138 AN ESSAY ON THE ROOTS OF INTEGERS, 



Here M = 396, m is the same as before, and r — 396 — • 64 = 332. And 



hence the approximate Root is 2 \\f. Then 2 f-ff- | 6 — 



332 332 2 332 3 332 4 332 5 332 5 



26 + 6-2 5 - ]. 15-2 4 - [- 20-2 3 - + 15-2 2 - + 6-2- | or 



665 665 2 665 3 665 4 665 5 665 6 



6-2 5 -665 5 -332 + 15-2 4 -665 + .332 2 + 20-2 3 -665 3 -332 3 -f 15-2 2 -665 2 -332 4 -f- 6-2-665-332S + 332 6 

 64 + — or 



665 6 



8,289,866,541,885,600,000 = 62 5 '665 5 -332 



5,173,375,360,725,600,000 = 15-2 4 '665 4> 332 2 



1,721,865,282,968,320,000 — 202 3 -665 3 -332 3 



322,364,252,224,896,000 = 152"-665 2 -332 4 



32,187,949,395,087,360— 6-2665'332 5 



1,339,147,769,319,424= 332 5 



665 6 = 86,482,825,840,140,625)15,540,998,534,968,822,784 (179 -f- 64 = 243 



15,480,425,825,385,171,875 

 •60,572,709,583,650,909 



60,572,709,583,650,909 



Hence the deficiency in this case is 396 — 243 ■ 



86,482,825,840,140,625 

 25,910,116,256,489,716 



a quantity no less than 1 52 



86,482,825,840,140,625 



Third. Let there be sought the approximate 6th Root of 397 — 



Here M = 397, m is the same as before, and r — 397 — 64 = 333. 



And hence the approximate Root is 2 -f-fj- . Then 2 |»§-f | 6 = 



333 333 2 333 3 333 4 333 5 333 s 



26 + 6-2 5 - + 15-2 4 . + 20-2 3 - + 15-2 2 + 6-2- + or 



665 665 2 665 3 665 4 665 5 665 6 



6-2 5 -665 5 -333 + 15-2 4 -665 4 -S33 2 + 20-2 3 -665 3 -333 3 + 15-2 2 -665 2 -3'33 4 + 6-2665-333 5 + 333 6 



64,_|_ or 



665 6 



