42 



L. EULERI OPERA POSTHUMA. 



Arithmetica. 



3V2. Numerus -*- ^ semper ad residua rcfertur, et de - 

 INumerus autem 5 reperitur inter residua, si divisor sit vel 20g 

 vel 20g -4- 19; at — 5 inter residua deprehenditur, si divisor sit vel 20g 

 20^-1-7, vel 20g-f-9. 



343. Colligamus haec, ut uni conspectui exponantur: 



k idem est judicium ac de — 1. 



1, vel 20^-^9, vel 20g-f-ll, 



1, vel 20g-*-3, vel 



Inter residua 

 erit numerus 



-H 1 



— 1 



-¥- 2 



— 2 



-4- 3 



— 3 

 -I- 5 



— 5 

 -f- 6 



— 6 



H- 7 



— 7 



H-10 



— 10 



-*-ll 

 — 11 



H-12 



— 12 

 -\-\k 



— ik 

 -+-15 



— 15 



si divisor primus fuerit 



hq 

 \q 

 Sq 

 Sq 

 i2q 

 i2q 

 20q- 

 20q 

 2kq 

 2hq 

 28g 

 28g 

 hOq 

 hOq 

 khq 

 kkq 

 kSq 

 kSq 

 56g 

 56g 

 QOq 

 60g 



i 



(«, 

 (1, 

 (», 

 (1, 

 («> 

 (», 

 («, 

 (•> 

 («, 

 (», 

 (•, 

 (1> 

 (», 

 (1> 

 («, 

 («, 

 (*, 

 (». 

 (1. 



3) 



7)n 



3) 



11) 



9, 



3, 



5, 



5, 



3, 



9, 



3, 



7, 



9, 



9, 

 11, 

 13, 



5, 



5, 



7, 

 17, 



/ th' 



11, 



7, 

 19, 



7, 



9, 



11, 



9, 



9, 



25, 



25, 



13, 



25, 



9, 



9, 



11, 



19) 



9) 

 23) 



11) 

 19, 



15, 

 13, 

 U, 

 5, 

 5, 

 23, 

 37, 

 13, 

 13, 

 17, 

 53, 



25, 

 23, 

 27, 

 13, 

 7, 

 37, 

 25, 



7, 



25, 



25, 



43, 



19, 



etc. 



27) 

 25) 

 31, 

 19, 

 37, 

 3, 

 35, 

 19, 

 45, 

 k5, 

 49, 

 23, 



37, 

 23, 

 39, 

 15, 

 37, 

 31, 

 11, 

 3, 

 53, 



31, 



39) 



37) 



19^5, 

 23, 27, 



47) 

 43) 

 31, 

 15, 



59) 

 47) (**) 



43, 

 19, 



43) 



31) 



k7, 

 23, 



^ 



51, 

 27, 



55) 

 39) 



344. Haec autem hactenus tantum inductionc nituntur, atque ad demonstrationem investig-andam 

 juvabit sequentia observasse. Primo, numerus quicunque zt n inter residua reperietur, si divisor 

 primus fuerit formae knq-\-i, vel adeo knq-t-iif denotante t numerum imparem quemcunquc. 



Scripturae ad marginem; 

 (*) a-cc — 2yy alios divisores primos non admittit, nisi formae Sq-t-{i,7)- 



(**) 1) Si xx=mn-i-r, tum quadratum xx tam per m quam n divisum, idem relinquet residuum r. Ergo 

 si residuum r convenil divisori m, conveniet etiam divisori n. 

 2)Si 



Hoc demonstrari potestj at si divisor 8n-i-l, inter residua est -f-2, quod autem hinc noa demonstralur. 



