The next issue of the Carnival of Mathematics, rounding up blog posts from the month of July 2024, is now online at Theorem of the Day.
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.
Hi! My name is Colin, and I am a PROPER mathematician now. I’ve made a contribution to the Online Encyclopaedia of Integer Sequences.
The next issue of the Carnival of Mathematics, rounding up blog posts from the month of June 2024, is now online at Cavmaths.
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.
I’m slowly working to (sort of) recreate Martin Gardner’s cover images from Scientific American, the so-called Gardner’s Dozen.
This time it’s the turn of the March 1964 issue. In the article ‘The remarkable lore of the prime numbers’, later included as chapter 9 in Martin Gardner’s Sixth Book of Mathematical Games from Scientific American, Gardner describes how Stanislaw Ulam in a boring meeting doodled a grid of numbers, spiralling out, then circled the primes. “To his surprise the primes seemed to have an uncanny tendency to crowd into straight lines.” These Ulam sprials, discovered the year before, contain lines related to prime-generating functions, which I have written about recently.
It used to live, unloved, in the A-level formula book: a mysterious result relating the area of a triangle to its sides. The most interesting thing about it was its name: Heron’s formula. (As far as I can make out, the chap’s name was Hero of Alexandria, and if you do a possessive in Greek it goes into the genitive case, which makes it Heron’s Formula. You might want to debate this; I regretfully decline.)
Love Triangle is available wherever good books etc. are, from June 20th – and signed preorders are available from Maths Gear
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.