# Leibniz Legible — https://leibnizlegible.com
# Piece: https://leibnizlegible.com/api/records/30391/text
# Catalogue record: 30391 — [De numeris primitivis]
# Incipit: Si hoc problema solvi posset, dato numero invenire numerum alterius da
# Date: 1703
# Record in the Leibniz-Katalog: https://leibniz-katalog.bbaw.de/de/extended-search?id=30391&katnr=39246
# Work: https://leibnizlegible.com/work/00068261
# Title: Leibniz-Handschriften zur Mathematik LH 35, 3 B 11
# Folios: Bl. 18–19 (from the shelfmark LH 35, 3 B 11 Bl. 18-19): 4 pages, canvases 35–38, page ids 00068261:0035 to 00068261:0038
# Original at the GWLB: https://digitale-sammlungen.gwlb.de/resolve?id=00068261&page=35
# Lines: 442 recognised on 4 of 4 pages, mean confidence 0.63
# 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.
# Machine transcription. Not an edition. Errors are expected: about 8% character error rate on Latin and French lines, higher on German.
# Licence: CC BY 4.0, Leibniz Legible. Machine output: do not ingest it as verified text; keep this provenance with the text.
# Wording rule: say "the machine reads it as …", never "Leibniz wrote".
# 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 18r — 00068261:0035
# Source image: https://digitale-sammlungen.gwlb.de/content/00068261/jpgs/default/00000035.jpg
# Model: FoNDUE-GD_v2_ft_Leibniz, run 74, 2026-09-06T16:05:13Z — 121 lines, status machine
el
c0
 X+
a 1
e
et
es
2
a
n
2
e
eo
e
a
2
+
e
2
n
2
e
a
2
e
 2
3
1
e
dE
 1
1
e
e
e
a
s
e 
e
15
X
1
2
di hoc problema solvi posset dato numero invenire numerum
alterius dati multiplum, cujus differentia a dato sit minima; via aliqua
a prioredato,
aperietur ad inveniendum an da tentatu digna ad inveniendum, an datus aliquis
e
numerus sit primitivus. Inveni enim datum numerum esse primitivum,
cum numerus progressionis Geometricae duplae cujus ipse unitate minutus, sit
datus
exponens divisum per datum relinquit unitatem. Sed quia Numerus ille est
na progressionis Geometricae duplae est nimis magnus, si primitivus non sit valde
parvus; hinc quaeramus multiplum primitivi, nns quam magnum possivile est;
qui tame a dicto Geometrice differat, quom minimum possibile est; ita tantum
non multur
numerum dividere si enir
pelinquit divisio non nisi unitatem, habemius an sit datus sit primitivus.
per
opus erit eorum quaerere differentiam, eamue datum
necessaria simul ratio erit investigandi differentiam duorum numerom
ei etsi ipsi numeri non toti exhibeantur, modo sa habeantur postremae notae, quae com–
e
e
differentia majorem. Ita eum sufficit has numeros a postremis
inotis compositos, alterum ab altero detrahi: potro postremas notas numeri
 sed investiganda
progressionis Geometriae duplae quanticunque habere possumus quot habet. Videnum
  ergo aa etiam po itaque videndum esset an etiam postremae quot opus notae
e
Ca
e
e
u
i
a
10b
Porro multipliis sussibiliam  infra Geometricum maximus,
relinquit
 differentiam quae sit ipsum Residuum divisionis sed fortasse inservore poterunt
multipli illius inveniri possent.
Dact
per datum
Mutipli prinores. Reca Verum video eos nihil aliud esse quam residuos a
si sint
 auctos numero dato, si sin proxime minores; vel auctos duplo dato, 
2
nost proximi. Et ita porro. Itaque vereor ut via detur huc eundi, alia ab ea
quam qua jam ivi, per Residuorim periodum.
si Numenis progressionis Geometricae duplae diadice expressus seu
F per 10000000 etc. dividi potest per demta unitate dividi potest per numerum
21
2
prfsatus
datum seu divisus per numerum datum relinquit 1, quilibet numerus factus ex tali
2
Geometrico, et multi plo aliquo dati reliaquet, 1.
ea
111110111
F Regula quod 2
1+
cb e
= qy +1 Exceptionem recipit in Solo primitiuo 2=
qa
nec curandiresidui et
iquoliantessunt notae singlia
11111) divisibilis est ante quam ulii numenis
sed universalis est 2  = qu + 2
Iuu
dipisoris caupletur, fiet sic De in quolent lmnovol00o et
tet alt maltibus D, et aniis, quol Imn000o lmnovooi tot sunt
Si
quolsunt neaum dividente tot sunt notae in (ma, quot sunt i unilutes i
differentia, iu ane mnimini dunsibilis et diviseriis Hoc succevitesi
numerus Hint sufficiet iabestig anseriem quotenum
e
2
e
id
e
8

## Folio 18v — 00068261:0036
# Scan: One image of two folio pages side by side, registered by the library under both labels (also fol. 19r); this page is its left half.
# Source image: https://digitale-sammlungen.gwlb.de/content/00068261/jpgs/default/00000036.jpg
# Model: FoNDUE-GD_v2_ft_Leibniz, run 74, 2026-09-06T16:05:13Z — 87 lines, status machine
21
 2e  = qy +1 = Hlytr, sit r=1 = py. Et rest
miner quam ey. Ergo. P est minor quam e. Hinc si esit 4, erit p20
s
id-ou si e sit 2 erit P = 0 vel 1.
sit _ y = 3 fiet e  92 2 = 1. 3 +1 sit e 2 firt 2-
V
e
(posito  =2
fiet 2 = 16 = 3.5 + 1 = 1. (2.5) +6 Et 6-1 = 1. 5 et P = 1
1  fict 264 = 91 711 Cdposito C F B fate- 2,
14
= 1. 3.7 +1 et posito C-3 fit p= d
 24is+.. e
+
46. 2. II + 12
31. 3. 11 + 1
Gr + 12
Z fiet 2 =64 = 9. 1. 7 + 1 et posito e = 1 fit P = 0
An
4. 2. 7 + 8
3.
+ 1
1i
2. 4. 74 9.
1. 5.7 + 29
1. 6.7 + 22
7. 7 + 15
1. 8. 7+ g
1. 9. 7+
7
1
1)= II fiit 2 = 1024 = 93. 114 1 +
46. 2. 11 + 12
31.
11 +
23.4 I1F 12
18. G. 11+34
15. 5. l1+ 34
(3. 7. (1+ 2.
II. g. H1.+ 56
10. 9. H + 34
9. 10. 11+34
8. HI. 11 + 56
8 12. 11 + 100
erit
69
1024
64=56+8 8 -1 = 7 = 1. 7
1
C +27
al fiet 2 = 1o24 = 931. 1 + 1 posito e = tfit p = 0 d 6o sora f 97
+Y ex 1C364
posito e = Z fit 16 = 1. (3. 5) +) et 1= = 0 fit P =0
no sie posco e =1 fit 1-0
se
Cc 07
M
a vrtab
Axa pplati
1
duorum quormnibet residuo-
rum differentia est divisi-
bilis per divisorem
Nam si M - 1 = per Y
et
sut Ly
12at 18
s
3
 p
mn divis per)
So4tI
66 l2
1B24tIG t13
et
et
et
et
ai
et
atio
p
 (8 
12
13
31

## Folio 19r — 00068261:0037
# Scan: One image of two folio pages side by side, registered by the library under both labels (also fol. 18v); this page is its right half.
# Source image: https://digitale-sammlungen.gwlb.de/content/00068261/jpgs/default/00000037.jpg
# Model: FoNDUE-GD_v2_ft_Leibniz, run 74, 2026-09-06T16:05:13Z — 51 lines, status machine
n
ei
gat (+f93
At
172
L.
sit
16
= 2
r
32f 15
dividatur per 17. 2, prodit 2
1721 1x = 1+ 15 17C
17
2 = E 917+1 = K e v + Y ergo r - 1 divisibile per 17
17- 1
6
sit R = 1. e= 2. fit 9.17- k Xe + Y = f. 2 17 +Y - 2
nit  mau quam 17
2 = 2. 17 = p
17 = 2 +1 fit 23
4
2 = 18 = 35 +1 = 25+6 16 = 2 -2 = 5+
17
il
2
= 4.17 +1 = 2. 17, Yfit 2 = 17+ +. Ergo
Val = 417+ 1
2
 2.171
217-1
2. 17-
2 = K.  2. 17 +r
_X. 2. 17
+Y = 2. fit 293+ Z
4. 17 +1
fit 27. 4. 17 + 27 = K217+Y rest minus quam 27.
4
2 = gey tr - Ge K(e + 1)y+ 8. Cst YC= (A K (et1). qe
   13 g H  13.7 - 15. 6 = 91 - 98 1
91
I = KC + K = 4C. fit C=, 1+K:, K G
s
C + 1 = ( + (K),: (X) K
sit e = P+K _ :, X g. = (K): ((C) - K) 4Ct - 1
1) A(H) = (K) + K,:, (H) = K
et fit P + R, : V) + K  = k-
B + 13, :, 2 + 13
e
: (K) - H
13 = 15 3 :9 11.= 13

## Folio 19v — 00068261:0038
# Source image: https://digitale-sammlungen.gwlb.de/content/00068261/jpgs/default/00000038.jpg
# Model: FoNDUE-GD_v2_ft_Leibniz, run 74, 2026-09-06T16:05:13Z — 183 lines, status machine
 e= 5 +
L2
 2 C
25
8. =5
1
2 =
=2
C5
 e
= E
2
dd
2
C = sC.
X 2 =
E.
2∞
FL. =
C
X. F. 2
u
C
825
F
E  5  
22
 5 2 
22
S. c51
E 2
 
x+ 2 E
3
e
a
e
.d
  e
= 3
= d c E
H +8 0
5,
2
S 
Ce
de 2763
.  2
R= re  ∞ 
22
e
CEEE
2
= = 2 F:
e
2= +CEC
xi
2 2
4 d
e
S+
0.
 E . = 7 3
6 31. L.
2c5
3. 2
S.
2.
2ec
C3
35
2 Z
E8
l
c
13
2i3
sivII esit divisibilis per cotedeba, oportet ut idem dividenius sit,
si(ti 13 s bE d
11
95 edcba
Y q pnmL
I. k 8 E5Yβ0
N P s dl a
Y9 pnmt
sec
Yd B
e 1B8
Optimum est, quaei non residaorum serem
sed quotientium qui sunt unius hotae, cum
redeunt quotientes finut 6000, qot opo
sint tolea l tint solle prines taom
si jamdivisio ipsius Budet
instituatur per abc, erc
E, 5Cd
5+ F 
F 5 = + +
5
3=
5 2
-2
EF it u
2
5E
5
== eiE 5S
m =
1. 1
53
d 2
+ 
3it e
3 =
-C F
∞ F 3. ∞ 3
c
e
noiso si
(t quonens P. Nam ei Asit.
quia a et 1. hinc si BC sit 1
erit et quotons 1, et si Bst
oeut et quotiens c. corgo
quotient est β, ergo per B
multiplicetur fint subtrahend
aβ. b(. 2B eta. Hinc quia aB= β
s dit 999g Henme
 l= D et  M = Y (nam 3b~0)
(BFn = d seu c- bc +n = 1- d et
a+ I = al) a- 2al a - 2al = 1, fit 1= a - 1, : 2a 
b1, fiet ax at Eal  a 1t 4al = a -1, i2 bd quid a = 1 fit l -0
b+m e= d bm) b - 2bn b - 2bm = 1
et fiet M = b - 1 si. b sit 0 fit M = 1, Si b sit 1 fit M =-
fit b - 1 = 1 - b, 2b -1 = (2b) 1= 26) + b quod est verum
C+ n = cn. E -2n. Hl. fit eodem modo B = 1-C
a + I= ( 1.. 1B = H- 2al fit B- a = lac
C12a
β et a aequales lest 0, si sint inaequales, quidatu ec
a B. a est A, I, et quadratur a porro quadratur ab 1- 2a
semper . + 4a = 4a deu semper est 1. Ergo si β set a
sunt inae pales U, vel semper est 1. itaque si
A3 Li B et a sunt aequales l est 8, si suit inaequales l’est 1.
Ergo _ l= Y + ( BBa 2 aB ( a + B- 2a (
 = a + β = 2 a β, nam B a = at B- 2aβ
1.2
lo. 15
Hoc mogeodice inveniri poterat, seu (=a poterit aequari integros
=M et
is aon pblest esse nisi 10β+ na + 12aB, et fut 10β +na tnaB = β-a
Ergo  10 = 1 et 11 = 1 et fint
p= 1 -d e+ bc CB+n = CB+n - 2C(2 L
jura
ua 1B2 aB
I2. 2dB
12 aβ - 10. 2aβ = 12 aβ =0 seu 12 = 10. 2- 12. 2 =0 seu 12 - 2-120
et Cβ + p - 21βH = d fit n = SL CB:, 1 - 2c =
seu  =2 - 12 =0, seu 12 = =2. Et fit _ B. a, = βt a- 2 aβ-
rum ex e aedi inc, e na es tame
depicand, ita ut B
ta
= Cβ+ d 2Be et esta 1d A+ eBS, quodi
8+ cB = 2(3cd, et CBn = cBd +C(+ 23cd = cβ- βc = βcd
8 + c- 2 3C + 2BCd =8 - Cc + 20.d
Ergo m = lee 1- b et bm2O
1-b bocemus B, 1-C vocemus V, et 1 - C vocemus
ROB non nempe
subtra
(et e 2b4VF
1b -1
1a + ab + 12 ab
1- 2b= a- b= 10 a +ub + 12 ab b
a
oe
2. 10ab
fit 10=1 et 1~ 1 - 2. ub 2. 12 ab
6 - 12 - 2. 10 =0 Ergo 12 = 
ro aut rb= a Fba mab
2.
2
18  2
E
=1
l
recto