The next issue of the Carnival of Mathematics, rounding up blog posts from the month of May 2024, is now online at Girls’ Angle.
The Carnival rounds up maths blog posts from all over the internet, including some from our own Aperiodical. See our Carnival of Mathematics page for more information.
A few weeks ago I heard someone casually refer to ‘that formula of Euler’s that generates primes’. I hadn’t heard of this, but it turns out that in 1772 Euler produced this formula:
\[ f(x) = x^2 + x + 41\text{.} \]
Using this, \(f(0)=41\), which is prime. \(f(1)=43\), which is also prime. \(f(2)=47\) is another prime. In fact this sequence of primes continues for an incredible forty integer inputs until \(f(40)=41^2\). It might generate more primes for higher inputs, but what’s interesting here is the uninterrupted sequence of forty primes.
This got me wondering. Clearly \(f(0)\) is prime because 41 is prime, so that much will work for any function
\[ f(x) = x^2 + x + p \]
for prime \(p\), since \(f(0)=0^2+0+p=p\). Are there other values of \(p\) that generate a sequence of primes? Are there any values of \(p\) that generate longer sequences of primes?
I wrote some code to investigate this. Lately, I’ve taken to writing C++ when I need a bit of code, for practice, so I wrote this in C++.
I figured the cases where \(f(0)\) is prime but \(f(1)\) isn’t weren’t that interesting, since \(f(0)\) is trivially prime. In fact, \(f(x)=x g(x)+p=p\) when \(x=0\) for any prime \(p\), but saying so doesn’t seem worth the effort.
So I kept track of the primes \(p\) whose functions \(f(x)=x^2+x+p\) generate more than one prime, and the lengths of the sequences of primes generated by each of these. This produced a pair of integer sequences.
I put the primes that work into the OEIS and saw that I had generated a list of the smaller twin in each pair of twin primes. I was momentarily spooked by this, until I realised it was obvious. Since \(f(0)=p\) and \(f(1)=1^2+1+p=p+2\), any prime this works for will generate at least a twin prime pair \(p,p+2\).
What about the lengths of the sequences of consecutive primes generated? The table below shows the sequences of consecutive primes generated for small values of \(p\). Most primes that generate a sequence produce just two, and \(p=41\) definitely stands out by generating forty.
| \(p\) | \(f(x)\) | Primes generated | Number of consecutive primes generated |
| 3 | \(x^2+x+3\) | 3, 5 | 2 |
| 5 | \(x^2+x+5\) | 5, 7, 11, 17 | 4 |
| 11 | \(x^2+x+11\) | 11, 13, 17, 23, 31, 41, 53, 67, 83, 101 | 10 |
| 17 | \(x^2+x+17\) | 17, 19, 23, 29, 37, 47, 59, 73, 89, 107, 127, 149, 173, 199, 227, 257 | 16 |
| 29 | \(x^2+x+29\) | 29, 31 | 2 |
I was pleased to see this sequence of lengths of primes generated was not in the OEIS. So I submitted it, and it is now, along with the code I wrote. (I discovered along the way that the version where sequences of length one are included was already in the database.)
Anyway, I amused myself by having some C++ code published, and by citing Euler in a mathematical work. Enjoy: A371896.
The next issue of the Carnival of Mathematics, rounding up blog posts from the month of April 2024, is now online at Ioanna Georgiou’s blog. The Carnival rounds up maths blog posts from all over the internet, including some from our own Aperiodical. See our Carnival of Mathematics page for more information.
This is a guest post by Elliott Baxby, a maths undergraduate student who wants to share an appreciation of geometrical proofs.
I remember the days well when I first learnt about loci and constructions – what a wonderful thing. Granted, I love doing them now; to be able to appreciate how Euclid developed his incredible proofs on geometry.
With the emphasis on occasionally, I’m occasionally working to (sort of) recreate Martin Gardner’s cover images from Scientific American, the so-called Gardner’s Dozen.
This time I’m looking at the cover image from the November 1959 issue. The column is ‘How three modern mathematicians disproved a celebrated conjecture of Leonhard Euler’, about the so-called Euler’s Spoilers, the story of three mathematicians – Parker, Bose and Shrikhande – who had disproved a conjecture of Euler’s about Latin squares. The column was reprinted as chapter 14 in his New Mathematical Diversions from Scientific American.
The next issue of the Carnival of Mathematics, rounding up blog posts from the month of March 2024, is now online at Tom Rocks Maths.
The Carnival rounds up maths blog posts from all over the internet, including some from our own Aperiodical. See our Carnival of Mathematics page for more information.
You know how loads of things in maths are named for the wrong person? In 1996, a fun quiz appeared in The Mathematical Gazette based on history of maths misconceptions. It contained a series of questions where the obvious answer is not correct, such as “Who discovered Cramer’s rule?”, “Did Pascal discover the Pascal triangle?” and “Who first published Simpson’s rule?”
I was looking for a demo to show my students that generative AI programs are not producing accurate knowledge when I thought of this quiz. I put its questions to ChatGPT to see how it did. The point of the exercise is that these systems just parrot back words from their training data without any concept of truth, so if the training data is full of misconceptions, so too will be the responses. But these are misconceptions from the 1990s, so how much influence will they have on the responses?
You can see how ChatGPT did when I gave it the quiz in a short, free to read, open access paper in the British Journal for the History of Mathematics: Generative AI and accuracy in the history of mathematics.