Number theory Definition and 475 Threads

Homework Statement How would you prove using number theory that C(n,m) is an integer where n => m =>1? Do you need the Chinese Remainder Theorem? It seems like it should follow easily from what C(n,m) represents but it is hard for me for some reason. Homework Equations The Attempt...

Homework Statement "Tell whether the statement is true and give a counterexample if it is false" "Let m > 0 and (m, ab) = 1. If neither x^2 congruent to a (mod m) nor y^2 congruent to b (mod m) is solvable, then z^2 congruent to ab (mod m) is solveable. Homework Equations Legendre...

Homework Statement Let a be a generator of F_q Prove that a^i is a generator if & only if i and q-1 are relatively prime. Homework Equations a is a generator of F_q means that a^(q-1) = 1 and a^i cannot be 1 for all i not q-1. relatively prime means that gcd(i,q-1)=1 fermats...

Homework Statement Prove that \frac{1}{p} c(p,n) = (-1)^{n-1}/n (mod p) I expanded that combination in every way I could think and I tried to use Wilson's Theorem and I couldn't get :(Homework Equations The Attempt at a Solution

Prove that 1/5*n^5+1/3*n^3+7/15*n is an integer for all integers n.

number theory problem For every prime p \ge 5, prove that p^2-1 is evenly divisible by 24(gives an integer answer). Example: for p=5, {5^2-1 \over 24} = 1

is there a fuzzy number theory, u know on the lines of the normal number theory? could someone pls tell me abt any book or website that deals with this!

Does elementary number theory have any applications in physics? If so how?

Homework Statement For what natural numbers n is (10^n)-1 divisible by 73? Homework Equations N/A The Attempt at a Solution I have already found that it holds when n is a power of two greater than 8. (That means when n is great than 8, not the eigth power of 2) What other natural...

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 a silly question from our book, that seems to be a pain to solve: ------------ Solve the following equation: 59^x \equiv 63 \ \text{mod 71} ------------ The idea is to use the discrete log (or index). Note that 7 is a primitive root mod 71. The two books I have looked at...

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 an odd prime. Show that x^2 \equiv 2 (mod p) has a solution if and only if p \equiv 1 (mod 8) or p \equiv -1 (mod 8) The Attempt at a Solution Ok, I figured the more of these I try, the better I'll get at them. Assuming that x^2 \equiv 2 (mod p) has a...

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...

Edit: I finally figured this problem out, although I didn't use residue systems, so a solution is no longer necessary. Homework Statement For this problem, I need to find a fomula for the largest unattainable score in a game where you can either score m or n points at a time (m and n are...

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...