m 



L. EULERI OPERA POSTHUMA. 



Arithmetica. 



;l>'it*"l"ll»'n)u 



■H 



Hinc p = 60-4- 12 = 72 =±=2. 36, ergo 2 est reisidutim. Porro j) = 60^^20 = 5.16, ergo 5 residiiiim, ideoqiie 

 etiam 3. Sive sumto p = 60-*- 30 = 90 = 10.9, ergo 10 residuum, hinc etiam 5. Sumto autem c=3 fit 

 2»»-i- 1=191 primus, ergo i'i»iMnoj .ii«i tmm\u^' .rtionbia-n Jn^» 



,uiiikii4»tuiHd^, Miii p = 60 — 12 H- 59 -+-?=i 48^59 H- 9 = 48 -»-(0, 2, 6, 12, 20, 30, 42). mino u 

 Hinc «tatim p = 4.8=3.16 dat 3 pro residuo; deinde jp = 50 = 2.25 dat 2 pro residuo. Pbrro 



'^ p= 48 -1-42 = 90= 10.9, ergo 10 residuum, ideoque et 5. -> ►-v.- ... 



i;jiduQuamq;uam haec prorsus certa videnlur, tatneii demoftstratio desideratur. 'J ijjlnoi/jjn/. r 



«fiild ,si b/ » uivumiii 'i?xterj»Jf'? iivriu-^oo t» nerrn iiifii;")! fiUn rifilq W{>'i\v. 



a^mf^ ^vn-ium.Bmmo mimm.r.iP'*'''^ comideratio formulae^ 2mH- 1 = 4abfj^u,.,j , .,„,„,{^^^9; 



ubi primo inquirendum, quibusnam casibus a inler residua reperiatur. Quia ii settiper est numerus formae 

 4r-*-l, nostra formula ita referetur 4»6db!:(4r-4-l), at formula 4r-f-l continet primo omnia quadrala imparia, 

 quae quidem cum 4a6 numeros primos dare possunt, majora aulem infra 4a deprimi possunt, dum ab iis sub- 

 trabitur 4a quoties fieri possit, hocque modo pro quovis casu numeri a, formula 4rH-l certos sortietur valores 

 minores, quam 4a, ac si a fuerit numerus primus, hoc modo omnes prodeunt idonei valores pro 4r-i-l, qui 

 aulem numeri hujus .formae non Qccurrunt, eos formula 4^-i-l indicemus, atque his numeris utriusque generis 

 4r-f-l et 4^-1-1 pro quovis numero primo a definitis, sequentia habebimus theoremata. 



^ ^ ^* Si fuerit 2m-i-l =4a6=i= (4r-i-l), tuui formula a'"— 1 semper erit divisibilis per 2m-Hl, ac casu 

 signi superioris tam -i-a quam — a inter nesidua quadratorum reperientur, casu autem signi inferioris, tantum 

 -+-a erit residuum, et — a non-residuum. 



II. Si fuerit 2w-f-l =4a6± (4^-1-1), tum semper formula a'"-!- 1 dividi poterit per 2m-f-l, tum vero 

 'Pt6 signo superiore -+~ neque a nec — a erit residuum, sive neque xx-\-ayy nec xx — ayy unquam per 2m-i-<l 

 dividi poterit, Prb signo autem inferiore — , inter residua erit — o, sive formula xx-t-ayy divisibihs erit pei* 

 2m-f-l; probe autem hic notetur, haectantum valere, si a fuerit numerus primus, numeri enim compositi 

 aliara requirunt evolutionem. Nunc igitur pro singulis numeris primis a exhibeamus numeros illos duplicis 

 generis in formulis 4r-t-l et 4^-f-l contentos. - ud ~v — i^V — Vi;-ir- e.J:.i: = ^\ > •}•; 



9, 17, 25, 33, U,%%W,'m' '■■' ** ' '^'» J?.^^4^ ,..u ,. -,. JA 



f4r-»-l= 1, 



a =2 < 



(4^-4-1= 5, 



m j4rH!^l= 1, 



« = 3 < 



(4^-1-1= 5i 



1 ,e.r— L (4r-i-l= 1, 

 **~" |4^H-1=13, 



M:-.':,!ri_. f 4r-|-l 



n oio.oq ;'iuli()' vH'~+^ 



' ^ ^^^J4r>f-1= 1, 5, », 25, 37, 

 \4^-Hl=13, 17, 21, 29, 41, 



.rS:i= I~! 



fwiposbi 



iini 



13, 21, 29, 37, 45, 53, 61, eto. ^ 



13, 25, 37, 49, 61, 73, etbi .m^V.t.:^ 

 17, 29, 41, 53^ 65, 77, etfe.' *- v 





9, 

 17, 



21, 29, 41, 49, 61, 69, 81, 89, etc. 

 33, 37, 53, 57, 73, 77, 93, 97, etc. 



•».'i!5:;i'. .? 1) ■n!!(.<T»5?.o'i 



:oe^«^ 



''Ci \ 



= 1, 9, 25, 

 = 5, 13, 17, 



29, 37, 53, 57, 65, 81, etc. 



33, 41, 45, 61, 69, 73, etc. 



45, 49, 53, 69, 81, 



57, 61, 65, 73, 85, 



53, 61, 69, 77, 



57, 73, 85, 89, 



69, 77, 



89, ^3^, 97, etc. 



101, 105, etc. 



81, 101, etc. 



93, 97, etc. 



81, 89, 93, 101, etc. 



73, 97, 105, etc. i- :. 



^_.»y^(4r-i-l= 1, 9, 17, 25, 29, 49, 



"~" (4?H-1= 5, 21, 33, 37, 41, 45, 



_ (4rH-l= 1, 9, 13, 21, 25, 33, 49, 53, 



"~~ (4q-i-1= 5, 29, 37, 41, 45, 57, 61, 65, 



a = 19|*-^^'^ ^' ^' ^' *^' ^'^' *^' *^' ^*' 



l4Q-f-l=13» 2t, 29, 33, 37, 41, 53, 65, 



0=23!*^*^"^*™ ^*? '^ ^*^^'^' *^ 49, 73, 77, 81, 85, etc. 



\ 4^-H 1 = 5, ,17, 21, 33s 37, 45, 53/ 57, 61, 65, 89, etc. 



73, 

 69, 



77, V 'Of f j o«,- -^C^. 



