431 



Now, if we write the number with the digits in reverse order on a slip of paper 

 (S) and place it above the number itself as shown above, we see that the 

 digits which occur in the third column of the partial products are the digits 

 which come together, or correspond, when the slip of paper (S) is placed so 

 that the first digit of the number, when digits are reversed, is placed over 

 the third column. This is possible in no other scale, because the product 

 of two digits in the binary scale does not give a number of more than one 

 place. We can give the following rule for squaring a number written in the 

 binary scale: 



Write the number ivith the digits equally spaced and write the same number 

 with the digits reversed on a slip of paper, using the same spacing. Place the 

 slip of paper above the number so that the first digit in the reversed order comes 

 above the last digit of the number. Move the slip of paper a single space to the 

 left each time. Count the correspondences at each step. The number of corres- 

 pondences at each step is the number which belongs in that place in the result which 

 is immediately beneath the first digit on the slip. Continue this until there are 

 no more correspondences . 



It is easily seen that by means of the above rule the process described 

 by Lucas can be followed out by counting the correspondences and will lead 

 to the result in the lines marked (B) in Figure I, without having to write 

 the part (A). 



It would be possible to construct a machine which would have two 

 parallel bars in which could be set pins for the places where 1 occurs in the 

 number. The pins on one bar would be in reverse order. The bars could 

 be turned over and the number of pins striking could be recorded automati- 

 cally. At the same time one bar could be moved along one place and be in 

 readiness for the next turn. From the machine then would come the data 

 for compiling the part (B) of Figure I. Or, a more complicated machine 

 could be constructed which would give the part (C) at once. This would so 

 shorten the work of testing the Mersenne numbers that it would be possible 

 to check the results on all of them again with a reasonable expenditure of 

 time. 



Purdue University, 



LaFayette, Indiana. 



