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 By the early twentieth century, it had been superseded by "number theory".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. D

    Help for Number Theory problem

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

    Proving Even Divisibility of p^2-1 for Primes p≥5 in Number Theory

    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
  3. P

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  4. P

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

    Interesting number theory question

    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...
  6. F

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  7. Gib Z

    Solving (p-1)(p^n+1)=4m(m+1) for odd primes p and positive integers m and n

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  8. B

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  9. mattmns

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  10. mattmns

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  11. mattmns

    Number Theory - Primitive Roots

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  12. T

    What Is the Smallest Integer N for Which 10N Is a Square and 6N Is a Cube?

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  13. T

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  14. mattmns

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  15. mattmns

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  16. E

    Can Odd Primes Solve x^2 ≡ 2 (mod p)?

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  17. E

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  18. F

    Number Theory - Largest Impossible Score

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  19. mattmns

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  20. mattmns

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  21. mattmns

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  22. L

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  23. F

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  24. A

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  25. M

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  26. D

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  27. C

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  28. D

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  29. O

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  30. M

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  31. J

    Solving the Mystery of Number Theory: 1/3+1/3^2+1/3^3... + 1/3^n=1/2(1-1/3^n)

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  32. T

    How Do You Prove the Relationship Between LCM and GCD in Number Theory?

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  33. B

    Number Theory Help: Find Solutions of 3x^5≡1(mod 23)

    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 +...
  34. R

    Master Number Theory Questions Easily | Proving Formulas & Theorems"

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  35. E

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    "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...
  36. B

    Number Theory Help: Showing 25 is Strong Pseudoprime to Base 7

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  37. murshid_islam

    Best Number Theory Books for Self Study | Ogilvy & Hardy

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  38. R

    Number Theory Proofs: Square Numbers and Irrational Roots

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  39. B

    Number Theory Help: Solving Problems & Understanding Repunits

    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...
  40. B

    Discover the Triple Peak of Your Life with Number Theory Help

    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...
  41. R

    Exploring Number Theory Questions: From Proofs to Existence

    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...
  42. V

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  43. A

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    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...
  44. B

    Any Good Literature or Books on Number Theory

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  45. T

    What is a good introductory textbook for Number Theory at the university level?

    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...
  46. S

    God, I hate Number Theory Proofs

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  47. B

    Can the Extended Euclidean Algorithm Solve Equations in Number Theory?

    Hi I got a question: I have the following problem ax + by = c. Where a,b are positive integers, and c is a known integer. If I calculate the gcd(a,b) is it then possible to find the x,y which makes the above equation true ? Best Regards, Bob
  48. S

    Help with 2 Number Theory Problems

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  49. E

    Is There a Relationship Between g(1-s) and g(s) in L-Dirichlet Series?

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  50. P

    Solve 99^99 congruent to (mod 47) using Fermat's Theorem

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