Integers Definition and 474 Threads

An integer (from the Latin integer meaning "whole") is colloquially defined as a number that can be written without a fractional component. For example, 21, 4, 0, and −2048 are integers, while 9.75, 5+1/2, and √2 are not.
The set of integers consists of zero (0), the positive natural numbers (1, 2, 3, ...), also called whole numbers or counting numbers, and their additive inverses (the negative integers, i.e., −1, −2, −3, ...). The set of integers is often denoted by the boldface (Z) or blackboard bold



(

Z

)


{\displaystyle (\mathbb {Z} )}
letter "Z"—standing originally for the German word Zahlen ("numbers").ℤ is a subset of the set of all rational numbers ℚ, which in turn is a subset of the real numbers ℝ. Like the natural numbers, ℤ is countably infinite.
The integers form the smallest group and the smallest ring containing the natural numbers. In algebraic number theory, the integers are sometimes qualified as rational integers to distinguish them from the more general algebraic integers. In fact, (rational) integers are algebraic integers that are also rational numbers.

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  1. L

    Find all positive integers n such that 3x^2-y^2=2018^n has an integer solution

    I need an idea. Thank you.
  2. Heisenberg7

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  3. chwala

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  4. chwala

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  5. M

    I Is this the correct way to quantify these integers?

    Does the above quantifier represent/symbolize that all of the integers ## a, b, c, d ## cannot be ## 0 ##? Is this correct?
  6. M

    I Planck’s constant: Why are atom vibration frequencies integers only?

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  7. PhysicsRock

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  8. anemone

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  9. S

    I Union of Prime Numbers & Non-Powers of Integers: Usage & Contexts

    Is there a name for the union of {prime numbers} and {integers that are not powers of integers}? For example, we would include 2, 3, 5, 7, 11... And also 6, 10, 12... But we exclude 2^n, 3^n, ... and 6^n , 10^n , etc. What are some interesting contexts where this set crops up?
  10. brotherbobby

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  11. M

    Determine all integers ## n ##

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  12. M

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  13. M

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  14. anemone

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  15. M

    How to obtain three consecutive integers?

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  16. D

    I Primes -- Probability that the sum of two random integers is Prime

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  17. M

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  18. M

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  19. M

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  20. M

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  21. M

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  23. M

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  24. M

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  25. M

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  26. arivero

    I Can integers be defined as N[[sqrt(1)]]?

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  27. Andrew1235

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  28. J

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  29. D

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  30. A

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  31. C

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  32. M

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  33. Arman777

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  34. maistral

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  35. C

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  36. A

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  37. PeroK

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  38. B

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  39. DanjoJojo

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  40. brotherbobby

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  41. LCSphysicist

    B What is the identity element in the group {2,4,6,8} under multiplication mod 10?

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  42. A

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  43. anemone

    MHB Prove that the sum of 6 positive integers is a composite number

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  44. anemone

    MHB Positive Integers: Evaluate $a+b+c$

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  45. Svein

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  46. D

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  47. anemone

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  48. J

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  49. e2m2a

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  50. ?

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