Number theory Definition and 475 Threads

Hello all, I probably should have posted this in a math forum but I don't know of any. Can anyone recommend a book/books on elementary number theory with exercises? My math background is not very strong with very little knowledge of set theory so it should be understood by me. We're covering...

Brocard's problem is a problem in mathematics that asks to find integer values of n for which $$x^{2}-1=n!$$ http://en.wikipedia.org/wiki/Brocard%27s_problem. According to Brocard's problem $$x^{2}-1=n!=5!*(5+1)(5+2)...(5+s)$$ (x,n) is the solution tuple of the problem. If there are infinite...

Brocard's problem is a problem in mathematics that asks to find integer values of n for which $$x^{2}-1=n!$$ http://en.wikipedia.org/wiki/Brocard's_problem. According to Brocard's problem ##x^{2}-1=n!=5!*(5+1)(5+2)...(5+s)## here,##(5+1)(5+2)...(5+s)=\mathcal{O}(5^{r}),5!=k##. So, ##x^{2}-1=k...

Homework Statement How many values of k can be determined in general, such that (k/p) = ((k+1) /p) = 1, where 1 =< k <=p-1? Note: (k/p) and ((k+1)/p) are legendre symbols Question is more clearer on the image attached.Homework Equations On image. The Attempt at a Solution I've tried...

The Diophantine equation below, $$ x_0^{2} - (x_1^{2}+x_2^{2}+x_3^{2}+x_4^{2}+x_5^{2}+x_6^{2}+x_7^{2}+x_8^{2})=1$$ 1. Does above equation have any specific name? 2. What are the solutions(a formula)?? 3. in the case,$$x_8^{2}=0$$ , does anything special happen?? 4. What is the general way...

$$x^{2}+1 \neq n! $$since $$x^{2}+1=(x+i)(x-i) $$so ,$$ x^{2}+1$$ has only prime of the form of (4k+1) , where n! has prime of the form( 4k-1) and (4k+1) . :oldbiggrin:

I have encountered the below problem- Given, ##z(z-1)## has all prime < ##\sqrt{z} <n## , Prove(or disprove)- ## π(z)-w(z-1)-A= π(2z-1)- π(z) ## where A={0 ,1}, π (z) is the prime counting function, π(2z-1)- π(z) is the number of primes in between z and (2z-1), ##\omega(z-1)## is the number...

https://www.quantamagazine.org/20150312-mathematicians-chase-moonshines-shadow/ Sorry for the long title but ST = string theory. Just thought it was interesting news personally since string theory has been elusively hard to prove or observe(at least the particles it claims to predict, notably...

x^2=n!+1⇒ (x+1)(x-1)=n! where (x+1)/2 and (x-1)/2 are consecutive integers and have consecutive primes as factor ,let ,y and z (respectively) so it can be written y-1=z. Consider prime counting function π(z),π(2z-1) that count primes less than the variable or argument. It can be seen that f(z)...

Homework Statement let m|d, n|d and gcd(m,n) = 1. show mn|d Homework Equations gcd(m,n) = d = mx + ny for x and y in integers The Attempt at a Solution d = mr d = ns 1 = mx + ny 1 = (d/r)x + (d/s)y I don't know, a bit lost, just moving stuff around and not making any real progress. Any tips?

Homework Statement Show that gcd(a+b,a-b) is either 1 or 2. (hint, show that d|2a and d|2b) Homework Equations d = x(a+b)+y(a-b) The Attempt at a Solution so by the definition of divisibility: a+b = dr a-b = ds adding and subtracting these equalities from each other we can arrive at where...

1) 3^(2^a) + 1 divides 3^(2^b) -1 2) If d > 2, d ∈ N, then d does not divide both 3^(2^a) + 1 and 3^(2^b) -1 Attempt: Set b = s+a for s ∈ N m = 3^(2^a). Then 3^(2^b) - 1 = 3^[(2^a)(2^s)]-1 = m^(2^s) -1 Thus, m+1 and m-1 divides m^(2^s) -1 by induction. If s = 1, then m^(2^s) -1 = m^2 -...

I am inquiring about a good introductory statistics book or books, that supplement each other well. My math background consist of calculus 2, linear algebra, and ODE. This is for a first course in statistics. Also, what would be a good introductory number theory book? Or should I complete...

First, I'm in need of a topic to write a paper about as part of my degree requirements. The paper is supposed to be 10-15 pages. (I only mention this because it limits the scope of the paper and thus the topic.) I took a course in number theory last semester and really enjoyed it, so I'm really...

Homework Statement Okay: How many numbers divide 1000 that are multiples of 5 I have seen you do 1000/5 = 200 But how does this mean there are 200 numbers that divide 1000 that are multiples of 5? This just says: 1000 divided into 5 equal pieces, is 200. So how does this give how many...

Homework Statement if 1 = gcd(a,b), show that gcd(ac,b) = gcd(c,b) Homework Equations None The Attempt at a Solution My attempt at a solution: Let d = gcd(ac,b), Let g = gcd(c,b), I want to show that g|d and that d|g. I then went on to make a bunch of circular writing and get nowhere... I...

Homework Statement Let m be the number of numbers fromantic the set {1,2,3,...,2014} which can be expressed as difference of squares of two non negative integers. The sum of the digits of m is ... Homework EquationsThe Attempt at a Solution I got a solution from a magazine but I didn't under...

Homework Statement Not actually for homework, but i didn't know where to post this. Problem: Show that any integer to the fourth power can be expressed as either 5k or 5k+1 where k is an integer. Homework Equations None. The Attempt at a Solution My starting point is to consider that all...

Homework Statement This is a problem I had as a margin note in an old notebook that I will recycle. I want write it using LaTeX. Problem is that I also want to write it using "proper" math notation instead of English words. Firstly, I got this: \textrm{Proof that }\nexists x, y \in...

Homework Statement Let p be a prime, k be positive integer, and m ∈ {1, 2, 3, ..., pk-1}. Without using Lucas' theorem, prove that p divides \binom{p^k}{m}. Homework Equations The definition of the binomial coefficients: \binom{a}{b} = \frac{a!}{b! (a-b)!} The Attempt at a Solution I've...

Pretend you own a printing press and you want to be able to represent any arbitrarily large natural number. You also want to store the fewest possible number of characters in your collection, just to save space. What base, if any, would yield the largest ratio of numbers you can represent to...

Greetings, I am looking for a accesable introduction to the field of number theory that leads up to primes eulers proof of infinite primes, goldbach proof of inifinite primes and their deriviations(the deriviations are the most important and should be clear if possible) and so on. I have a...

I have recently been doing some reading (skimming really) some books on number theory, particularly algebraic number theory. While number theory seems to draw heavily on rings and fields (especially some special types of rings like Euclidean rings and domains, unique factorization domains etc)...

Let $x$ and $y$ be positve integers such that $xy$ divides $x^2+y^2+1$. Show that $$\frac{x^2+y^2+1}{xy}=3$$

Isn't it amusing ?What could be the probable explanation for this?Also when operated by division operator gives the rest of the number as the quotient (Note only when the divisor is 10)

Let L be the level number of a bipartite graph G, and so L1 be the first level of n1 vertices, L2 be the second level of n2 vertices, ... Lk be the kth level of nk vertices. Then a bipartite graph G12 is created by a combination of L1 and L2, G23 is of L2 and L3,...,Gij is of Li and Lj. The...

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Hi! (Cool) I am given the following exercise:Try to solve the diophantine equation $x^2+y^2=z^2$ , using methods of elementary Number Theory. So, do I have to write the proof of the theorem: The non-trivial solutions of $x^2+y^2=z^2$ are given by the formulas: $$x=\pm d(u^2-v^2), y=\pm 2duv...

Let $S$ be a nonempty set of natural numbers, equipped with the following membership rules: $$\text{if} ~ ~ x \in S ~ ~ \text{then} ~ ~ 4x \in S \tag{1}$$ $$\text{if} ~ ~ x \in S ~ ~ \text{then} ~ ~ \lfloor \sqrt{x} \rfloor \in S \tag{2}$$ Show that $S = \mathbb{N}$, and find all the natural...

Why are perfect numbers important? What is the best way of introducing these numbers on a first course in number theory? I could not find any application apart from their connection to Mersenne primes. Are there any applications of such numbers?

Homework Statement Basically, I'm working on a problem where I'm supposed to find a missing digit in a social security number. The number is as follows: 301 X91 - 2005. where X is the missing digit. Now, how these numbers are constructed, is that the first six numbers are the persons...

1. show that the sum of. The reciprocals of the primes is divergent. I am reposying this here under homework and deleting the inital improperly placed post 2. Theorem i use but don't prove because its assumed thw student has already lim a^1/n = 1. The gist of the approach I took is that∑1/p =...

Problem 1 Suppose ab=cd, where a, b, c d \in N. Prove that a^{2}+b^{2}+c^{2}+d^{2} is composite. Attempt ab=cd suggests that a=xy, b=zt, c=xz. d=yt. xyzt=xzyt. So (xy)^{2}+(zt)^{2}+(xz)^{2}+(yt)^{2}=x^{2}(y^{2}+z^{2})+t^{2}(z^{2}+y^{2})=(x^{2}+t^{2})(z^{2}+y^{2}) Therefore this is...

Which course do you think is more important or interesting to take for someone interested in theoretical computer science or theoretical mathematics, number theory or abstract algebra? I am mainly interested acquiring skills and knowledge that will enable me to prove something significant...

This is my first time posting anything on the forum so I apologize if I do anything wrong. I have enrolled myself into elementary number theory thinking we would be taught how to do proofs however it is apparently expected that we already know how to do this. And so since I am a beginner at...

Does anyone know of a reference work that lists natural numbers with unique properties? Like 26, for example, being the only natural number sandwiched between a square (25) and a cube (27). Does such a reference book exist? IH

Hello :) That's my 2nd year in Math, and I want to start writing an article on NT or Group Theory. I know most of the basic GT and some NT. I still don't know residues/congruences completely, I face problems about understanding the theorems. There are a lot of theorems in these chapters and...

Homework Statement What does triangle line mean? What is "+" for sets here? Once I know that, if I need assistance, I will show an attempt. Otherwise I will be satisfied. :) Homework Equations The Attempt at a Solution

Does anyone know any recommended Number Theory textbooks for independent study? I have a few lecture notes/eBooks, but I always prefer having a physical textbook which I can read and learn from. I would prefer a textbook with exercises + answers, so I can check if my answers are correct. I'm...

So could someone please clarify these: a|b and a|c then a|bx+cy for any x,y integers? a|b and b|c then a|bx+cy for any x,y integers? seems the two are very similar, but are those both theorems?

If a|b then ac=b; now does c always divide b as well?

Conjecture: suppose n is an integer larger than 1 and n is not prime. Then 2^n-1 is not prime. Proof attached. Could someone please explain to me how they got to xy= 2^(ab)-1. I see the -1 part. Also I think I do not understand the concept of 2^((a-1)b) I mean is it some index or some...

Conjecture: suppose n is an integer larger than 1 and n is not prime. Then 2^n-1 is not prime. Proof attached. Could someone please explain to me how they got to xy= 2^(ab)-1. I see the -1 part. Also I think I do not understand the concept of 2^((a-1)b) I mean is it some index or some...

Okay if a/b then doesn't a/nb for any integer n?

Hey Guys, This is my first post to Physics Forums. If I posted this question in the wrong area or am violating some other etiquette, please let me know! I'm working on a proof and am currently stuck. I'm trying to prove that x (a weird number, an infinite product to be specific), is NOT...

Find the least positive integer x such that x=5 (mod 7), x=7 (mod 11) and x=3(mod 13). How to proceed?

Show that 2x^3+x^2+3x-1 = 0 (mod 5) has exactly three solutionsHow to proceed with it?

An integer is divisible by 9 if and only if the sum of its digits is divisible by 9 Proof by induction?

I find that in number theory, number theorists (and mathematicians more generally) generally prefer elementary proofs over any other kind of proof. Am I right about this? If so, why is this? Is this something to do with the content of number theory itself? Thanks!

How does one show that a prime element in a Ring is irreducible and how does one show that ##|| x || = 1## iff x is a unit. okay from my knowledge I know that units are invertible elements, so how does the norm of x make it 1... maybe I am not too sure about this