Number theory Definition and 475 Threads

Recall the definition of n! (read n factorial"): n! = (n)(n-1)(n-2) ….(2)(1) =∏(k) In both (a) and (b) below, suppose p≥3 is prime. (a) Prove that if x∈ Zpx is a solution to x square ≡1 (mod p), then x ≡±1 (mod p). (b) Prove that (p-1)!≡±1 (mod p) Zpx x shoud be above p a and b looks...

Im really good at number theory but how to show this statement has me stumped! "Show that among the positive integers greater than or equal to 8, between any two cubes there are at least 2 squares"

Homework Statement Find the number of roots for the equation x^2+1=0 \mod n \: for \: n = 8,9,10,45 Homework Equations The Attempt at a Solution I have no idea where to start. Could someone help me understand?

Can you help me with this problem: if a^2 divides b^2, show that a divides b. This was a homework question that I had a while ago and it was solved by using the fundamental theorem of arithmetic. I instead tried to solve it with proof by contradiction: a^2 divides b^2 implies a divides...

b>1, p an odd prime b2 = -1 mod p if p|b2 + 1 show p = 1 mod 4 I know that the order of b is 4 (mod p)

Homework Statement Suppose that gcd(a, b) = 1 with a, b > 0, and let x0, y0 be any integer solution to the equation ax + by = c. Find a necessary and sufficient condition, possibly depending on a, b, c, x0, y0 that the equation have a solution with x > 0 and y > 0. Homework Equations...

I did a little searching but couldn't find anything concrete. Reading Fermats Last Theorem has sparked some interest in studying the mathematics the book includes. Could anyone point me in the right direction of getting started in number theory? Thanks

I've been stuck on this for a while now, and I was wondering if anyone could help me out. The problem is: If ax^{2}+bx+c=0, prove that all integer roots divide b I'm fairly new to number theory, but this is the one problem that's been really tough for me. If someone could even give me...

Hi, I'm basically looking for a book that is very approachable and written for someone who knows very little about math but that goes through actual math, i.e. starts with set theory, then constructs the natural numbers, the integers, yada yada and strongly emphasizes how math is...

I'm taking the class next semester, and I heard that number theory is usually a difficult subject. Is that true? If so, how should I approach it?

What have I done wrong here? Define f: A -> B (a,b) -> f(a,b) ≡ a + bc mod p Let A = {(a,b): a,b integers, 0≤a,b≤√p} B = {0,1,2,..,p-1} By pigeonhole principle there are distinct (a1,b1), (a2,b2) in A with f(a1,b1) = f(a2,b2). => a1+b1c ≡ a2+b2c mod p => (a1-a2) ≡ (b2-b1)c mod p =>...

Wilson's Theorem: (p-1)! ≡ -1 mod p Statement: As an immediate deduction from wilson's theorem we see that if p is prime with p ≡ 1 mod 4 then the congruence x2 ≡ -1 mod p has solutions x = +-(r!), where r = (p-1)/2.How do I plug in p ≡ 1 mod 4 into wilsoms theorem so I can see this? I'm...

Homework Statement a > 1, k > 0. Show that k divides \phi(a^k - 1), where \phi is Euler's totient function (Hint: use some group theory). Homework Equations If n = p_1^{a_1}p_2^{a_2}...p_m^{a_m}, then \phi(n) = n(1 - 1/p_1)(1 - 1/p_2)...(1 - 1/p_m) The Attempt at a Solution I guess...

Hello, :) Does anyone have any good ebooks or pdf's about number theory? If so is there anything like Paul G. Hewitt's Conceptual Physics? I loved the book :)

Homework Statement Let a,b,c be integers. If (b,c)=1, then (a,bc)=(a,b)(a,c) Homework Equations This is difficult to answer because some theorems that we haven't proven yet, we can't use. The Attempt at a Solution Let g=(a,b) and h=(a,c), g and h are integers. That means g|a and g|b...

Hi, i have just registered to the forum, because this time i study number theory and in some problems i can't figure out how to solve them. This time i have to prove: If two integers x,y doesn't divided with 3 then the (x^2 - y^2) always is divided with 3. Does anyone has a clue how to...

Homework Statement Let n and k be integers with n>=2 and k>=2. Prove that (n-1)|(n^k - 1). Hence prove that if n^k - 1 is prime then n=2 and k is prime. Homework Equations The Attempt at a Solution I think you go about this question by using proof by induction. However I am...

Homework Statement 1. Homework Statement If n is a nonzero integer, prove that n cam be written uniquely in the form n=(2^k)m, where It is in the primes and unique factorization chapter so maybe that every integer n (except 0 and 1) can be written as a product of primes Homework...

Homework Statement Prove that any integer n >= 2 such that n divides (n-1)! + 1 is prime. Homework Equations The Attempt at a Solution I'm having trouble getting started, I have no idea how to approach this, can someone give a hint on where to begin maybe because I'm just not...

Homework Statement A given number x, if divided by 31, the remainder is 10, if divided by 73, the remainder is 35, if divided by 111, the remainder is 29. Then, what's the number x? Homework Equations x = 31k_1 + 10 = 73k_2 + 35 = 111k_3+29, \tex{ then?} The Attempt at a...

[b]1. This is a problem involving public key cryptography [b]2. 16^31 is congruent to 081 (mod 247) [b]3. I would first evaluate 16^31 and the divided by 247 to find the remainder. I know how to work with congruences, but 16^31 is a very huge number I don't know how to evaluate it into...

Homework Statement Find all solutions in Z12 to the equation X^2=4 Homework Equations The Attempt at a Solution This is my attempt, since it is in Z12, i wrote X^2=4(modulo 12) and by this the answers are trivial, X= 2, 4,8,... and also -2,-4... this just seems to easy so i...

Use Gauss Lemma (Number theory) to calculate the Legendre Symbol (\frac{6}{13}). I know how to use Gauss Lemma. However we use the book: Ireland and Rosen. They define Gauss Lemma as: (\frac{a}{p})=(-1)^n. They say: Let \pm m_t be the least residue of ta, where m_t is positive. As t ranges...

I posted this question but I am not getting anywhere with this question, any help would be very appreciated: 1. let p be odd prime explain why: 2*4*...*(p-1)\equiv (2-p)(4-p)*...*(p-1-p)\equiv(-1)^{(p-1)/2}*1*3*...*(p-2) mod p. 2. Using number 2 and wilson's thereom [(p-1)!\equiv-1 mod p]...

I am currently taking a course in number theory, and I stumbled upon a book on amazon which relates number theory to quantum and nuclear physics... Is this possible? In what areas and how? As previously mentioned I am taking a course on it so if anyone can give the specific mathematical...

I am an undergrad taking my first course in number theory. For some reason, this is the hardest course I have ever taken in my life. It puts Calculus and Differential Equations to shame in my honest opinion. My question is, am I the only one who thinks so? I mean, I go over the chapters...

Homework Statement neither my professor nor my TA could figure this out. so they are offering fat extra credit for the following problem Let n be a positive integer greater than 1 and let p1,p2,...,pt be the primes not exceeding n. show that p1p2...pt<4n The Attempt at a Solution...

Homework Statement Show that for any fixed a and b, there is an algorithm to convert an n-digit number from base a to base b with O(n^2) operations. The Attempt at a Solution Really i am completley lost here. Working backwards, i know to convert from base a to a base b you must use...

I am currently a college student who just started taking my first number theory course this week, and with all the stuff I am learning my only question is... are there practical applications to number theory? I mean there are many theories in physics yet many of those have been put to...

Anyone here who enjoys maths in general, dislikes number theory? If so why? Have a vote.

Homework Statement If n>4 is a composite number, show that n|(n-1)! Conclude that (n-1)! not congruent -1(mod n). (This shows that Wilson's theorem can be used as a proof of primality. It is unfortunately not practical for large numbers) Homework Equations The Attempt at a Solution I...

I've never studied the number theory before, and its not something I can study as an elective. What good book could I use for self study?

I'm sure that for many of you this class is old news; but I just started elementary number theory this summer and, as much as I love the challenge of the course, and doing these proofs; I feel like an amateur boxing Mike Tyson here. These things are chewing me up and spitting me out. Granted I...

Homework Statement If a, b < c, and d are positive integers, prove the following inferences. 1. a|b \wedge c|d \rightarrow ac|bd 2. a|b <=> ac|bc Homework Equations The Attempt at a Solution 1. a|b = x, then b = ax c|d = y, then d = cy bd = axcy thus ac|bd =...

For any 10 digit natural number N in which the first digit corresponds to the total no of 1's. the 2nd digit corresponds to the total no of 2's. . . . the 10th digit corresponds to the total no of 0's. determine, with proof, if the number of such natural number N is finite, and if...

1) Suppose 2^k + 1 is a prime number. Prove that k has no prime divisors other than 2. (Hint: if k=ab with b odd, consider 2^k + 1 modulo 2^a +1) First of all, I have a little question. k=ab with b odd. Is this always possible for any natural number k? Why? Assuming it's always...

what is probabilistic Number theory ?? , i mean i think is the fact that you 'work' with the probability of a number being prime or square or something like that if possible i would like more info.

Let n=12^12. Find sigma(n). sigma(n) = the sum of the positive divisors of n.

Hi all, Consider the the number of distinct permutations of a collection of N objects having multiplicities n_1,\ldots,n_k. Call this F. Now arrange the same collection of objects into k bins, sorted by type. Consider the set of permutations such that the contents of anyone bin after...

So I'm in a number theory course this semester but we aren't using a textbook (the professor gave the explanation that they'd never found a textbook that does exactly what they wanted or something, but it seems from my investigations that our course follows a pretty standard introduction)...

Is there any good "basic" books on number theory?

what exactly is number theory i have heard of it but i don't know what it is

what exactly is numder theory i have heard of it but i don't know what it is

I'm currently taking a course, "Abstract Algebra I & Number Theory" and I'm wondering: what is the difference between abstract algebra and number theory? the two topics seem meshed together. i tried googling both of them and it doesn't really help. it's hard to tell the differences between...

Homework Statement Hi guys, i have never taken number theory yet now I am forced to quickly understand it as it was required for a class i signed up. I need help with these problems and would greatly appreciate any hints or help in the right direction. Thanks. 1)Find with proof, all n such...

Okay I hope it's okay if I have a couple question. I've been strugelling a bit with this problem set. About a quarter of the questions I just don't seem to see how to start them. Any hints would be greatly appreciated. Thank you kindly I Homework Statement Give that p\nmid n for all...

I was working on a problem with numbers of the form 2^k+n for k (potentially) large and n fixed. I pre-sieve candidate primes in the range by finding a value f such that whenever k=f\pmod{p-1} it holds that p|2^k+n, and test a large range of k values. 1. The search for such an f is...

What does the symbol Î mean in number theory? As in... Prove if r,s Î Z, then 4r + 6s is even... Also, where can I find a website with a comprehensive math symbol index?

Homework Statement http://math.stanford.edu/~vakil/putnam07/07putnam2.pdf I am working on number 2. So I want to find 2^70 + 3^70 mod 13. I can use Fermat's Little Theorem to reduce the exponent to 10, but I do not know what to do next... Homework Equations The Attempt at a...

How is measure theory associated with number theory, if at all. If they are connected, can anyone give a link?