ON CERTAIN ACQUIRED HABITS. 15 



1 7 r >395 being proposed, he named 5 x 34,279, 

 7x24,485, 59x2905, 83x2065, 35x4897, 

 295 x 581, and 413 x 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 dis- 

 covered 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 X 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 1 5. 

 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 



