The unedited version of my Irish Times'

'An Irishman's Diary' column of 22^{nd} Feb. 2000

An e-mail of 8^{th} January 1999 to a
young niece and nephew - explaining in simple terms my serendipitous discovery of a *beautifully
constructed* 2000-digit prime number - has just resulted in a painting of *that
prime number* being bought by Turlough Sheehan of Consolidated Distribution Services
to the benefit of the Irish Cancer Society (ICS) and the NSPCC (UK). This is the story of
that year long link.

About 2300 years ago,
Euclid, the renowned Greek mathematician, contemplated this question: how many prime
numbers are there altogether? *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
themselves. The first several, and some later ones are: 2, 3, 5, 7, 11, 13, 17, ... , 127,
... , 1009, ... , 8191, ... , 65537, ... .

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 2000 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. Pocklington’s seminal paper was rescued from oblivion in 1927
by the renowned US mathematician Derrick Henry Lehmer (1905-1991). 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*, and at the same time I wrote a lengthy e-mail to niece Jo and nephew Ben, explaining in
simple terms my discovery of the 2000-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 html text of a
public lecture I gave last October about the history of Fermat numbers may be viewed
here).
That discovery - which resulted in many sleepless nights! - attracted a lot of interest and much friendly
correspondence, amongst which was a delightful one from 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 Máireád,
seeking my consent to publish the text of my explanatory e-mail under the imprint of their
*Folding Landscapes*,
and I immediately agreed. The resulting booklet - *A Prime For The Millennium* -
designed by Simon Cutts and Tim Robinson, was recently published, and the ICS agreed to
accept my author royalties.

The *Guardian*
of November 22^{nd} contained a lengthy article by David Ward about the booklet,
together with a visually stunning display of all 2000 digits of the millennium prime.
There followed several interesting responses: besides hearing from two composers (one of
whom wanted to use all 2000 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, Tom Marine.

In recent months the
British NSPCC launched its *Full Stop* campaign ("Put a full stop to child
abuse"), and Tom Marine contemplated a work with sale price donated to them. On
reading the *Guardian* article, Tom Marine conceived the idea of a 40 by 50 inch
canvass, with the entire 2000 digits entered into the unit squares, each digit represented
by a different shade of red, with one exception: a single green (representing the Full
Stop of the NSPCC campaign) in a square of my choice. I opted for the 27^{th}
square in the 49^{th} 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 ICS. Coming to hear of
this, Turlough Sheehan immediately offered to be the carrier and purchaser. I contacted
Tom Marine and he accepted Turlough Sheehan’s generous offer. Bravo to Tim,
Máireád, Tom and Turlough.

John Cosgrave