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# Piece: https://leibnizlegible.com/api/records/1367/text
# Catalogue record: 1367 — [Aufz. zur Zahlentheorie]
# Incipit: Quaeritur in integris llvv = s4 + z4
# Date: 1676
# Record in the Leibniz-Katalog: https://leibniz-katalog.bbaw.de/de/extended-search?id=1367&katnr=58995
# Work: https://leibnizlegible.com/work/00068000
# Title: Leibniz-Handschriften zur Mathematik LH 35, 4, 3
# Folios: Bl. 31–32 (from the shelfmark LH 35, 4, 3 Bl. 31-32): 4 pages, canvases 61–64, page ids 00068000:0061 to 00068000:0064
# Original at the GWLB: https://digitale-sammlungen.gwlb.de/resolve?id=00068000&page=61
# Lines: 273 recognised on 3 of 4 pages, mean confidence 0.80
# 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.
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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 31r — 00068000:0061
# Scan: One image of two folio pages side by side, registered by the library under both labels (also fol. 32v); this page is its right half.
# Source image: https://digitale-sammlungen.gwlb.de/content/00068000/jpgs/default/00000061.jpg
# Model: FoNDUE-GD_v2_ft_Leibniz, run 73, 2026-08-27T15:28:38Z — 115 lines, status machine
quaeritur an possit ex esse X + 3 = gu. ita et X sitractus fit a + Bs -. 1
fit a 5 + q = vY et X 2 a vV : 9q
4 5 6 7 8 9 10 1 1 n quolientes si quadrati dpudantur per 3, sunt B 9 10 10n
1b 27 36 49 64 81 100 1ut 1193 0 1 3 5 8 12 16 21 27 33 40 4856 65 25 804
2 0 1 2 o en
1b B1 256 625 1096
v - 4 = v
0 5 27 85208 64 - 16 4
8 -4, 814 3
X = 
sit v = 8. et 9 = 2, g0 v =   _ _ 4 :
12 + (a - 4 181 1
V. + gu = D ag - 3. 1 v = ag - 49
8- 2 =53 8+2 = 3. 4 8- 2. 3.  v- 9 = 5  8- 2 =
261 256
si un et st divedentur per a, debent residui esse eqales t isi vet quidi
si 4294 et B + dividan ter per a, iebent, vel residui este aequiles,  bll simul facere a
tantque q et  primi inter se
v+ q=  v est major quam que
2 2 3 4 4 3 C 6 y 8 8 910 10 1 12 12 B 14 1115 16 6 17 18 18, 19 20 20 21 21 19 14 24
 33 8 12 1b 21 27 33 4u 48 56 65 75. 85. 9)b. 108 120 13 H76] 175 192 28 125 243261280. 30 320341 36)  q)
si a primition fit
va - : a est intere Cio fgg = a fi v = act + 4q n ∞ + 49 = V:cofiet
ac, L 299 = 5: 9 1 4 299 = a8 Erg res d reodita est fucilior
H v. ex SEB, 8= 2 prSE fini aptis ou per et
vidend an possibiem sit 1 24q pos diviti per a
sunt pudemi cuter se, fiot s
f2 3 3. 2 = 2. 2 = 
quis
b2 B
Residui quadi
quoidoh dim bex tut 6, 2 3 3 0292
lon
1s07
oc 0 i
125
al
Nucmndiidus per 2 078g. 10
V, Residmi Numeroru 01 01 01, 0 1
quadralam 0 1 0 101010
dibplam Eonctula 0 V v 00b 00000
biquadiatorm 0 1 0 P 0 1 0 101 0
per 3) Numer, 0 1 2 3. 4 5 6
Rresidui numeram 0 1 2 0 1 2
quadulam 0 1
diflaistam 0 2 2 0 2 2 0
biqualtatam 0- 1. 1 0 - l--
Resieni apadior
quadralor
01 2 3 4 0 1
0 14 4 1 9 1 4
duplor contorum 02 3 3 20 2
biquabutom 0 1
pr TRlesidii Nlama 0 1. 2- 3. 4.
quadralom
ducam qu
tiquadratom
2
4 
0 2 2 4 4 2 7
E et 5 non possent esse a
si postuletur, ut X sit fractus
et sit X+ = = yy
9al2 822. 4 X = 4 sit 4 + 3. 19 - 1 -2. 3
Resisui quadratorum per Sdibisom 0. 1. 4 4. 1. 0. 1. 4. 41 0
sumira vel sisfeantiae
per.3dale
duplam quadrolam
3 1 11 1
023320
ee Bimii R s Nen possunt componi duo retisui
unn biquadrati per 5 divisi alter bini dupliquadrat per 5
divisi, ut summa vel differentia sit S. Ergo. Aunem 5 non qula rat
. 16
I
256
V, qu. primi inter se
, et q pui, iem β01 9 prmm intli se
9 10 I1. 12 13114. 15 16 17. 18 1920
89 + 14 = 100
0 1 2 0 1 quia residut et duplo quadrato eum
0
chesitro a  biquadrato componit 3, etsi
0 2   2 ipsi sunt primi inter 
0 1 1
0
exeluditur 3
11641
quia summa et differentia pesidui a biquadento et a diulo
quadiato, non componit 0 vel 5, quamo radices sunt
spiimi inter se, ideo non quadat 5
on
+
14  2 Hvisib. per 3 14 + 2 = 2
q= 2 v = BB ab - 299 - a
vn - 9, divisib per a
19-  divisib per 3
d
vt q. l  10g2n -8 i
H tus a
19-3 = 2
Bv - q divisib. per a v 449 - a β v + Hi
20 = aB + A+
+4 desta tab, aus pera, v+qq a bd v
108
29
2
B = 34. - n
resi ma a n biquadruli
et in duplo quadulo non facit D1, Q
quando non sunt primi inter se
0
Residui duplolu
73+16
3

## Folio 31v — 00068000:0062
# Scan: One image of two folio pages side by side, registered by the library under both labels (also fol. 32r); this page is its left half.
# No recognised text on this page (recognized).

## Folio 32r — 00068000:0063
# Scan: One image of two folio pages side by side, registered by the library under both labels (also fol. 31v); this page is its right half.
# Source image: https://digitale-sammlungen.gwlb.de/content/00068000/jpgs/default/00000063.jpg
# Model: FoNDUE-GD_v2_ft_Leibniz, run 73, 2026-08-27T15:28:38Z — 79 lines, status machine
qraeritur in integris lvo = s+ + Z, posito  l = 1 seu Wov - 52 - 21. Itaque
dibet esse Z divisibilis per lo- s. politari et per lx +s. punc
ponamus primum l∞ - ss = l. et fiet lB = l + ss, et A WV + Uss = Z, seu
seu ( + EUs) = Z seu 2Us= 2- l Necesse est Z esse imparem seu Z = 28 + l unde fiet
a
sunt primi inter se
EUlis = 16Z 169 + 4. 85L, 6. 495l + 4. 291 17 - l) et ff di seu t et necesse est ces esse
parem sit S= 20, et f divisis omnibus per 8 siet Rcoco = 28 + 49  + 392 Il +913. unde
c. a est divisibilis per  fietque cd = mmp: Il et _ = mmn: l, et coco: g = n p p: H,
et pudit Il nop  = 2m Z 2m  t4 t + 3m ay  A 2 m n + a m nul, 2 mun 1 + B3
U seu a 16 app = 2m n + 4 lm un + 3 le mmn +  1) sed hinc sequitur
esse n = l et hiil (pp = 2m + aUm + 3 ltum + 16. seu siet
App. s - l = 2 m + ael m t + 3 ( mm. Ergo Epp -d dividi potest per mni
ubi p et B sunt primi inter se p + L, p- l, divisibilis per A nim. jam p + l, et
P-l suit primi inter se, si _ p est par ex non habent nisi 2 dinisorem communem, si prtim
si m sit par, uit n mep p impar contra si m impar, erit p. par, eo ipso
quia sunt primi inter se, quanquam hoc et aequatio ostenat
si sint sit p, par, erunt P+1, p - 1, et en impar impares, et productum etiam
impar, divisibile per tum, Ergo f sit M = qr: 1, (quanquam altetatum possit esse 1)
et sit p.  p - l,  = qgh. et - P+ B = 1YK : I et p = 9gh + l: l, et rursus
 qgh +2I3 = rrK et (hK = 29r + 499rr + 3(E
idrtiam aliande pat t
Ex 2IIss = 24 ( pecss est Z esse imporem, ergo selle parem
sari qua sunt set Z primi inter se, necesse est uno existenti impare, alterum esse pare
vidindur an in interus biquurum ventu unitale possit esse drplus quadratus
L, et p- 1
sprimi inters
pisi quod possunt
habere 2 commune
sint impares,
sn sut pares, non
4X 211, 2-1
5+1 6-
Redeam
49
nodu
in
impares
videndum quandonum quadrati Bequudiati omne per numerum aliquem astumtums
drvisi reliaquat residuum
d2s 2401
fietu
Residui biquadiulorum per aliquem numerum divisium sunt biquadiati
repo residui a biquadiatis residucrum a radicaibus pereundem nunem
divisis itaque numdrus si Z) per 5 divitus debet relinquere
residuum debet necessario s’esse numerum, cujus multiplus ab
aliquo biquadrato tantum excedatur unitate. id est quaerendi
sunt divisores bigrudratorum unitate multator, quod si
contingat ejusdem dividendi divisores esse tales uut binos confactores esse
tates, ut unus sit duplus alterias, habetur quaesitum vel sic Biquadhatus
imparonnitate minutus dimidietur, debet habere diviforem prodire quadiatus
Hoc non video quomodo obtiper possit nsi vel tentando serem
quot longun biquadrilorum quod longum, vel Methodo dyadica
t
si non ponaenus lv - s = H, sid velimus solum Z esse divisi
ponatuur Z = mapuY et lv - s- nmn: l, et fiet
c
bile per (y - (5
2. L: mMn . l
et lv- 11, = m n pp 9 et lv +  =  nppq r, et fiet uetique
e
m npq + H = n PPg3+ - s) seu M  pE Y quen = m 1 pq = 25)
Z est impor sed quia ( et Z sunt promi inter se, nccessano n, P, 4, sunt aequales
unitati, et fit au lZ = n Y et  P = M AIs = r l55 seu 255 = 1- m8
sit  sunt primi inter se Aora sol v sunt tates. Eode modo set Y sunt primi inter se
Ergo etiam Y et m non prossunt nisi binarim habere communem, si modo habeant
et vel ambo sunt impares, vel ambo pares.  andous sint impores torul primi inter se
imo ineo si gubo sint pares, fore etiam Sparem; et im non essent sit m, vel set Y
primi inter sei Ergo Y et m sunt impales itaque r et in sunt primi inter se, seu
omnes tres, S, x mi sunt primm inter se sid Hcnl quia F Hinc 1 + in par, et
Y - n par et Y+ +  rr + mm par, itaque 255 dvidi potist per 8, seu s est par 1
s=2o fit Bcoco = H int M = 2B + 1 _ r - 2) a 1. Eiit 1 - m = 27. 2(3
erro bis pivisibiterer Y+ C et per  - B et per _Mmm
 β, s, prim inter se
881
1G=C
+ 15. = 34
Wo = 5 + Z
ope

## Folio 32v — 00068000:0064
# Scan: One image of two folio pages side by side, registered by the library under both labels (also fol. 31r); this page is its left half.
# Source image: https://digitale-sammlungen.gwlb.de/content/00068000/jpgs/default/00000064.jpg
# Model: FoNDUE-GD_v2_ft_Leibniz, run 73, 2026-08-27T15:28:38Z — 79 lines, status machine
sit s = dED. H et l y - β = SsE :ll, fint ss:, V - β3 = cBB: I = 1 + β, mm + rr : 
Rursus s - A nV: H _ At V + B = AN n A: I et s5:, ) + β = p rV: l = 7 - β, mm + rr
V = SdG + B I: I1 = 2X M = Bl. fint 2 βIl = + n - ddE H
U, EBB = AN, mm LrY U1 nVr = dSE. mm+ Vr denique dEB = A p
l, E0b: AM = U prr : SE fint SSEEDB = XAnurr, seu SCD = X V. ut patet
r = 26dE + 2 Bll + (3, i
M = AN  - ddE +13,: U
co, M, Y puim inter se
co= fgh. l  r + m = +(d, seu r = 2ffg. + mI,.
V = M, retmm = glih siet 2Atg 4 EUm, rrtmm = 4ghh seu funt
a et
 A sit
249 +. EUm, 4  ifgg L a ffgmil + um = a gha, ego gin quia
m sunt poium inter se non potest m, dadi per g ergo g = l et fit (F5 + 2 m, (4 f (afsim falum
fet 3 pricui inter se, et a m
fiet h = BVS  et B um = B(3). I fint
deberet htr dividi posse per fir
2f  fflm + iimm YSsl
et Am = A L (3e): l fiit 2f t LIs( (+ l +  + lf +  + alff BBY - 813)5d
 In
2HF (- I13) (lH (3) (+ UF3) a) F d (5 (33)  (29, 55. Eg. f posset duidi per 6)
fiet B f et h sunt primi inter be Erupo et 5, Y et S,
sed fer n, adioq f et Y sud prini ader s, lugo V= , et fit, 25 4 +.   + 3 f ( = lldd
r= 2A = H + ββ et Y = s5+ BB in = diff. inter hoc conta
Et M = disterenliae quadrulor f et B
- 5 + + f 1 +65 p + Eft ( +  sen r m = 8f pB + gff B = loco
= n = - f +4se BB - 6 fa (s + affβ
itaque posito un = s
iter a 2et +2p = cocd: 4f) BB
Et ditur f.  5-
4
3 Eugo si  4E = quadrato
necesse est ut detur f + B= dimidio quadrato, est actum har posterius minus priore
cu et  to qudrato
si datur du duptam quadratum differentia, quad biquadculorm, datur minus liquadrate
dimidum quadratur summa si datur summa quadbiquadrulo aequalis quadinto
datur differentia biquadrulorm aequalis duplo quadrato, et summa biquadralorm
aequalis dimndro quadrato inio et duplo
X =  bZ - y2 13. 1
et y = qu v: c0  4: 1) L 2. 3,ta b = 13 C23 X = 13 Z -1 y = 4 v =  Co -1
A- 3g X = 13abilit y  16. 11.
Y a un bc = 13. 3
3 = 16 - 13, = 4 + 113, 4- Y13
b7 = 16 -9. = 4. +3, 4-3
v6, coco  C + e
V p c00o Vb
v - c0c 1b = B 1- 5
vrp ctb v F cocol. AX
coco
(bc +. C) = cono Yb = d . (V0 +) c = 99 + 2gcocodb
19. 16 + 1
Ex his
sed in C pro Y et β ponatur valores 2) = r - 1 et 2β = m- 1 et fiet
2EDD_ = Y+ M = 2, mm+r et 2 a V Y = Y - m, mm + YY
( E EPB() = MaH, matm U, uV = SSE, mm+rY
eleoco = X= v- m. = rr+mm rr mm
Coco= rremme, rr- umA
-3 3= bc
133
B
AB Cd + : Cod F C
10b
2P0c0 0
P qucd + bc = on
beore
bcoco + E  V
bco +C
c m. 99
cidio
Cdco, cou
coc
er
alcae
co
edt  vn
aco H.  v
a = 600 - bbco = v rb + bac, vob- B aco
V = H(bcorC) su Abbc + a) = v et sit
 fit a = s(b co + a) + bcon, tibl c 7 a) - b cc