a high school senior in my first semester of Calculus, so my math is pretty limited at the moment. I was wondering if you guys could recommend any introductory number theory books that you think are about at my level. Any suggestions would be really appreciated. Also, sorry if I...

Hello all, here is a problem I am working on that is giving me some problems. p,q, and N are defined as in RSA i.e. {p,q} in (Z_p,*), N = pq a in (Z_n,*) g in (Z_{N^2}) s.t. g=aN+1 mod N^2 The problem is to show that the discrete log problem base g is easy in Z_{N^2}, i.e. : given...