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# Catalogue record: 55899 — 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=55899&katnr=35756
# Work: https://leibnizlegible.com/work/00068207
# Title: Leibniz-Handschriften zur Mathematik LH 35, 15, 5
# Folios: Bl. 8 (from the shelfmark LH 35, 15, 5 Bl. 8): 2 pages, canvases 11–12, page ids 00068207:0011 to 00068207:0012
# Original at the GWLB: https://digitale-sammlungen.gwlb.de/resolve?id=00068207&page=11
# Lines: 78 recognised on 2 of 2 pages, mean confidence 0.85
# 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 8r — 00068207:0011
# Source image: https://digitale-sammlungen.gwlb.de/content/00068207/jpgs/default/00000011.jpg
# Model: FoNDUE-GD_v2_ft_Leibniz, run 74, 2026-09-06T16:05:13Z — 61 lines, status machine
Cs
Problema Leibnitianu.
Esto Aequatio: bx +1 = vy. quaeritur ejus solutio in integris,
sosito b esse numerum datum, X et y quaesitos, y autem mino
rem quam b. Letitur autem solutio ea lege, ut dissimulentur
factores numeri dati b.
Solutio.
Cum petatur y esse minorem quam b.
ponat b -2 = y
Ergo b2- 2bZ +3 = y 2 = bx + 1
Unde bx = b - 2bZ +27- 1
et X = b- 2b7 +E=1
Proinde si Z sit unitas, evanescent +2 -1
Atque ita habetur solutio in integrus, quantus cunque detur B. b. dummodo
major quam 2. Frit enim b - Z = b = 1 = y
= 2 = X
atque ita bx + 1 = b = 2b + 1 = y2
cujus S. b - 1 = y. O. F. F.
Si vero Z assumatu major unitate, tunc quoque b datus debet esse a
e major ternario, et (ut habeatu solutio in integrii) Z ea cautione su
mendum est, ut ejus quadratu unitate minutu (22- 1) dividi possit
sine fractione per b.
Sista b = t 3. 2- 2. erit b-2 - b - 2 = 32 = 1 = y
b= 2bZ 1 E 1 = qu 12  4 ut
Esto b = 8. 2= 3. erit b - Z = 6-3 = 5 = y
b= 262 +X2 = 25 = y
b= 262 +22-1 = 24 =3 =x
Hic assumpsimus Z = 3. quia Z =1 = 9 - 1 = 8 dividi pot est per b = 8
Esto rursus b = 8. 7 =5. erit b -Z = 6-5 = y = 3
b - 2b2 + 2-1 = b
5
Nic 2= 1 = 25- 1 = 24
= 1 = x. dividi potest pb = 8
Atque ita. si b sit = 8. habentu tres solutiones in integris
1. P Z = 1. Quadit enim y = 7. X = 6
da
e
etia
c
2 2 3.
2 = 5.
y = 5. X = 3
y = 3. X = 1
Lterud sit b = 12. 2 - 5. erit b - 2 = 12 -5 = 3 = y
b=2b7 +7-1 = 4b = 4 = X divisibile p b = 12
2
Iem b = 12. Z-7. erit b - 2 = 12 -7 = 5 = y. 22- 1 = 49 -1 = 48 dividit
b2- 262 + 221
 24 = 2 = X
Iterud tres solutioneinintegrii posito b = 12
ta p
e
2daF Z  5
3ti a p2 =7
12
= - evadit y = 11. X = 10
y = 3. X = 5
Y = 5. X=2
Fb = 12
22 1 = 25 -( = 24
Vertatur.

## Folio 8v — 00068207:0012
# Source image: https://digitale-sammlungen.gwlb.de/content/00068207/jpgs/default/00000012.jpg
# Model: FoNDUE-GD_v2_ft_Leibniz, run 74, 2026-09-06T16:05:13Z — 17 lines, status machine
orollariu F Sequitur 1. Si Z sit unitas, cum quovis dato numero b
n
3. 4. 5. 6. 7. etc. etc. haberi solutionem integris, est enim perpetuo
b = dato. y = b - 1. X = b - 2.
Corol. 2. Cum nullo dato numerobhaberi possunt plures, so.
futionet in integru, quam tres, ut suora insinuatu est.
Coroll. 3. Si Utu E= 1 reperiri non possit, quod dividi queat sine
fractione per b. tunc 1. b admittit tantu unam solutionem pZ = 1.
Coroll. 4. Duo valores Z sorater unitatem) simul additi
exhibent 2B. datu b. ut b = 8. erit Z = 3 et = 5.
= 12
Unde habito uno valore Z (praeter unitatem) cujus 22- 1. dividi queat
sine fractione per b, statim habetu alter valor 7 p detractionem.
Tac interea sufficiant super insigni hoc Problemate;
es
in quo an mentem Lerit. CQti Tua assecutus sim, in
formari seto, et in eo. quod mihi deest, erudim.