What is Number theory: Definition and 471 Discussions

Number theory (or arithmetic or higher arithmetic in older usage) is a branch of pure mathematics devoted primarily to the study of the integers and integer-valued functions. German mathematician Carl Friedrich Gauss (1777–1855) said, "Mathematics is the queen of the sciences—and number theory is the queen of mathematics." Number theorists study prime numbers as well as the properties of mathematical objects made out of integers (for example, rational numbers) or defined as generalizations of the integers (for example, algebraic integers).
Integers can be considered either in themselves or as solutions to equations (Diophantine geometry). Questions in number theory are often best understood through the study of analytical objects (for example, the Riemann zeta function) that encode properties of the integers, primes or other number-theoretic objects in some fashion (analytic number theory). One may also study real numbers in relation to rational numbers, for example, as approximated by the latter (Diophantine approximation).
The older term for number theory is arithmetic. By the early twentieth century, it had been superseded by "number theory". (The word "arithmetic" is used by the general public to mean "elementary calculations"; it has also acquired other meanings in mathematical logic, as in Peano arithmetic, and computer science, as in floating point arithmetic.) The use of the term arithmetic for number theory regained some ground in the second half of the 20th century, arguably in part due to French influence. In particular, arithmetical is commonly preferred as an adjective to number-theoretic.

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  1. T

    What are the possible values of m in the equation 3m + 3n - 8m - 4n! = 680?

    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...
  2. T

    B Math olympiad basic number theory problem

    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...
  3. K

    Prove there are infinitely many primes using Mersenne Primes

    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...
  4. K

    Finding the orders of 1, 2, .... , 16 (mod 17)

    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...
  5. X

    (Number theory) Sum of three squares solution proof

    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 +...
  6. S

    I How Does Summing Cubic Expansions Reveal the Formula for Sum of Squares?

    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...
  7. T

    I Proof of Ramanujan's Problem 525 with A=5, B=4

    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...
  8. G

    MHB Number Theory for Electronic Signal Frequency Synthesis

    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...
  9. donaldparida

    B Can modular arithmetic help us find remainders and unit digits?

    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?
  10. S

    Hexadecimal and factorial problem

    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...
  11. Ling Min Hao

    I Solve Diophantine Eqn: 7x+4y=100 | X & Y Answers

    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
  12. Ling Min Hao

    Number Theory Is Introduction to Theory of Numbers by Hardy good ?

    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...
  13. Rectifier

    Number theory - fields, multiplication table

    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...
  14. Rectifier

    Efficient Method for Finding Units in Number Theory Rings (Z12, ⊗, ⊕)

    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...
  15. N

    I can't understand this problem

    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...
  16. N

    Can Natural Numbers a, b, c with a Dividing bc Imply a Divides c?

    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...
  17. R

    I Is this proof of the least common multiple theorem in number theory valid?

    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...
  18. L

    Number theory quick calculation problem

    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...
  19. M

    MHB Solve Number Theory Problem to Find Time for Express Bus

    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...
  20. B

    Number Theory What Are Some Recommended Books for Learning p-adic Numbers?

    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?
  21. e2theipi2026

    A Does this imply infinite twin primes?

    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) =...
  22. B

    Number Theory Concise Introduction Book to Number Theory?

    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...
  23. matqkks

    I Group Theory: Unlocking Real-World Solutions for First-Year Students

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  24. J

    A Is this product always greater than these sums?

    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...
  25. C

    I Understanding Number Theory Proofs: Order of Elements in Finite Groups

    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...
  26. K

    What is the Theorem regarding the number of polynomial zeros modulo p and H?

    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...
  27. R

    How Many Ways Can a Positive Integer Be Represented as a Sum of Two Squares?

    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...
  28. W

    Fascinating number theory relationship

    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...
  29. A

    How many ways a number can be written as components sum?

    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...
  30. K

    Number Theory Introductory number theory book

    Hello, I'm looking for an introductory number theory book for high school. Any recommendations are welcome. Thanks.
  31. T

    Proof That 10-adic Number B is Divisible by A

    The question at hand: Let A be a 10-adic number, not a zero divisor. Proof that a 10-adic number B is dividible by A if 2^q*5^p*B has ends with p+q zeroes. My work so far: Because A is not a zero divisor, it is not dividible by all powers of 2 nor 5, so it follows from a theorem that A =...
  32. T

    10-adic number proof: A^10 has the same n+1 last digits as 1

    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...
  33. Callmejoe

    Discrete Math/Introductory number theory problem

    The instructor to my discrete mathematics course gave this question to us. How do you find the smallest achievable value(V) for which all greater values are achievable using only A and/or B, when A and B are relatively prime(coprime). For example for 5 and 7 the answer is 24 (7+7+5+5). Playing...
  34. S

    Number Theory: Difference of Perfect Squares

    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 -...
  35. K

    Are there prime numbers n for which S=/0?

    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...
  36. B

    Discrete Books on the introductory number theory?

    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...
  37. Y

    {Number theory} Integer solutions

    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...
  38. Y

    {Number Theory} Smallest integer solution

    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...
  39. M

    Value of studying number theory?

    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...
  40. Shackleford

    Number Theory (2) Homework: Find Integer Orders Modulo 19 & 17

    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...
  41. S

    Condition for Power Diophantine Equation

    **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...
  42. P

    Difficult number theory problem proofs

    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...
  43. S

    If a and b are both quadratic residues/nonresidues mod p & q

    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...
  44. B

    How to do mathematics research as undergraduate?

    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...
  45. UncertaintyAjay

    Number Theory & Calculus Theorems: Looking for Interesting Ones to Prove

    So, I like proving theorems in number theory and calculus. I'd like some interesting ones to prove. Recommendations?
  46. moriheru

    Proving Equivalence of Decimals in Number Theory

    I have a question, in the field of number theory (Hardy and Wright chapter 9 representation of numbers by decimals) concerning the prove by contradiction of the statement: If Σ1∞ an/10n Σ1=∞bn/10n then an and bn must be equivalent, for if not then let aN and bN be the first pair that differ then...
  47. Mastermind01

    Number Theory Book/Books on elementary number theory

    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...
  48. S

    Is Brocard's Problem the Key to Solving Infinite Solutions?

    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...
  49. S

    An argument for "Brocard's problem has finite solution"

    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...
  50. C

    How many values of k can be determined, such that

    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...
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