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# Catalogue record: 55504 — [Problema Leibnitianum]
# Incipit: Esto Aequatio: bx + 1 = yy. quaeritur ejus solutio in integris,
# Date: 1716-05
# Record in the Leibniz-Katalog: https://leibniz-katalog.bbaw.de/de/extended-search?id=55504&katnr=35757
# Work: https://leibnizlegible.com/work/00068207
# Title: Leibniz-Handschriften zur Mathematik LH 35, 15, 5
# Folios: Bl. 7 (from the shelfmark LH 35, 15, 5 Bl. 7): 2 pages, canvases 9–10, page ids 00068207:0009 to 00068207:0010
# Original at the GWLB: https://digitale-sammlungen.gwlb.de/resolve?id=00068207&page=9
# Lines: 73 recognised on 2 of 2 pages, mean confidence 0.82
# Catalogue: Leibniz-Katalog / Arbeitskatalog der Leibniz-Edition (https://leibniz-katalog.bbaw.de/), ed. Arbeitsstelle Potsdam I der Berlin-Brandenburgischen Akademie der Wissenschaften, CC BY 4.0.
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# After the first blank line, page by page: a "## Folio <label> — <page id>" line ("## Canvas <n>" where no folio label is recorded), "# " lines naming its source image and model, then its recognised lines in reading order, one per line.

## Folio 7r — 00068207:0009
# Source image: https://digitale-sammlungen.gwlb.de/content/00068207/jpgs/default/00000009.jpg
# Model: FoNDUE-GD_v2_ft_Leibniz, run 74, 2026-09-06T16:05:13Z — 33 lines, status machine
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

## Folio 7v — 00068207:0010
# Source image: https://digitale-sammlungen.gwlb.de/content/00068207/jpgs/default/00000010.jpg
# Model: FoNDUE-GD_v2_ft_Leibniz, run 74, 2026-09-06T16:05:13Z — 40 lines, status machine
Atqu ita, si b sit = 8. habentur tres solutiones in inteptis
+a
pZ = 1. evadit erum y = 7. X = 6
Zda p Z-7
top 77.
L=5.
tera sit b = 12. Z= 5. erit b - Z = 12 ,5-
b= 2bZ+ 22. 1 =
5. X = 3
I = 9
48
X =
Z2 1-
= 4 =x d divistbule f0 = 12
Item b = 12. Z = 7. erit b-Z = 12 -7 -5 = 9. Z21 = 49 = 1 = 48 dividitur
b= 2bZ +2
2
Ieru tres solutiones in integris posito b = 12
24
12
Z = 1  - - - evadit y = I1. X = 10
2da  Z = 5
otiap
1
L
T. X = 5
y = 5. Z = Z
Corollario Fm sequtur Fo si Z sit unitas, cum quo vis dato numeto b 3. 4. 5. 6. 7. et. etc.
habete folutionem integris, est enum perpetuo b = dato. y - b- 1. X = b - 2.
Coroll. 2. Cum nullo dato numero b. haberi possunt plures solutiones in
integris, quam tres, ut supta insinuatu est.
Coroll. 3. Si Otu Z= I repetiri non possit, quod dividi queat sine fractione perb.
tunc H9b. admittit tanti unam solutionem pZ =
Coroll. 4. Duo valores Z (praeter unitatem) aqua simul additi
exhibent n. datuo b. ut b = 8. erit Z = 3 et = 5.
b = 12
Unde habito uno valore Z (praeter unitatem) cujus Z= 1. dividi queat sine.
fractione perb; statim habetur alter valoi Z P detractionem.
15
Ab = 12