**In his book “The Man Who Mistook His Wife For A Hat” (1985) Oliver Sacks describes an intriguing case of savant syndrome. He tells the story about his encounter with the twins John and Michael, who had been in institutions since childhood, variously diagnosed as autistic, psychotic or severely retarded. Others before Sacks had already investigated these boys and found out that they were very good in calendar calculating. For a random given date they could quickly tell which day of the week it is. Sacks however discovered a far more unusual ability of the twins when he observed them in 1966. On a certain day he notices that they are exchanging six-figure numbers and seem to be very pleased with the numbers they receive from each other. Sacks notes down the numbers and when he comes home, takes out his ‘ tables of powers, factors, logarithms and primes’ and finds out that all the numbers, mentioned by the twins, are in fact prime! Truly remarkable, because the twins were not able to perform simple calculations and did not seem to understand multiplication and division.**

The next day Sacks returns with a book full of prime numbers and starts to play the game along. But he starts out with a prime number consisting of eight figures. After half a minute thinking (or longer) the twins start to smile. This is to Sacks a clear sign that they are both pleased with this new ‘toy’ (a bigger prime) and happy to find out that Sacks had understood their game. In return John mentions, after five minutes thinking, a nine-figure number and his brother follows suit. Sacks now picks a ten-figure prime from his booklet, which in a while gets responded by a twelve-figure number. As Sacks’s precious book only contains primes up to ten-figure numbers, he retracts from the game. One hour later the brothers are exchanging twenty-figure numbers!

I had already encounterd this story years ago, but when I came across it again recently (while reading the book “We are our brains” by Dick Swaab), I noticed that there were some sceptical questions raised considering the credibility of Sacks’s story. More than enough reason for me to dig a bit deeper into this!

In 2006 Makato Yamaguchi wrote an article in which he questions the report of Sacks. The most important question he asks is which book Sacks used. If the book would indeed contain all the primes consisting of ten figures or less, it would have to list more than 455 million numbers, which is of course infeasible. Maybe his booklet contained only a few ten-figure primes? Yamaguchi couldn’t find any book available in 1966 which contains any list of this sort. Also Sacks couldn’t tell him which book it had been. Nor was he able to recollect which numbers were involved. All his original annotations and the book are lost. Sacks would admit however that his book may only have contained primes up to eight figures.

If you read the description of events by Sacks carefully (read it yourself), you’ll notice he actually only mentions the primeness of the six-figure numbers which he checked at home and the numbers he chose himself from the book. He further states that he assumes that the twenty-figure number is a prime. But he didn’t check whether the numbers called out by the twins larger than six figures were actually prime. Of course this wouldn’t have been an easy task in 1966 without the access to ready available computer programs we have nowadays. But it seems that Sacks didn’t even try to get certainty; he just states it would have been difficult.

In a postscript to the chapter about the twins (at least in the Dutch version I have), Sacks mentions another algorithm to test numbers for primeness. But his description only shows that he has no real mathematical knowledge that matters in this case. Sacks sticks to a romantic idea that the twins had a special sensibility to numbers, that they could somehow pick out prime numbers in vast ocean of ordinary numbers. The idea that someone could ‘see’ the primeness of a big number without doing some calculations, is something I don’t buy, however. Perhaps some idiot savants might be faster in recognizing prime numbers, but that doesn’t imply that they are using other methods. Let alone methods, which cannot be described as an algorithm.

Can we rely upon the first part of the story: were the twins really exchanging six-figure primes? Or is there another more reasonable explanation for this at first sight astonishing performance? Let’s start by pointing out that the calculation skills of the twins were not as bad as Sacks writes: the researchers (Horwitz a.o.), who established (in 1965) that they could hardly perform calculations, in 1969 wrote they had at least the capability of adding three-figure numbers. As far as I know Horwitz only describes the calendar calculation. Unfortunately I didn’t have access to their articles to see if they mention anything about their skills with prime numbers.

How hard is it anyway to give six-figure primes? Between 100,000 and 999,999 there are 68,906 primes. What is the chance of choosing a prime while keeping away from those numbers which are obviously not prime? It would be silly to try an even number or one which ends in ‘5’; those are of course divisible by 2 and 5 respectively. Many will recall the trick to find out if a number is divisible by 3: you just add the figures of the number, repeat that over en over again with the result, until you can easily see if the remaining number is divisible by 3. If so, the original number is also divisible by three. An example: 561,251 gives as sum of its figures 20, in the next step this gives 2, which is not divisible by three. And therefore we can conclude that 561,251 is not divisible by 3.

These kind of tricks can also be found for the other small (prime) divisors 7, 11, 13, 17, etc. Because the two brothers were good calendar calculators, they must almost certainly have known how to divide by 7. Or at least have known how to determine the remainder of a division by 7. One of the ways of doing this with 7 goes as follows: take the last two figures of a number and add the double of anything in front of these two figures. The new number is divisible by 7 only if the starting number was divisible by 7 as well. You can repeat this procedure until the point you can easily make out if the remaining number is divisible by 7. Let’s consider our example number 561,251 again: first you get 2 x 5,612 + 51 = 11,275, then 2 x 112 + 75 = 299, then 2 x 2 + 99 = 103 and at last 2 x 1 + 3 = 5, which is not divisible by 7. And therefore 561,251 is also not divisible by 7.

If you exclude the numbers, which are divisible by 2, 3, 5 and 7 using these tricks, you’ll keep 205,714 six-figure numbers. Picking a number at random from this set, gives you a chance of about 33 percent that the chosen number is prime. That’s not a really small chance! If you also master the trick for divisor 11 (which is in fact easier than the trick for 7) this chance rises to 37 percent. If the twins used this approach and Sacks only took a few numbers home to check, the chance that those numbers were all prime by coincidence is not really that small. And we should also think about the way Sacks checked whether these numbers were prime if this unique book of his doesn’t exist.

It would be interesting to know in what way the numbers were called. Probably not as *‘five hundred sixty one thousand two hundred and fifty one’*. At least this is not imaginable with the twenty-figure numbers! More likely is that they called out the numbers figure by figure like a telephone number *‘five-six-one-two-five-one’*. That could also mean that the twins saw the huge numbers more as a row of figures than as a single number. For applying the tricks for division by small factors it doesn’t really matter and it could be an explanation for the ‘fact’ that they stepped without much trouble from six-figure numbers to eight-figure numbers and even further.

A curious aspect of the story is that John after getting the ten-figure prime from Sacks, answered with a twelve-figure number, and in doing so skipped one with eleven figures. Possibly he thought of an eleven-figure candidate, but found out that it failed one of the division tricks. In that case you can try to add an odd figure at the end and see if this new number works, without too much recalculation. Again an example: does 13,725,097,771 work? No, it’s a pity, it can be divided by 7. Let’s try to add a ‘1’ at the end, which gives us 137,250,977,711. And that number is not divisible by 2,3,5 &7. OK, fine! But note that the number is not prime, because 137,250,977,711 equals 19 x 41,893 x 172,433.

A study by Hermelin and O’Connor (1990, *Factors and primes: a specific numerical ability*. Psychological Medicine, Vol. 20) shows another *‘idiot savant*‘ using a strategy like the one I describe here. He used it to figure out whether four- and five-figure numbers might be primes. He excluded numbers divisible by 3 and 11 and while doing so still made quite some judgment errors.

What could Sacks have done to give his story more credibility? At least he could have tried to call six-figure numbers which are not that simple to unmask as ‘fake’ primes. If he tried for example 254,539 which equals 331 x 769, it would be interesting to see the twins reactions (for clarity 331 and 769 are prime). Maybe they would have been as happy with this composite number as with real primes. That would have been an indication that they were in fact only ruling out small divisors.

We’ll probably never know what the twins were actually capable of, if Sacks cannot explain in more detail what really happened. It’s hard not to get the impression that the story was a bit ‘spiced up’. The calendar calculation skills of the twins were quite impressive, but not unique. Maybe Sacks was tempted to present an even more impressive feat than what already had been written down in existing literature. Why did he actually wait to publish about it until 1985, almost twenty years later? In any case Sacks was clearly not that interested to get better confirmation of his hypothesis. Or maybe he didn’t understand enough about the math involved to test it properly.

Just when I finished writing the original Dutch version of this article, I stumbled upon a comment on a blog concerning the same story. There is in fact a likely candidate for the book Sacks claims he used: D. N. Lehmer, *List of Prime Numbers from 1 to 10,006,721*. Washington, D. C., Carnegie Institution of Washington, 1914. xvi+133 pp. The book by Lehmer would suffice to do the checks on the six-figure numbers of the twins and there are some 8 figure primes in the book Sacks could have used. From the title of the book I cannot make up if there are any ten-figure primes mentioned, but it might have some appendices for known primes with more figures than the list for which it claims to be complete. I’ve send Sacks an e-mail with this suggestion. Until now I didn’t hear back from him, except for a kind reply by his assistant that my suggestion was passed on to him (he doesn’t use a computer himself it seems).

Finally, Makato Yamaguchi pointed me to a YouTube video of the twins showing their calendar calculation skills. You can check the dates and days of the week yourself, I found these to be correct in all cases.

*original version in Dutch: De priemgetal tweeling van Oliver Sacks published December 22 ^{nd}, 2011*

**Published on May 9 ^{th} 2012**