Mathematics

Does someone know this? While I was making a small study on circle packing recently, I came across this problem that left me scratching my brains. I found this problem interesting and wanted to share. It goes something like this: Q) We want to reduce the size of circles and as we do so, fill them up into a different and bigger circle as much as possible (maximum). As the radius of the small circles approaches to zero and fills up the mother circle, 1.Does the packing density approaches to 1 (filling up the entire mother circle)? OR 2.Does the packing density approaches to that of the hexagonal lattice arrangement (i.e. 0.906...) and therefore not filling up the entire mot…

I have developed a new way to understand the behavior of numbers under the Collatz Conjecture. My approach involves classifying numbers into three distinct groups based on their behavior during the Collatz sequence, and I propose that all natural numbers eventually reach the first group, leading to 1. Here is a breakdown of the three groups: Group A: This consists of even numbers that, when divided by 2, always result in 1. These numbers are powers of two (e.g., 2, 4, 8, 16, ...). When repeatedly divided by 2, they eventually reach 1. Group B: This group consists of even numbers that, when divided by 2, result in an odd number. These numbers are reduced to odd numbers and…

Hello. I study quite wide and have had a bunch of opinions on 'false mathematics' and it's prevalence in science and everyday people. I spent time after years thinking and studying wondering how to practically respond and handle approaching 'real' maths when I got to put ideas into chatgpt. I want to place this on this thread to get constructive feedback and I ask bare minimum read the first page and the last article, addendum, to understand why it can help. I believe it quite conclusive for practical application. It is meant to be by design practical and highly useful so. Forgive me for uploading a mentioned chatgpt prime 'original' so to call 'alpha'.

Hello, my name is Phil, and I'm currently working on a robotic arm for a school project. The goal is to draw a circle on paper. My arm is a 3-axis arm with control on a 3D scale. For coding, I need to find the exact distance from the ground to Point C (the tip of the pen) and the 2D coordinates X,Y of Point C. To draw a circle, I also need to find the angles a and B of the robot motor using only the CS, D, Sol and X, Y values. Please help! The image describes the rest. Note that there may be errors in the formulas shown in the image. Make sure to tell me if you find any. Thank you!

In classical constructions with ruler and compass, a construction proceeds from point to point where the points are intersections of lines and circles, i.e., where the lines and the circles cross each other. Are the points where the lines and the circles touch each other rather than cross, allowed as well? More specifically, if I have constructed two circles and I know that they touch at some point, can I proceed with the construction using this point?

Dear colleagues, From 1990 to 2025, I worked on restoring Fermat's original proof of his theorem. There was a 23-year break in my work from 1993 to 2016, but in the end, I believe I have succeeded. But did I really succeed? Please take a look!

Me encontré con este problema geométrico mientras trabajaba en las propiedades de los triángulos. Dados ciertos ángulos y una igualdad entre dos segmentos, apliqué la propiedad de la suma de ángulos y el teorema del triángulo isósceles, pero no llegué a una conclusión clara. También consideré los ángulos exteriores, pero no pude establecer una relación directa. Este problema parece conectar varios principios geométricos, y me encantaría entender el mejor enfoque. ¡Agradecería cualquier idea o sugerencia!

So, using the area formula for a circle: A = πr² and plugging in A = 78.5, solving for r gives: r = √(A/π) = √(78.5/π) Using π ≈ 3.1416, that turns into: r = √(78.5/3.1416) ≈ √25 = 5 Looks right, but if pi isn’t rounded too early, does it change the final answer much? And in real-world stuff like construction or engineering, how much does rounding pi affect accuracy?

y = (((pnp^2/ x ) + x^2) / pnp) pnp = x * y (((((pnp^2 / x) + x^2)) / x) / pnp) where y = 0 If all of these are true in the factors we wish to find, x and y, is there a limit; a range; that could be computed that said if x is this big then y is that big? It wouldn’t be a differential equation that solves a spring. But how do I find and x that is true by testing if y is also true in these 4 constraints? It is a simple idea, but what is the math that completes it? I know that where y on the graph equals zero x has the value approximate to the smaller factor. I have an equation that will tell me y factor knowing x. If yo…

Hey everyone, I’m working on an app (Math Journey) designed for those of us who appreciate beauty of mathematics. Unlike existing applications, it will dive into a bit more advanced topics (like the Riemann Zeta function and so on) but will maintain level of popular mathematics. I’m looking for your suggestions on: Which advanced topics are rarely covered in typical apps but deserve more attention? Specific features you’d find exciting (e.g., interactive widgets, puzzle-based explorations, real-world case studies, etc.)? Any learning approaches that you’ve always wanted to see in an app? My hope is to craft an experience that matches the cu…

Hi Everyone, could anyone here help me comfirm that this calculation verifies prime numbers? This is the calculation: DSum(Mod((x^2−A),(2B−2x))==0,x,0.0,(B−2)) A is a reminder of a number to be tested while B is a factor of a top closest square that is used to subtract the test number. For example, If you want to test 7, the top closest square is 3^2 or 9. B is 3 as it is a factor of 3^2. By using the square to subtract the test number, 3^2- 7=2, u get 2. Plug into the formula, u get DSum(Mod((x^2−2),(3⋅2−2x))==0,x,0.0,1.0) If you want to test 19, the top closest square is 5^2 or 25. B is 5 as it is a factor of 5^2. By using the s…

\(\cos(\frac {\pi} {4})=\frac {1} {\sqrt{2}}\) \(\cos(\frac {\pi} {5})=\frac {1+\sqrt{5}} {4}\) \(\cos(\frac {\pi} {6})=\frac {\sqrt{3}} {2}\) Is cosine of any fraction of π an algebraic number?

This is a short guide to using the new LaTeX system that has been implemented on the boards. First off, for those who don't know what LaTeX is, a short description. LaTeX is, to all intents and purposes, a fully fledged math typesetting system - basically put, you can write math with it. It's a very flexible and hence very advanced piece of software, and the syntax for it is quite complex, but fairly easy to learn for typesetting smaller equations. On scienceforums.net, we've implemented a small LaTeX system to allow you to typeset equations (in other words, cut out all the x^2 stuff and make things easier to read for everyone). The basic principle behind it is this: you…

For any prime number n > 3, n mod 6 = 1 or 5. any prime number n > 3, n mod 3 = 1 or 2. The same prime numbers are in column 1 either way and the primes from column 2 (mod 3) are in column 5 (mod 6). Are there 2 kinds of prime numbers? Is there a name for these primes? The attached png is the primes mod 2,3,6, and 7.

* = multiplication i = inverse (of) multiplication 2 * 2 = 4 4 i 2 = 2 The reason I post this is because I don't see it appearing as part of arithmetic operations elsewhere apart from some references to inverse (e.g. link: https://en.wikipedia.org/wiki/Multiplicative_inverse) but probably not exactly what I am talking about.

As far I can discern when inventing combinatorics I also found the product rule for differentiation and derivatives. Not "Isaac Newton" or some other name that I can't confirm ever existed. When pertaining to the scientific method, empirical evidence is based on observation which I don't rely on as heavily as mathematics for separating fact from fiction. I know I don't get anything out of sharing an arithmetic that can literally be used to earn a person money for cracking a password combination or solving any type of Rubiks cube. I should be retired for this alone. I only share it because I know some things that are potentially worth a lot more. Just realize wheneve…

I have a question related to this topic. If we create a three dimensional matrix, and give each element of the matrix a name that would represent another three dimensional matrix, similar to how computers do it, wouldn't it be possible to visualize n-dimensional matrices? Inside each element of this matrix there would be other 3D elements. This makes visualizing, in this case, six dimensional matrices possible, no?

Hi I give here the Geometric Model and representation of Walker's Equation and proposal of an Equation od Infinite Series which I call 'Walker's Series' which appears to have escaped many ! Proof of Walker's equation.docx - Google D Dear Moderator, I tried to remove the duplicate images in this post but couldn't succeed Plz remove the repeating pics TY !

In short, I am curious about how this even works and I want to know because I enjoy math, so I decided to ask people with a higher education than mine and who definitely know more about math than I do. So my understanding of graphing in three dimensions to give you an idea of my level of understanding of the concept so you can explain it better is essentially this: In order to graph in more than two dimensions we must simply add a third axis: Z. Z represents the depth axis and intersects X and Y at the origin vertically relative to the plane that X and Y form. The Z axis forms a plane with the Y axis vertically and the X axis horizontally. Graphing…

(x^3/N) approximately= (x^2/y) (25/17) approximately= (85/58) Thoughts?

I have a paper that, IMHO, proves the Collatz conjecture. I will be looking at where to get it confirmed. I welcome comments. Abstract: The Collatz tree can be formed naturally, resulting in a good looking tree, but the numbers seem chaotic. By reforming the tree, the tree becomes weird, but the numbers look nice. And it becomes clear that the hailstone algorithm and its inverse can be used to traverse down and up the tree, from the root to all integers and beyond, and back to the root (the number 1). The link is http://dbarc.net/yr2024/collatz1.0.pdf

Can anyone explain in one paragraph or equation how the Riemann hypothesis implies a pattern in Prime numbers? The hypothesis an easy enough to follow the problem, but the pattern in Primes is not clear to me. I’m sure a Google search may be helpful, but I wanted to work it through on my own. I find it helpful to take one small part if the problem and see what it does. Take the book Practical Cryptography. It explains all the ciphers. They are all open source. Doesn’t mean you can solve them but it lets you see the inner workings. That is what I need an explanation of the patterns in Primes in the Riemann hypothesis. I hope you understand why I just don’t g…

I recently came across this equation on social media: At first I thought it would the answer after simplifying the equation would be and thus 12 + 6 = 18, but inputting into my calculator gets me 8 instead. Inputting the reverse () gives me my expected answer of 18. Is it an inherent property of percentages to take what's to the left?

Greetings to all. How to calculate the differential in hardware needs (a radiator size and coolant mass and ...) ? A plain simple automobile uses a number of litres of water as coolant on a radiator of such and such dimensions. If the water specific heat capacity is 4,181 J/kg°C , a very high number compared to many other fluids- is replaced by plain motor oil / transmission fluid; what does it translate to ? --> Doubling the area/volume/fanning of its radiator ? Tripling ? As replacement of water based coolant in ICE engines, transmission fluid can: - Eliminate rust progress or prevent it, - Eliminate boiling at operating temperatures, - Eliminate …

I have watched a video about the p-adic numbers: Then, there is the Ramanujan's summation which is used in quantum mechanics (Casimir effect): 1+2+3+4+5+6...=-1/12 https://en.wikiversity.org/wiki/Quantum_mechanics/Casimir_effect_in_one_dimension This forum does not support Latex maths? Then, we have one more such summation: 1+x+x*x+x*x*x+x*x*x*x+x*x*x*x*x...=-1/(1-x) This is simple for x<1 and for x>1 we get: 1+2+4+8+16+32+64...=-1 1+10+100+1000+10000...=-1/9 1+20+400+8000+160000...=-1/19 Can we obtain all these sums with the p-adic numbers?