Number theory Definition and 460 Threads

So I am kind of lost... I don't really know how to ask this. Project Euler is a website that hosts multiple programming contests and I am interested in this problem https://projecteuler.net/problem=608 but my question isn't truly about this problem but a more solution. I know that the Divisor...

In the last part of https://en.wikipedia.org/wiki/Riemann_zeta_function#Mellin-type_integrals, I read two expressions of Riemann's zeta function ζ(s) in terms of s and of integrals of the prime-counting function π(x) (the second one using Riemann's prime-counting function J(x) from which, the...

Ten mutually distinct non-zero reals are given such that for any two, either their sum or their product is rational. Prove that squares of all these numbers are rational. I tried using 3 of those numbers - a, b and c. And I checked each of the possible situations but I'm not sure if my maths...

Hi everyone, I am trying to learn the underlying number theory concepts behind cryptography, and I was wondering if anyone knows of good resources for learning about number theory as applied to cryptography. I was hoping to practice writing proofs as well. Thanks!

Homework Statement please see the image Homework Equations I'm not sure if this is relevant: ##r_2 \leq \frac{1}{2}r_1## ... ##r_n \leq (\frac{1}{2})^nr_1## The Attempt at a Solution i have shown that ##r_{i+2} < r_i## by showing the ##r_{i+2} - r_i## is negative, but how do I show that the...

RULES: 1) In order for a solution to count, a full derivation or proof must be given. Answers with no proof will be ignored. 2) It is fine to use nontrivial results without proof as long as you cite them and as long as it is "common knowledge to all mathematicians". Whether the latter is...

Can anyone help me with this divisibility problem. My approach:- 24 = 2*2*2*3 Now, This can be written as (r-1)(r)(r+1)(3r+2) There will be a multiple of 2 and a multiple of 3. But how to prove that there are more multiples of 2. PLEASE REPLY FAST!

Hi , everyone! This is my first post/thread/anything on this forum so first I apologise if I slip up or make any mistakes. Anyway, my question is about recommendations for textbooks for Undergraduate Number Theory. So far, I have studied Calculus 1-3 (including things like line integrals...

Homework Statement this problem came out in the math olympiad i took today and i got completely wrecked by this consider the following equation where m and n are positive integers: 3m + 3n - 8m - 4n! = 680 determine the sum all possible values of m: Homework Equations not sure which The...

so this is the question: let a and b be real numbers such that 0<a<b. Suppose that a3 = 3a -1 and b3 = 3b -1. Find the value of b2 -a. initially my line of thinking was that just solve the equation x3 - 3x +1 = 0 and take the roots which are more than 0 and then after that i got stuck ok that...

Homework Statement Prove that there are infinitely many primes using Mersenne Primes, or show that it cannot be proven with Mersenne Primes. Homework Equations A Mersenne prime has the form: M = 2k - 1 The Attempt at a Solution Lemma: If k is a prime, then M = 2k - 1 is a prime. Proof of...

Homework Statement So basically for n ∈ {1, ... , 16} Find the lowest t to satisfy nt ≡ 1 (mod 17) Homework Equations Euler's Theorem tells us that the order, t, must be a divisor of φ(17), which is Euler's Phi Function. φ(17) = 16 t ∈ {1, 2, 4, 8, 16} The Attempt at a Solution n = 1 11 ≡ 1...

Homework Statement Find all integer solutions to x2 + y2 + z2 = 51. Use "without loss of generality." Homework Equations The Attempt at a Solution My informal proof attempt: Let x, y, z be some integers such that x, y, z = (0 or 1 or 2 or 3) mod 4 Then x2, y2, y2 = (0 or 1) mod 4 So x2 +...

I found a deduction to determinate de sum of the first n squares. However there is a part on it that i didn't understood. We use the next definition: (k+1)^3 - k^3 = 3k^2 + 3k +1, then we define k= 1, ... , n and then we sum... (n+1)^3 -1 = 3\sum_{k=0}^{n}k^{2} +3\sum_{k=0}^{n}k+ n The...

Hi everyone. This is my proof (?)of ramanujan's problem 525: http://www.imsc.res.in/~rao/ramanujan/collectedpapers/question/q525.htm (link to problem) [![enter image description here][1]][1] $$ \sqrt{A^{1/3}-B^{1/3}}=\frac{(A*B/10)^{1/3}+(A \times B)^{1/3}-(A^2)^{1/3}}{3} \Leftrightarrow \\ 9...

I am new to this forum. I am an electrical engineer designing frequency synthesizers for electronic test and measurement equipment. I have a design problem and I think that number theory could help me solve it. I'm not a mathematician, so I will state the problem the best I can. Definitions...

I am new to number theory and I heard from my friend that we can use modular arithmetic to conveniently find the unit digit of a number or the remainder obtained on dividing a number by another number such as the remainder obtained on dividing (x^y) by a. Is it possible?How can we do this?

Homework Statement Hello all, I am trying to determine the last hexadecimal digit of a sum of rather large factorials. To start, I have the sum 990! + 991! +...+1000!. I am trying to find the last hex digit of a larger sum than this, but I think all I need is a push in the right direction...

The given question required me to solve 7x+4y =100 by using diophantine equation . I get an answer for x = 100 - 4t and y = 200 - 7t . But his given answer is x = 4t and y = 25-7t . I think both of mine and the answer given is correct but I can't figure out how he get another solution

I am currently an undergraduate students at university and i am keen on learning some mathematics that is not taught in school and i have chosen number theory as my main topic . Recently I have picked number theory by Hardy but I found it is quite hard to understand sometimes as I have quite a...

The problem Consider field ##(F, +, \cdot), \ F = \{ 0,1,2,3 \}## With the addition table: Find a multiplication table. The attempt Please read the most of my answer before writing a reply. My solution was $$ \begin{array}{|c|c|c|} \hline \cdot & 0 & 1 & 2 & 3 \\\hline 0 & 0 & 0 & 0 & 0...

The problem Consider the ring ##(Z_{12}, \otimes, \oplus)## Find all units. The attempt I know that I am supposed to find units u such that ##gcd(12,u)=1## But how do I do it the easiest way? I am not very keen to draw a multiplication table, calculate the terms and search where the...

Homework Statement Ok so this isn't really a problem, more like a problem set, I'm not sure if I'm able to understand it yet. The context is determining all the primitive pythogrean triples Letting x = a/c and y = b/c, we see that (x, y) is a point on the unit circle with rational number...

Homework Statement 1. If a,b and c are natural numbers and a, b are coprime and a divides bc then prove that a divides c 2. Prove that the lcm of a,b is ab / gcd(a,b)Homework Equations if a is a divisor of b then a = mb for a natural number m if a prime p is a divisor of ab then p is a divisor...

Hello I'm reading through George Andrews' Number Theory at the moment and I spent the last day working on this proof. I wanted to know if anyone could tell me how legitimate my proof is because I was pretty confused by this problem. The problem is to prove that the least common multiple of two...

Homework Statement You may use pen-and-paper and mental calculation. You have 6 minutes time. Give final digit of $$ (22)^3 ~+ (33)^3~ +(44)^3~+(55)^3~ +(66)^3~+(77)^3 $$ Homework Equations 3. The Attempt at a Solution [/B] I'm not terribly good at mental arithmetic myself. I was never...

On a particular bus line, between Station A and Station J, there are 8 other stations. Two types of buses, Express and Regular, are used. The speed of an Express bus is 1.2 times that of a Regular bus. Regular buses stop at every station, while Express buses stop only once. A bus stops for 3...

Dear Physics Forum friends, what are some good books for learning the p-adic numbers? What are the necessary pre-requisites? Do I need to know introductory number theory or basics of algebraic/analytic number theory?

I can prove the twin prime counting function has this form: \pi_2(n)=f(n)+\pi(n)+\pi(n+2)-n-1, where \pi_2(n) is the twin prime counting function, f(n) is the number of twin composites less than or equal to n and \pi(n) is the prime counting function. At n=p_n, this becomes \pi_2(p_n) =...

Dear Physics Forum advisers, Could you recommend me some brief, introductory books on the number theory I can read for few weeks before jumping into the analytic number theory? Big part of my near-future research project will involve a lot of the analytic number theory, so it is needed to read...

What is the most motivating way to introduce group theory to first year undergraduate students? I am looking for some real life motivation or something which has a real impact.

I've been working on a problem for a couple of days now and I wanted to see if anyone here had an idea whether this was already proven or where I could find some guidance. I feel this problem is connected to the multinomial theorem but the multinomial theorem is not really what I need . Perhaps...

I just want to make sure I understand these number theory proofs. b^{\phi (n)}=1mod(n) \phi (n) is the order of the group, so b to some power will equal the identity. so that's why it is equal to one. b^p=bmod(p) b^p=b^{p-1}b b^{p-1} produces the identity since p-1 is the...

Hello I am currently learning some of the basics of number theory, and struggling to understand this Theorem. Could someone please explain it with maby a simple example? :) THRM:(Number of polynomial zero mod p and H) Let p be a prime number and let H be a polynomial that is irruducible modulo...

Homework Statement : Recently, a group of fellow math nerds and myself stumbled upon an interesting problem. The problem is stated: "Find the average number of representations of a positive integer as the sum of two squares." Relevant equations: N = a^(2) + b^(2), where a and b can be 1 or...

Recently I noticed something odd about the triangular numbers. The basic definition is \displaystyle\sum_{x=1}^{n}x=T_n A short time after playing around with T_n values I discovered something very odd-another formula for triangular numbers involving the root of the sum of cubes from 1 to n...

If we have a positive integer, how many ways can this number be written as a sum of its components? By components, I mean all numbers less than that number. For example, 5 has 6 ways to be written; 5x1, 3x1+2, 2x2+1, 2x1+3,1+4 and 2+3. In digits form; [11111, 1112, 221,113, 14, 23] So there are...

Hello, I'm looking for an introductory number theory book for high school. Any recommendations are welcome. Thanks.

The question at hand: Let X be a 10-adic number. Let n be a natural number (not 0). Show that A^10 has the same n+1 last digits as 1 if A has the same n last digits as 1 (notation: A =[n]= 1) My work so far: (1-X)^10 = (1-X)(1+X+X^2+...+X^10) A =[n]= 1 1-A =[n]= 0. I think I can also say that...

Homework Statement :[/B] Determine whether there exists an integer x such that x^2 + 10 is a perfect square. Homework Equations :[/B] N/A The Attempt at a Solution :[/B] Assume x^2 + 10 = k^2 (a perfect square). Solve for x in terms of k: x = ±sqrt(k^2 - 10) Since k is an integer and k^2 -...

We have the set:S={1<a<n:gcd(a,n)=1,a^(n-1)=/1(modn)} Are there prime numbers n for which S=/0?After this, are there any composite numbers n for which S=0? (with =/ i mean the 'not equal' and '0' is the empty set) for the first one i know that there are no n prime numbers suh that S to be not...

Dear Physics Forum advisers, I am currently looking for an introductory textbook that covers the number theory without being too focused on the algebraic and analytical aspects of NT. My current underaduate research in the theoretical computer science and the Putnam preparation led me to the...

Homework Statement ##x_1+x_2 \cdots x_{251}=708## has a certain # of solutions in positive integers ##x_1 \cdots x_{251}## Now the equation ##y_1+y_2 \cdots y_{n}=708## also has the same number of positive integer solutions ##y_1, \cdots y_n## Where ##n \neq251## What is ##n## Homework...

Homework Statement Let ##x,y,z## be positive integers such that ##\sqrt{x+2\sqrt{2015}}=\sqrt{y}+\sqrt{z}## find the smallest possible value of ##x## Homework Equations Not even sure what to ask I'm trying to learn number theory doing problems and look up information by doing the problems...

It seems that no matter how unrelated two subjects of mathematics appear to be, there are always ways to use techniques from one area of math and use it to prove many useful results in the other, and vice versa. However, from my (inexperienced) point of view, number theory seems to be the only...

Homework Statement 1. Find an integer modulo 19 with each of the following orders of 2 and 3. 2. Find all integers modulo 17 such that its order modulo 17 is 4. Homework Equations The multiplicative order of a modulo n, denoted by ordn(a), is the smallest integer k > 0 such that ak ≡ 1 (mod...

**Observations:** Given a power Diophantine equation of ##k## variables and there exists a “general solution” (provides infinite integer solutions) to the equation which makes the equation true for any integer. 1. The “general solution” (provides infinite integer solutions) is an...

The following is a repost from 2008 from someone else as there was no solution offered or provided I thought id post one here 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...

Homework Statement If a and b are both quadratic residues/nonresidues mod p & q where p and q are distinct odd primes and a and b are not divisible by p or q, Then x2 = ab (mod pq) Homework Equations Legendre symbols: (a/p) = (b/p) and (a/q) = (b/q) quadratic residue means x2 = a (mod p) The...

Dear Physics Forum friends, I am a college sophomore in US with double majors in mathematics and microbiology. My algorithmic biology research got me passionate about the number theory and analysis, and I have been pursuing a mathematics major starting on this Spring semester. I have been...