430 



and the white squares the zeros. The first line is the residue of V 3 ; the 7 

 lines numbered to 6 represent the residues (mod 2 7 - — 1) of the partial prod- 

 ucts in 



L 



s 



f 



i 



3. 



1 



o 





% 





% 



'/, 



6 



'// 



ti 





7 



v> 



/ 









/A 



<?> 



i 



// 





# 



/A 



7/ 





fy 





r// 









V 











'/) 



7/ 





& 





^7/ 

















f(A) 



w 



ra 



Z t 



mi. 





squaring V 3 ; the lines below, numbered and 1 represent the addition of the 

 partial products above and reduction modulo 2 7 — 1, and line 2 represents the 

 addition of and 1; the single line below gives the residue of the square of 

 V 3 with 2 subtracted, which is the residue of V 4 . In order to complete the 

 test of 2 7 — 1 it is necessary to find V 6 . If the residue of V 6 is zero, then 2 7 — 1 

 is prime. This briefly is the plan as given by Lucas except that he used a 

 different illustrative example. 



In order to test a large number, say 2 127 — 1. it would be necessary to 

 make 126 of these square tables such as Figure I, each having 127 2 small 

 squares. This would entail considerable labor and require a great deal of 

 time. A simple device will reduce the labor of writing each line in the large 

 square (A) to counting. Let us set down the work of squaring a number 

 written in the binary system, for example, 13, which in the binary system 

 is 1101. 



(S) 



10 11 



1 



1 1 



1 



1 1 



1 



1 1 



1 1 



1 



110 1 





10 10 1 



1 



