An e-mail I sent just over a year ago to my young niece and nephew - explaining in simple terms my serendipitous discovery of a 2,000-digit prime number - has just resulted in a remarkable painting being sold in aid of the Irish Cancer Society and the NSPCC (UK). The story gives a whole new meaning to the phrase "painting by numbers".
Prime numbers - a fundamental goldmine for profound mathematical questions and real-world applications (being a cornerstone of modern public-key cryptography) - are those whole numbers which are evenly divisible only by 1 and by themselves. The first several, and some later ones are: 2, 3, 5, 7, 11, 13, 17. . .127. . . 1009. . .8191. . . 65537. . .
Euclid
About 2,300 years ago, Euclid, the renowned Greek mathematician, contemplated this question: how many prime numbers are there altogether? Part of Euclid's fame rests on his giving a definite proof that there are infinitely many prime numbers, but a fundamental difficulty with his "existence proof" is that, although it provides a guarantee that there are infinitely many primes, it does not provide an easy means of finding a particular prime.
A completely unresolved question is: given a definite prime number, what is the one immediately following? The next prime after 8191 happens to be 8209, whereas the one following 65537 is 65539 - gaps of 18 and 2 respectively. Again, what is the billionth (say) prime number? That number actually "exists", but it is by no means easy to produce it. In short, there is no ready means for producing a particular prime, or an especially large one.
In January last year, wishing to illustrate for my students a beautiful idea of the English mathematician Henry Cabourn Pocklington (1870-1952), I discovered a prime number with exactly 2,000 digits. Pocklington wasn't a professional mathematician; he was a physicist, who enjoyed the distinction of being a Fellow of the Royal Society, but he produced one short (only two pages!), magnificent mathematical paper during the first World War which has guaranteed his place in history.
I dubbed my lucky discovery a "millennium" prime, and informed professional colleagues world-wide. Within a few days it featured in Ivars Petersen's wonderful MathTrek column in the US weekly Science News. At the same time, I wrote a lengthy e-mail to niece Jo and nephew Ben, explaining in simple terms my discovery of the 2,000-digit prime, and that, at the time, was that.
Then, in late July, I was the fortunate joint discoverer with Yves Gallot (Toulouse) of the (then, and still) largest known "composite Fermat number", a number so large that it could not be written out in the entire universe, even if the universe was filled with paper of atomic levels of thinness, and the digits written as micro-dots. (The text of a public lecture I gave last October about the history of Fermat numbers may be viewed at my website: www.spd.dcu.ie/johnbcos.)
Correspondence
That discovery - which resulted in many sleepless nights! - attracted a lot of interest and much friendly correspondence, among which was a delightful letter from the author/cartographer Tim Robinson. Tim - who studied mathematics in Cambridge - sent me a copy of an article he had written about prime numbers for inclusion in Marie Heaney's Sources, and I reciprocated with my January millennium prime e-mail.
Within a few days I heard from Tim and his wife Mairead, seeking my consent to publish the text of my explanatory e-mail under their Folding Landscapes imprint, and I immediately agreed. The resulting booklet - A Prime For The Millennium - designed by Simon Cutts and Tim Robinson, was published recently, and the Irish Cancer Society agreed to accept my author royalties.
The Guardian of November 22nd last contained a lengthy article about the booklet, together with a visually stunning display of all 2,000 digits of the millennium prime. There followed several interesting responses: besides hearing from two composers (one wanting to use all 2,000 digits as the basis of a composition, the other seeking a 216-digit prime for his purposes), I also heard from a Leeds-based painter called Tom Marine.
Painting
The British NSPCC was launching its "Full Stop" campaign ("put a full stop to child abuse"), and Tom Marine hoped to produce a painting that could be sold for the benefit of the charity. After reading the Guardian article, he had conceived the idea of a painting measuring 40 inches by 50 inches, with the entire 2,000 digits entered into squares, each digit represented by a different shade of red, with one exception: a single shade of green (representing the "Full Stop" of the NSPCC campaign) placed in a square of my choice. I opted for the 27th square in the 49th row.
I got the idea of finding a carrier who would bring Tom Marine's work to Ireland to find a buyer here, with the sale price to be divided equally between the NSPCC and the Irish Cancer Society. Hearing of this, Turlough Sheehan of Consolidated Distribution Services immediately offered to be the carrier and purchaser. I contacted Tom Marine and he accepted Mr Sheehan's generous offer.
Bravo to Tim, Mairead, Tom and Turlough.