430 
and the white squares the zeros. The first line is the residue of V;; the 7 
lines numbered 0 to 6 represent the residues (mod 27—1) of the partial prod- 
ucts in 
TR 3 “(8 ) 
Twain ©) 
Fie. 1 
squaring V;; the lines below, numbered 0 and 1 represent the addition of the 
1, and line 2 represents the 
partial products above and reduction modulo 27 
addition of 0 and 1; the single line below gives the residue of the square of 
V; with 2 subtracted, which is the residue of V,. In order to complete the 
test of 2’—1 it is necessary to find V,. If the residue of Vs is zero, then 27—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!*7—1, it would be necessary to 
make 126 of these square tables such as Figure I, each having 127? 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. 
1014 (S) 
POE Oa Ore 
