Prime Spirals - Numberphile
Why Mathematics Stack Exchange recommends it
Mathematics Stack Exchange users link it when explaining why certain quadratic expressions generate unusually many primes and as background for questions about the Ulam spiral.
Recommended by 1 people across 1 different months, 2013 to 2021.
What it is
A short Numberphile video in which Dr James Grime shows the Ulam spiral, where primes are marked on a square spiral of integers, and explains why they line up along diagonal lines. He ties this to quadratic expressions like 4x^2 - 2x + 1 that produce many primes. Filmed and published by Brady Haran.
What people said
Recommends Winther · 2021-11 · Mathematics Stack Exchange · on “$P^3 - (P-1)^2=q$ How do I know that this guess might be mathematically significant?”
There are many such functions. The key is to make sure the expression is designed such that its never divisible by the first few primes (like $2,3,5$ in our case). If so then its a pretty good chance you are going to get a lot of primes in the sequence as you have excluded the smallest possible factors so any factor much be fairly big. A simpler example is $4p^2-2p+1$. See e.g. [LINK] (which links to [this video] which you might find interesting )
Cites Jeel Shah · 2013-07 · Mathematics Stack Exchange · on “Quadratic expression that generate primes”
I recently learned that there exist quadratic expression that generate some primes and some of these equations generate more primes than others. In the following video [this video] , the person shows the following expression $$4x^2 -2x+1$$ that generates some primes. So I was wondering, are there infinitely many expression that exist that can produce primes and what makes one quadratic produce more primes than another? In other words, do some quadratic expressions have a special property that causes them to generate more primes than others? Thanks!
Cites jan · 2017-03 · Mathematics Stack Exchange · on “How to find the next diagonal number in the Ulam spiral?”
[LINK] That's the Ulam spiral. I'm doing some programming with primes, and I'm wondering how to efficiently find diagonal numbers. Basically what I'm trying to do is for some number $n$, find the numbers that are diagonally next to it on the Ulam spiral. For example: in the picture the numbers that are diagonal to $3$ are: $11, 13, 15$, $1$ (there is no reason why I circled just one of them on them on the picture). Since I'm making a program I could just tell the computer to get the $(x+1, y+1)$ number (and other diagonal numbers), but that would be extremely slow with big numbers, so I'm trying to find a mathematical way of doing it. If you need more info on the Ulam spiral, take a look at