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.)

So there I was, peacefully proofreading Matt Parker’s forthcoming book Love Triangle when I was shocked by a vicious — and frankly unprovoked — assault on the very idea of Heron’s formula.

Following on from the series of ‘Pascal’s Triangle and its Secrets‘ posts, guest author David Benjamin shares another delightful piece of mathematics – this time relating to prime numbers.

At the time of writing the largest known prime number has $24862048$ digits. The number of digits does not reflect the true size of this prime but if we were to type it out at Times New Roman font size 12, it would reach approximately $51.5$ km, or about $32$ miles. Astonishing!

Patrick Laroche from Ocala, Florida discovered this Mersenne prime on December 7, 2018. I was surprised to discover that it’s exponent $82589933$ is the length of the hypotenuse of a primitive Pythagorean triple where $82589933^{2} = 30120165^{2} + 76901708^{2}$ as indeed are 8 of the exponents of those currently ranked from 1 to 10.

The Greek mathematician Euclid of Alexandria ($\sim$325 BC-265 BC) was arguably the first to prove that there are an infinite number of primes – and since then, people have been searching for new ones. Some do it for kudos, for the prize money, to test the power of computers and the need to find more of the large primes used to help protect the massive amount of data which is being moved around the internet.

Mersenne primes, named after the French monk Marin Mersenne, are of the form $2^{p} -1$, where the exponent $p$ is also prime. Mersenne primes are easier to test for primality, which is one reason we find so many large ones (all but one of the top ten known primes are Mersenne). When Mersenne primes are converted to binary they become a string of $1$s, which makes them suitable for computer algorithms and an excellent starting point for any search.

Since generally testing numbers for primality is slow, some have tried to find methods to produce primes using a formula. Euler’s quadratic polynomial $n^2+n+41$ produces this set of $40$ primes for $n = 0$ to $39$. When $n=40$, the polynomial produces the square number $1681$. Other prime-generating polynomials are listed in this Wolfram Mathworld entry.

The French mathematician LejeuneDirichlet proved that the linear polynomial $a+nb$ will produce an infinite set of primes if $a$ and $b$ are coprime for $n=0,1,2,3,4,…$. Then again, it also produces an infinite number of composite numbers! However, this gem: $224584605939537911 + 1813569659748930n$ produces 27 consecutive primes for $n=0$ to $n=26$ – and of course, all the primes are in arithmetic progression.

14 fruitful fractions

The primes are unpredictable, and become less common as they get larger. Consequently there is no formula that will generate all the prime numbers. However, there is a finite sequence of fractions, that – given an infinite amount of time – would generate all the primes, and in sequential order.

They are the fruitful fractions, created by the brilliant Liverpool-born mathematician, John Horton Conway (1937–2020) who, until his untimely death from complications related to COVID-19, was the John von Neumann Emeritus Professor in Applied and Computational Mathematics at Princeton University, New Jersey, USA.

The fruitful fractions are

$\frac{17}{91}$

$\frac{78}{85}$

$\frac{19}{51}$

$\frac{23}{38}$

$\frac{29}{33}$

$\frac{77}{29}$

$\frac{95}{23}$

$\frac{77}{19}$

$\frac{1}{17}$

$\frac{11}{13}$

$\frac{13}{11}$

$\frac{15}{44}$

$\frac{15}{2}$

$\frac{55}{1}$

A

B

C

D

E

F

G

H

I

J

K

L

M

N

The first time I encountered this set of fractions was in the wonderful book, The Book of Numbers, by Conway and Guy. I was so intrigued as to how Conway came up with his idea, I emailed him to ask. I was delighted to receive an outline of an explanation and even a second set of fractions, neither of which I can now find – it was 1996 and pre-cloud storage! But no worries… Conway explains everything in this lecture, which also demonstrates his passion for mathematics and his ability to express his ideas in a relaxed and humorous way, even when he searches for an error in his proof on 26 minutes. The lecture also includes an introduction to Conway’s computer language, FRACTRAN, which includes the statement:

‘It should now be obvious to you that you can write a one line fraction program that does almost anything, or one and a half lines if you want to be precise‘.

Using the fractions to find prime numbers

Here’s how the fractions are used to generate primes.

Start with the number $2$

Multiply by each of the fourteen fractions until you find one for which the product is an integer

Starting with this new integer, continue multiplying through the fractions until another integer is produced. (If this process reaches fraction $N=\frac{55}{1}$, the integer’s product with N is guaranteed to be another integer as N has a denominator of $1$; the process continues with this new integer being multiplied by fraction A)

Continue multiplying through the set to create more integers

When the integer is a power of $2$, its exponent will be a prime number.

The 19 steps needed to produce the first prime number are:

The successive primes are produced almost like magic – but the number of multiplications needed to produce each new prime becomes larger and larger, and so the method, though wonderfully inventive, is not at all efficient.

Edit: Since this article was first published, the exponent $82589933$ of the Laroche prime has been accepted as the next term in the sequence http://oeis.org/A112634