I study thinking, but this is all personal armchair theorizing on my part.
I think with autistic savants, or savantism of any kind, the person's brain is naturally inclined to involve more, or unusual parts of the brain in the pattern formation and recognition process. Autistic savants probably have deficiencies in being able to recognize and process social behavior, and those parts of the brain that recognize and interpret social behavior are reassigned(or more accurately naturally designed) to recognizing and interpreting sounds, and numbers, and patterns that aren't used normally in socializing.
I don't think savants necessarily make lightning fast calculations, I mean some do, and some people who arent savants can think pretty quickly, but I think what allows savants to do such amazing things is that they have pattern recognition abilities that are better or better suited to a particular task. When you were first learning math, you probably did multiplication tables, and perhaps counted to make sure you were correct, but eventually you learned that 9X9 equals 81, and you didn't have to think about it, and 900 X 900 probably has an answer related to the previous. And when you look at 9X9 you just intuitively know it equals 81, you don't need to check. And when you aren't sure about an answer, say, what is 9X6?, you might be able to say it is 54 because 9X5 equals 45.
I think savants who excel at math know dozens of these small patterns intuitively, and consider larger combinations of these small patterns with ease. For instance you know 9X9 is 81, and 90X90 is 8,100, but what about 99X99? The answer to each individual computation is 81, and will be no matter how many digits the number is, the question is, what digits is it placed in. Now most people cannot consider large numbers with ease, their short term memory does not allow them to hold on to too many individual calculations at once. Some people have such short term memory that even if they can't see that 99X99 is just a series of 81's being added at specific digit markers, they can still work out the calculation just the way you'd do it by hand. Which ones savants do, I am not sure. I think it is more likely that savants learn intuitively that double number combinations like 99 come out with predictable and similiar answers.
9X9 = 81, 99X99 = 9801, 999X999 = 998001, 9999X9999 = 99980001
So essentially I think what produces savantism or genius or whatever is better pattern formation and recognition, or better suited. Sometimes one pattern recognition skill is incompatible with another. And if you want my guess on how that really works then I will have to start getting into biology and my suspicions on the nature of intelligence which I am using to formulate ideas on how to build true artificial intelligence.
