In the world of mathematics outreach, it’s always prime time
By Keith Devlin @KeithDevlin@fediscience.org, @profkeithdevlin.bsky.social
Best wishes to all for 2025, which happens to be the product of the proper divisors of its square root.
Figure 1. Scientific American article, November 1, 2024. The article is excellent; the headline is idiotic. Mathematicians’ reactions surely ranged from a yawn to a dismissive “Cute!”. An attention-grabbing headline is important in science outreach. But don’t shoot yourself in the foot.
Last year saw the announcement that a new “largest known prime number” had been discovered; an integer with 41,024,320 digits.
Note to popular science writers: Notice that I just avoided putting a modifier such as “massive”, “mind boggling”, etc. in front of that number. So, it can be done. I agree it’s tempting. But really, there are clear reasons why we should resist the urge!
First of all, “massive” is an entirely relative term. To Amazon boss Jeff Bezos that number is tiny; it’s the amount he makes in US dollars every five hours. At the other end of the spectrum, the average monthly Social Security payment to a retiree is around $2,000, so it would take over 1,700 years to bring in $41M. So, to the average American retired person, $41M in five hours (or even an entire lifetime) is a mind-boggling figure.
Secondly, the primes go on forever, so from a mathematical perspective, the only meaningful modifier you can add to any particular prime is “tiny”, and even that’s a stretch.
Still, I have to admit that for anyone writing about mathematics for a general audience, such terms are effective; they do convey that the size has some significance. And, although I would not use them in writing for the MAA, for the general public, they really are fine. (For Scientific American, I think the issue is somewhat murkier. The author of the prime article does use “mind boggling”, and I’m sure I have made similar calls in articles I’ve written, and very likely have used that very expression. Popular science writing is a juggling act performed on a knife edge. I’ve written about that before as well, for instance in the May 2004 Devlin’s Angle.
The headline for Scientific American’s story of the new prime discovery, however, is just plain, over-the-top wrong. There is not a single mathematician in the world whose “mind was blown” by the announcement. Reactions among mathematicians surely ranged from a yawn to a dismissive “Oh, that’s cute”, and more likely the former. (I guarantee you that headline was not supplied by the writer, who has a Ph.D. in theoretical computer science from Harvard. Many times when I saw the published version of a piece I had just written for popular media over the years, I cringed when I saw the headline that had been placed at the top.)
To the best of my knowledge, I was the first ever writer of a “largest known prime” story in the mainstream press anywhere. Before my piece, which appeared in the British national newspaper The Guardian on May 12, 1983, stories on large primes were restricted to math & science hobbyist publications. The headline given to my piece by the Guardian’s headline writer was “The biggest prime number in the world.” If I were to quibble (I didn’t and won’t) I’d say you should read-in a silent “known” before “prime” but including it would greatly diminish the headline’s impact to grab the reader at the breakfast table. It’s enough for a headline to be (taking my lead from Billy Crystal in The Princess Bride) “mostly right”.
Figure 2. The photo shows me in front of a DEC Vax minicomputer from which I printed out the decimal presentation of the then largest known prime number in the world.
In fact, that story was my first ever math piece (on any topic) in a national newspaper, and it’s what launched me onto a second career as “science writer”. My article was quickly picked up by other media outlets, and BBC Television reached out to me that very day for help doing a piece on their regular children’s news program John Craven’s Newsround (I always thought a children’s news program was a great idea) and later another segment for their popular tv series Record Breakers. The paper printout of the new prime (which was really the last prime discovery for which you could meaningfully do that) I generated for Record Breakers to show on air proved useful to me some time later as a cover photo on a new book I was about to publish on computers and number theory. (See Figure 2.)
For the record (and at the time this was the record for known primes), the discovery I wrote about was a prime with 25,962 digits. (“Tiny” compared to the recent one.)
In any event, the instant success of my first Guardian post resulted in the paper’s science editor phoning me up and asking if I would write more math articles for them. And suddenly, I was a “science writer” for a national newspaper. The assistant science editor who was assigned to me became the science editor not long afterwards (when the original one retired), and he used my first few articles to coach me on how to improve my style to work better for a newspaper audience. I learned a lot, including how to identify stories that can be made to work, and how to set about making them work.
One lesson I learned quickly was that if you write articles in a way that makes them appeal to a general newspaper reader, some of your fellow academics will rejoice in pointing out that you have failed to fully understand the basics of the particular area of mathematics you are writing about. I quickly decided that the best response was to ignore them.
Anyway, to get back to the newly identified prime, the Scientific American piece about it is well written, and follows the same overall format I adopted for my piece. So I won’t repeat here what you can find in that article.
I will note that the GIMPS program to oversee the search for large primes, which the article references, was not started until 1996, thirteen years after my Guardian piece appeared.
For a few years, I wrote follow up pieces whenever a new “record prime” was discovered, but knowing that Euclid had proved there are infinitely many primes, I eventually tired of doing so. You can find all my pieces in the Guardian-articles compilation book All the Math That’s Fit to Print that the MAA published in the Spectrum series in 1994.
To end, and also for the record, I’ll reprint here my original “record prime” Guardian article from 1983. (As an actual column of newsprint, it had to be a specific length, roughly 700 words, but was really character count that mattered. Meeting that target was a skill I had to master quickly.)
Both articles illustrate, features you need to consider including in order to both keep the reader interested and talk about mathematical ideas and developments that are way more important than the attention-grabbing “discovery” the article is ostensibly about.
Stories about prime numbers are particularly well suited to that because the basic definitions and ideas are well within the capacity of your average reader, and moreover you can include some remarks about how prime numbers are useful to society. (You lose points if your story does not give at least one everyday application. In fact, the editor may well reject your piece. That’s just a basic law of nature about newspaper articles on math.) Stories that involve the Fibonacci sequence and the Golden Ratio offer similar “low hanging fruit” benefits on a tree that could reach to the clouds, though with that example you have to take care not to cite as an application one of the many false claims that litter the Web. [See the May 2007 Devlin’s Angle.]
The biggest prime number in the world
The Guardian, May 12, 1983
A prime number is any whole number which can only be divided (without recourse to remainders or fractions) by the numbers 1 and itself. (For example, 2, 3, 5, 7, 11 are all primes.) Although the prime numbers have been studied by mathematicians (both professional and recreational) since ancient times, it is only over the last few years that interest has been shown from other quarters.
Recent developments in cryptology (the science of making and breaking secret codes) have tended to involve more and more aspects of the branch of mathematics known as number theory, and in particular the properties of prime numbers. Not surprisingly therefore, large communications and data organizations such as IBM and AT&T now provide extensive funding for research into prime numbers, and, perhaps less well-known to the public at large, one or two less academic agencies such as America's National Security Agency (NSA) are also highly involved in such matters. So, it is unlikely that only a handful of ivory-towered mathematicians will show interest in the recent announcement that a new prime has been discovered, a prime immeasurably larger than any known beforehand.
So large is this number that it would be pointless trying to represent it in the way numbers are usually expressed, using a string of the digits 0 to 9. If the editor of the Guardian were to decide to print this number in the normal way, using regular sized type, with no headlines, advertisements, or pictures in the way, the number would take up just over an entire half page of the newspaper.
Fortunately, mathematicians have a special notation for describing numbers of this magnitude. Using this notation the number in question looks quite tame. It is 2^86,243 – 1. That this number is prime was discovered by David Slowinski of the USA. As you might imagine, he had more than a £5 pocket calculator to help him with his calculation. In fact, he made use of the world’s most powerful computer, the giant Cray–1 machine at the Cray Research Laboratories. Even with this incredible computing power, it took the machine 1 hour 3 min and 22 sec simply to check that the above number is indeed a prime. Months of computing were required to find this number in the first place.
It is not hard to explain what the notation used above means. To obtain Slowinski’s number you take the number 2 and multiply it by itself 86,243 times, and then, as a final fillip, you subtract 1. The result is a number with precisely 25,962 digits when written out in the normal way. How can we begin to comprehend the size of such a monster? To get some idea, let’s look at the apparently insignificant number 2^64. This can be visualized as follows. Imagine an ordinary chessboard. If we number the squares on this chessboard starting in the top left-hand corner and proceeding row by row down to the bottom right-hand corner, using the numbers 1, 2, 3 and so on, the last square we number will get the number 64.
Now imagine that we start putting ten-pence pieces on the squares of the chessboard. On square number 1 we put 2 ten-pence pieces, on square 2 we put 4, on square 3 we put 8, and so on, on each successive square putting exactly twice as many coins as on the previous one.
On the last square we will form a pile of exactly 2^64 ten-pence pieces. How high do you think this pile will be? Six feet? Fifty feet? More? Wait for it. The pile will be about 37 million million kilometers high! So, the pile would stretch way beyond the moon (a mere 400,000 kilometers away) and the sun (150 million kilometers from Earth), and in fact would reach the nearest star, Proxima Centauri. And that is only for the 2^64. To reach Slowinski’s new prime you have to double up the pile of coins a further 86,179 times. You would have left the entire universe long before you got there.
Why should anyone be interested in such huge numbers? There are various answers to such a question. To the mathematician, the way the prime numbers are distributed amongst all the numbers is an extremely interesting question in its own right. No one can say, just where the next prime number will be. With small numbers, there appear to be lots of primes about, for instance of the numbers less than 25 the numbers 2, 3, 5, 7, 11, 13, 17, 19, 23 are all primes. But as soon as you start looking at much larger numbers, the primes become much less frequent, though they do not appear to follow any particular pattern.
Besides this perhaps esoteric interest, like almost all pure research there are various useful offshoots from the work. For instance, simply to get the computer to handle a number with 86,243 binary digits, like Slowinski’s, an entire discipline of computer science known as multi-precision arithmetic has had to be developed, and you can bet your last prime that the NSA (amongst others) are interested in that.
Postscript: For subsequent discoveries of large primes, see also Chapters 2, 4, 8, 63, 105, 130, and 142.