https://en.wikipedia.org › wiki › Table_of_Gaussian_Integer_Factorizations
Table of Gaussian integer factorizations - WikipediaThe article is a table of Gaussian Integers x + iy followed either by an explicit factorization or followed by the label (p) if the integer is a Gaussian prime. The factorizations take the form of an optional unit multiplied by integer powers of Gaussian primes.
https://en.wikipedia.org › wiki › Gaussian_integer
Gaussian integer - WikipediaGaussian integers share many properties with integers: they form a Euclidean domain, and have thus a Euclidean division and a Euclidean algorithm; this implies unique factorization and many related properties. However, Gaussian integers do not have a total ordering that respects arithmetic.
https://www.alpertron.com.ar › GAUSSIAN.HTM
Gaussian integer factorization calculator - AlpertronThe Gaussian integers are complex numbers of the form a + bi, where both a and b are integer numbers and i is the square root of -1. The factorization is unique, if we do not consider the order of the factors and associated primes.
Vidéos
https://stackoverflow.com › questions › 2269810
What's a nice method to factor gaussian integers?The Gaussian integers form a unique factorization domain. Some of them act as units (e.g. 1, -1, i, and -i), some as primes (e.g. 1 + i), and the rest composite, that can be decomposed as a product of primes and units that is unique, aside from the order of factors and the presence of a set of units whose product is 1.
https://math.libretexts.org › Bookshelves › Combinatorics_and_Discrete_Mathematics...
1.13: The Gaussian Integers - Mathematics LibreTextsIn this chapter, once we have a few fundamental concepts, we will see how the Gaussian integers satisfy a division algorithm and a version of unique factorization. We will also see the Gaussian integers pop up a few times in later chapters.
https://circles.math.ucla.edu › circles › lib › data › Handout-4022-3710.pdf
Prime Factorizations Part II - Gaussian IntegersWe’ll now study how Gaussian integers factor - as it turns out, they will factor uniquely. To prove this, we once again call back to the proof for the integers, the Fundamental Theorem of Arithmetic, which uses the size of integers. So we introduce a notion of the size of a Gaussian integer. Definition 7The norm of a Gaussian integer α= a+ b √
https://brilliant.org › wiki › gaussian-integers
Gaussian Integers | Brilliant Math & Science WikiGaussian integers are complex numbers whose real and imaginary parts are both integers. The Gaussian integers, with ordinary addition and multiplication of complex numbers, form the integral domain \ (\mathbb {Z} [i]\). Formally, Gaussian integers are the set.
https://www.math.uci.edu › ~ndonalds › math180b › 6gaussian.pdf
6 Gaussian Integers and Rings of Algebraic IntegersThe Gaussian integers are the set Z[i] = fx + iy : x,y 2Zgof complex numbers whose real and imaginary parts are both integers. Z[i] is a ring (really a subring of C) since it is closed under addition and multiplication: (x +iy)+(p +iq) = (x + p)+i(y +q), (x +iy)(p +iq) = (xp yq)+i(xq +yp)
https://kconrad.math.uconn.edu › blurbs › ugradnumthy › Zinotes.pdf
THE GAUSSIAN INTEGERS - University of ConnecticutIn particular, induction on the norm (not on the Gaussian integer itself) is a technique to bear in mind if you want to prove something by induction in Z[i]. We will use induction on the norm to prove unique factorization (Theorems6.4and 6.6). The norm of every Gaussian integer is a non-negative integer, but it is not true that every
https://mathworld.wolfram.com › GaussianInteger.html
Gaussian Integer -- from Wolfram MathWorldA Gaussian integer is a complex number a+bi where a and b are integers. The Gaussian integers are members of the imaginary quadratic field Q (sqrt (-1)) and form a ring often denoted Z [i], or sometimes k (i) (Hardy and Wright 1979, p. 179).
