168 



L. EULERI OPERA POSTHUMA.;\ 



Arithmetica. 



CoROLLARiUM 2. Ilinc si exponens e divisorem habcat n, ut sit e = dn, tum semper dari poterit forma 



x" y" per A divisibilis. Cum enini a*'"— 6^ sit divisibilis, sumatur xr=.a'^ et y=zh^, vel etiam x = c^dtzad 



et y = b^±pd, vel adhuc generalius x =.fa^±aA et y = fh'^±^A.,,,^y,,^. ...^ 



NB. In his formulis, ubi productum (jp^ — \)[q — 1) (r — 1) occurrit, sufficit ejus loco minimum commune 



dividuum numerorum p — 1, q — 1, r— :!, scribere. 



, .';:■'?-, : \ — 'J.i> \\a tjr r,!i . . ( itH irtfil. 



Quoniam formula a/* — y" praeter x — y nuUos habet divisores, ,nisi in forma A«-+-l contentos, sic casu 



n = 5 formae x^ — y^ praeter x — y, divisores sunt 5A-f-l hoc est: 11, 31, 41, 61, 71, 101, 131,. etc. Si ergo 



proponatur formula x^ — 1, eaque casu x=za divisorem habeat 5A-+-1, eundem divisorem habebit casibus 



af=a*, x = a^, x=:a^, etc, sicque ex uno casu reliqui omnes deduci possunt, cunoi sit 

 ^iruU'!' ^Mij>ii'iv 



a; = a^±M(5A-i-l), 

 unde sequens tabula est confecta: 



htn?.aoq j^f»9t vnol 



Div. pr. p. 



oamim *tt|^ ^ 



(113 



11 



31 



41 



61 



71 



101 



131 



121 



1331 



Valores x 



1— 2-1- 4-t- 3-H 5-Hetc. 



1-+- 2-*- 4-4- 8-+- 16-1- etc. 



1— 4-1- 16-H 18-*- 10-f-etc. 



1— 3-H 9— 27-+- 20-1- etc. 



1-4- 5-f- 25— 17— 14-1- etc. 



1— 6-f- 36— 14- 17-1- etc 



1-f- 53-f- 58-f- 61— 42-f-etc. 



1-H 3h- 9-f- 27-4- 81-f-etc. 



generatim 



(— 2f±.UM 

 {-+- 2f:±z 31 iW 

 (— 4fzt MM 

 (— 3f± 61M 

 (-f- 5f± 71 M 

 (- 6)^zbl01M 

 (— 42)^±131M 

 (-f- 3)^±121M 

 (-f-124)^±1331M 



1 — 161 -f-632 — 596-i-124-f- etc. 

 minimus aulem valor ipsius x ex proprietate supra allata reperilur. Ita si divisor =31, quia a'*' — 1 divisorem 

 habet 31, sumatur x=:a^, fiet x^-^i- Sumatur a = 2, erit a;=64±2.31, unde minimus =2. Ita sij)=101, 

 quia a^^^ — 1 divisibile per 101, sumatur x^=a^°°, sive a; = a^°± 101 M. 



Ut formula x^-h-y^ divisibilis fiat per 37, numeri x el y ex sequenti schemate: 

 hinrKKw. r H, 10, 11 ( 8, 6, 14 



X /2, 17, 15 j/ |l6, 12, 9 



(3, 7, 4 (13, 18, 5 



scih*cet ex eadem linea horizontali sumi debent. 



At ut a?®-i-y^ divisibile fiat per 61, x et y ex sequenti schemate sumuntur 



X 



1, 13, 14 



2, 26, 28 

 4, 9, 5 



7, 30, 24 



8, 18, 10 



11, 21, 32 



,22, 19, 3 



17, 23, 6 



16, 25, 20 



,27, 15, 12 



singulis autem -his numeris adjici intelligenda est dt 6lM Hinc casus simplicissimus est 2'-l-3*. Singuli autcm 



hi terniones in unica forma comprehendi possunt, quae simplicissima est 4n, 5n, 9n, vel in hac In, 13n, 14«. 



Problema. Ut formula x^ — 1 divisibilis fiat per divisorem idoneum 4, valoreS ipsius x definire. 



' SoLUTio. Divisor A necessario debet contineri in hac formula A = — — — , cujus factor quicunque dabil 



valorem idoneum pro A; tum autem Ires habebuntur valores principales pro x, qui sunt 1, rta, zizaa, quibus 



Q-t-| 



adjici potest ±MA. Ita si s^malur a=2, erit 4=^—7, ideoque vel 4=3, vel 4=7, et tum erit a;=l,2, 4. 

 27±1 ., , . _ ?^* _ . , _ „ „ 64=tl 



Si a = 3, erit 4 = -——, i^eo^ue vel 4 = 7, vel 4 = 13, eritque x=l, 3, 9. Si o = 4, erit 4: 



4±i 



