Recent content by Ventrella

Are all complex integers that have the same norm associates of each other? I have seen definitions saying that an associate of a complex number is a multiple of that number with a unit. And I understand that the conjugate of a complex number is also an associate. But I am looking for a...

Thank you. I'm not sure what some of those variables refer to, but I believe you are saying that I can characterize this problem in terms of polar coordinates, which allows me to see that the argument changes in regular steps, and therefore z consistently remains either on or off an axis...

I am exploring the behaviors of complex integers (Gaussian and Eisenstein integers). My understanding is that when a complex integer z with norm >1 is multiplied by itself repeatedly, it creates a series of perfect powers. For instance, the Gaussian integer 1+i generates the series 2i, -2+2i...

I identified what appears to be a partitioning of all integers > 1 into mutually disjoint sets. Each set consists of an infinite series of integers that are all the powers of what I am calling a "root" r (r is an integer that has no integer roots of its own, meaning: there is no number x^n that...

The first comment suggested that being 89 years old makes Sir Atiyah's claim less credible. I would like to believe that one's math insights steadily improve and that while age may slow the brain, it does not make one less insightful.

Hi andrewkirk, According to Wikipedia, "The norm of a Gaussian integer is ... the square of its absolute value as a complex number ... a sum of two squares." I believe you are referring to the "Euclidean norm", which I understand to be the same as the complex modulus, or the absolute value...

I am exploring Gaussian integers in terms of roots, powers, primes, and composites. I understand that multiplying two integers with norm 5 result in an integer with norm 25. I get the impression that there are twelve unique integers with norm 25, and they come in two flavors: (1) Four of them...

Hi dagmar, Could you elaborate? Perhaps you could decompose your definition so I can understand the component parts. Specifically, I don't understand: "...the exponents of all the powers of the primes in their prime decomposition..." I don't see how prime numbers are involved here. But if you...

Hello mfb, I believe you are incorrect. You say that every integer has a perfect root. When you say "perfect" I assume that means integer. Of course every integer has roots, but not all of them are perfect (integers). The roots of integers are either integers or real (in fact, if they are not...

Maybe I didn't explain myself. I'm talking about integers that do not have perfect roots.

Thanks for the suggestion, Simon! "Perfect nth root" makes sense. It's better than "root". I would still like to find a way to express this class of numbers without having to reference an integer (i.e., "nth") of which it is a perfect root - since there are infinitely many n's. I think of...

I would like to know if there is an official name for the class of integers that are (not) perfect powers. A perfect power is a number that can be expressed as xn, where x and n are both integers > 1. I have been calling these integers "roots" - since they do not have any integer roots of their...

Hi jedishrfu, I am planning on writing an algorithm, for sure :) I am familiar with Processing, but I would prefer to do it from scratch in javascript/canvas. I just wanted to share the idea beforehand in case anyone had some insights about the math involved. I am wondering if it can be...

Does there exist a binary fractal tree… (reference: http://ecademy.agnesscott.edu/~lriddle/ifs/pythagorean/symbinarytree.htm ) …whose leaves (endpoints) lie on a circle and are equidistant? Consider a binary fractal tree with branches decreasing in length by a scaling factor r (0 < r < 1) for...

Hi Willem2: Thanks for sharing that video - I understand the fractal structure better now. My interest in this topic is in regards to the art of visualizing patterns in large integers. I suspect that the patterns that Ono and colleagues have revealed could be visualized in a way that brings out...