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# Title: Leibniz-Handschriften zur Mathematik LH 35, 15, 5
# Folio 7r, page id 00068207:0009
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Maji 1711
Esto sequatio: bX + 1 = gy. quaeritur ejus solutio in integris, porito b est
numerum datum, x et y quaesitos, y autem minorem quam b. Detitur autem
solutio ea lege, ut dissimulentur factores numeri dati b.
Solutio.
Cum petatur y esse minorem quam b.
ponatur b - Z = y
ergo b0- 2bZ +Z8 = ye = bX +
Unde bx = b  ∞bZ + 72
et X = b 2 -2 bZ + Z 2- 1, =b
Proinde si Z sit unitas, evanescent + 2 2
Atgs ita habetur solutio in integris, quantuscunqu detur D 2 b. Dummodo
major quam 2. erit enim b - Z = b - 1 = y
C
atque ita bX +1 = bZ- 2b. + 1 = 9Z
Cujus V. b. 1 = y. a. C. f.
Si vero Z assumatur majoi unitate, tunc quoque b datus debet esse majoi
ternano, et (ut habeatui solutio in intepris) Z ea cautione sumendum
est, ut ejus quadratu unitate minutu (22- 1) dividi possit sine
fractione per b.
Cstob = 8. Z = 3. erit b - Z = 8 - 3 = 5 = y
b= 2bZ + Z Z = ∞5 = yZ
b2
ZbZ
22
24 =3 =X
Hic assumpsimus Z= 3. quia Z 2 - 1 =9 - 1 = 8 dividi potest per b = 8
esto rursus b =b. Z = 5. etit b - 2 = 8 - 5 = y = 3
26Z + 221 =
Hic Z= 1= 25 1 = 24
dividi potest pb = B
=X
vertatui