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Gaussian Vertex Prime Labeling of Some Graphs Obtained from Origami Models
Published in N. P. Shrimali, Nita H. Shah, Recent Advancements in Graph Theory, 2020
Gaussian integers are complex numbers of the form γ = x + iy where x and y are integers and i2 = −1. The set of Gaussian integers is usually denoted by Z[i]. A Gaussian integer γ is said to be an even Gaussian integer if γ is divisible by 1 + i and otherwise is called an odd Gaussian integer. The norm on Z[i] is defined as d(x + iy) = x2 + y2. One can easily see that the only units of Z[i] are ±1, ± i. The associates of Gaussian integer γ are unit multiples of γ. In Z[i], two Gaussian integers are relatively prime if their common divisors are the only units of Z[i]. A Gaussian integer γ is said to be a prime Gaussian integer if and only if ±1, ± i, ± γ, ± iγ are the only divisors of γ.
Pseudo-Random Pixel Rearrangement Algorithm Based on Gaussian Integers for Image Watermarking
Published in Frank Y. Shih, Multimedia Security, 2017
Aleksey Koval, Frank Y. Shih, Boris S. Verkhovsky
A Gaussian integer is a complex number: Z[i] = {a + bi : a, b ∈ ℤ}, where both a and b are integers. Gaussian integers, with ordinary addition and multiplication of complex numbers, form an integral domain. The norm of a Gaussian integer is a natural number, defined as |a + bi| = a2 + b2.
Remainder and quotient without polynomial long division
Published in International Journal of Mathematical Education in Science and Technology, 2021
Let be the commutative ring of Gaussian integers (i.e. the ring of all complex numbers of the form a + bi, where a, and , with the usual operations). We want to determine the remainder and the quotient of Using the previous code, we can check that r = 0, and .