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  • ...than <math>1</math> and less than <math>n</math>. One only needs to check integers up to <math>\sqrt{n}</math> because dividing larger numbers would result in === Gaussian Primes ===
    6 KB (1,052 words) - 18:22, 1 January 2025
  • ...math>\mathbb{Z}[i]</math>). Note that of these, the integers and Gaussian integers do not have inverses; the rest do, and therefore also constitute examples o Among the finite commutative rings are sets of integers mod <math>m</math> (<math>\mathbb{Z}/m\mathbb{Z}</math>), for any integer <
    6 KB (994 words) - 05:16, 8 April 2015
  • ...also exist quadratic reciprocity laws in other [[ring of integers|rings of integers]]. (I'll put that here later if I remember.) ...primitive <math>q</math>th root of unity in <math>K</math>. We define the Gaussian sum
    7 KB (1,182 words) - 15:46, 28 April 2016
  • ...l and physics terms named after him including the [[Gaussian integer]]s, [[Gaussian distribution]]s, and [[Gauss's Law]].
    816 bytes (118 words) - 10:17, 27 September 2024
  • * The ring of [[integers]] <math>\mathbb Z</math> with norm given by <math>N(a) = |a|</math>. * The ring of [[Gaussian integers]] <math>\mathbb Z[i]</math> with norm given by <math>N(a+bi) = a^2+b^2</mat
    2 KB (357 words) - 14:28, 22 August 2009
  • Prove that there are infinitely many positive integers <math>n</math> such that <math>n^{2} + 1</math> has a prime divisor greater The main idea is to take a gaussian prime <math>a+bi</math> and multiply it by a "twice as small" <math>c+di</m
    5 KB (1,002 words) - 00:09, 19 November 2023
  • The theorem, with some study of the Gaussian integers <math>\mathbb{Z}[i]</math>, resolves these questions for the form <math>x^2 We use the fact that the set of [[Gaussian integer]]s <math>\mathbb{Z}[i]</math> has a [[Euclidean algorithm]]. Hence
    4 KB (612 words) - 11:10, 30 May 2019
  • A '''Gaussian integer''' is any [[complex number]] of the form <math>a + bi</math> where
    196 bytes (34 words) - 19:05, 23 January 2017
  • * The [[Gaussian integer]]s, <math>\mathbb Z[i]</math>
    6 KB (1,217 words) - 22:05, 23 August 2009
  • Find the last three digits of the number of 7-tuples of positive integers <math>(a_1, a_2, a_3, a_4, a_5, a_6, a_7)</math> such that <math>a_1 \, | \ ...e{1}1\underline{1}110</math>. Find the last three digits of the sum of all integers <math>N</math> with <math>1 \leq N \leq 81</math> such that <math>N</math>
    8 KB (1,377 words) - 12:20, 2 August 2024
  • ...d <math> m_i=10m_{i-1}+1 </math> for <math> i>1 </math>. How many of these integers are divisible by <math> 37 </math>? ...nteger factors of smaller absolute value. Factor <math> -4+7i </math> into Gaussian primes with positive real parts. <math> i </math> is a symbol with the prop
    3 KB (464 words) - 10:02, 27 May 2012
  • ...nteger factors of smaller absolute value. Factor <math> -4+7i </math> into Gaussian primes with positive real parts. <math> i </math> is a symbol with the prop
    1 KB (224 words) - 07:43, 28 May 2012
  • ...theory]] that, in a nutshell, extends various properties of the [[integer|integers]] to more general [[ring|rings]] and [[field|fields]]. In doing so, many qu ...particular, leads to [[factorization]] in the [[Gaussian integer|Gaussian integers]].
    10 KB (1,646 words) - 14:04, 28 May 2020
  • ...th one thing which without it no arithmetic can survive: Counting Positive Integers (whole numbers). Addition is combining these integers. The symbol for combining numbers is +. <math>a+b=b+a</math>
    35 KB (5,884 words) - 09:25, 7 December 2024
  • ...is defined to be a complex number whose real and imaginary parts are both integers.) ...s are all <math>i</math>-multiples of each other, so they are all Gaussian integers, and none of them lie in the same quadrant. Furthermore, because, from the
    2 KB (440 words) - 11:06, 2 August 2024