In the book
The Man who Mistook his Wife for a Hat and other
Clinical Tales by Oliver Sacks there is a true story
about two twin brothers (John and Michael), both autistic, who have the following
properties (Sentences in italics are direct quotes from the books.)

They cannot do simple addition or subtraction with any accuracy,
and cannot even comprehend what multiplication and division mean.
(page 197)

John would say a number a six figure number. Michael would
catch the number, nod, smile and savour it.
(page 201). Oliver Sacks wrote down their numbers and, following
a hunch, found out they were all primes.

Oliver Sacks joined them and spoke an 8digit prime.
There was a long pause it was the longest I had ever known them to make,
it must have lasted half a minute or more and then suddenly, simultaneously,
they both broke into smiles.... An hour later they were swapping 20 figure
primes, at least I assume this was so as I had no way of checking.
(Page 203. Note that this happened in 1966.)
This raises several questions.

How are they doing it? These brothers were unable to tell Oliver Sacks.
However, in other essays in books when Oliver Sacks gets to talk to
savants that are not autistic they also can't explain how they do it.

Since these twins do not know basic arithmetic they are not using
the Sieve of Eratosthenes. Nor are they using the AKS Primality
algorithm to test their primes.
I speculate that they are using a different model of computation
then we work with.
One is tempted to say Neural Nets! or Analog Computers,
but I suspect it is something completely unfamiliar to us.

If we could figure out what they are doing could it lead to
a solution to the
polymath
problem on finding primes?
Alas no, since I doubt what they are doing would fit into what we are doing.

Prediction:

Someone will devise a model of Savant Computing.
The Polymath problem referred to above will be in SAVANTP, giving the field
a push (not as big a push as FACTORING IN QP gave Quantum).

People will come up with 1 or 2 more real problems in SAVANTP, and dozens of
complexity classes and theorems about SAVANT computing.

There will be some results in lower bounds on classical models (which my then may include quantum)
that are claimed to be easy to prove with SAVANT concepts but not otherwise.
People will argue about is it really easier?.

I will write a blog Is Savant the next Quantum?.

Could the twins get a job at the NSA?
Today no, since they need primes far bigger than 20 digits.
But back in 1966...

Posted By GASARCH to
Computational Complexity at 9/17/2009 11:46:00 AM