The case of Zerah Colburn is told this way by Samuel Butler in Life and Habit (1878).
Some learn to play, to read, write, and talk, with hardly an effort - some show such an instinctive aptitude for arithmetic that, like Zerah Colburn, at eight years old, they achieve results without instruction, which in the case of most people would require a long education. The account of Zerah Colburn, as quoted from Mr. Baily in Dr. Carpenter’s “Mental Physiology,” may perhaps be given here.
“He raised any number consisting of one figure progressively to the tenth power, giving the results (by actual multiplication and not by memory) faster than they could be set down in figures by the person appointed to record them. He raised the number 8 progressively to the sixteenth power, and in naming the last result, which consisted of 15 figures, he was right in every one. Some numbers consisting of two figures he raised as high as the eighth power, though he found a difficulty in proceeding when the products became very large.
“On being asked the square root of 106,929, he answered 327 before the original number could be written down. He was then required to find the cube root of 268,336,125, and with equal facility and promptness he replied 645.
“He was asked how many minutes there are in 48 years, and before the question could be taken down he replied 25,228,800, and immediately afterwards he gave the correct number of seconds.
“On being requested to give the factors which would produce the number 247,483, he immediately named 941 and 263, which are the only two numbers from the multiplication of which it would result. On 171,395 being proposed, he named 5 × 34,279, 7 × 24,485, 59 × 2905, 83 × 2065, 35 × 4897, 295 × 581, and 413 × 415.
“He was then asked to give the factors of 36,083, but he immediately replied that it had none, which was really the case, this being a prime number. Other numbers being proposed to him indiscriminately, he always succeeded in giving the correct factors except in the case of prime numbers, which he generally discovered almost as soon as they were proposed to him. The number 4,294,967,297, which is 2^32 + 1, having been given him, he discovered, as Euler had previously done, that it was not the prime number which Fermat had supposed it to be, but that it is the product of the factors 6,700,417 × 641. The solution of this problem was only given after the lapse of some weeks, but the method he took to obtain it clearly showed that he had not derived his information from any extraneous source.
“When he was asked to multiply together numbers both consisting of more than these figures, he seemed to decompose one or both of them into its factors, and to work with them separately. Thus, on being asked to give the square of 4395, he multiplied 293 by itself, and then twice multiplied the product by 15. And on being asked to tell the square of 999,999 he obtained the correct result, 999,998,000,001, by twice multiplying the square of 37,037 by 27. He then of his own accord multiplied that product by 49, and said that the result (viz., 48,999,902,000,049) was equal to the square of 6,999,993. He afterwards multiplied this product by 49, and observed that the result (viz., 2,400,995,198,002,401) was equal to the square of 48,999,951. He was again asked to multiply the product by 25, and in naming the result (viz., 60,024,879,950,060,025) he said it was equal to the square of 244,999,755.
“On being interrogated as to the manner in which he obtained these results, the boy constantly said he did not know how the answers came into his mind. In the act of multiplying two numbers together, and in the raising of powers, it was evident (alike from the facts just stated and from the motion of his lips) that some operation was going forward in his mind; yet that operation could not (from the readiness with which his answers were furnished) have been at all allied to the usual modes of procedure, of which, indeed, he was entirely ignorant, not being able to perform on paper a simple sum in multiplication or division. But in the extraction of roots, and in the discovery of the factors of large numbers, it did not appear that any operation could take place, since he gave answers immediately, or in a very few seconds, which, according to the ordinary methods, would have required very difficult and laborious calculations, and prime numbers cannot be recognised as such by any known rule.”
I should hope that many of the above figures are wrong. I have verified them carefully with Dr. Carpenter’s quotation, but further than this I cannot and will not go. Also I am happy to find that in the end the boy overcame the mathematics, and turned out a useful but by no means particularly calculating member of society.
