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Leonardi Euleri Commentationes Arithmeticae Collectae (Tomus 1)

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fullscreen: Leonardi Euleri Commentationes Arithmeticae Collectae (Tomus 1)

Multivolume work

Persistent identifier:: 1736276433

Author:: Plato

Title:: Platons Werke

Sub title:: Griechisch und Deutsch mit kritischen und erklärenden Anmerkungen

Year of publication:: 1841

Place of publication:: Leipzig

Publisher of the original:: Verlag von Wilhelm Engelmann

Identifier (digital):: 1736276433

Language:: Greek, Ancient (to 1453); German

Additional Notes:: Bände 1-26 erschienen 1841-1857

Printer:: Engelmann, Wilhelm

Document type:: Multivolume work

Volume

Persistent identifier:: 1737053217

Author:: Plato

Title:: Theätetos

Sub title:: Griechisch und Deutsch mit kritischen und erklärenden Anmerkungen

Scope:: XXVI, 240 Seiten

Year of publication:: 1855

Place of publication:: Leipzig

Publisher of the original:: Verlag von Wilhelm Engelmann

Identifier (digital):: 1737053217

Signature of the source:: a 1414(20)

Language:: Greek, Ancient (to 1453); German

Usage licence:: Public Domain Mark 1.0

Printer:: Engelmann, Wilhelm

Document type:: Volume

Collection:: Philosophy

Section

Title:: [Die in diesem Dialoge vorkommenden Personen sind: Eukleides, Terpsion. Sokrates, Theodoros, Theätetos.]

Write comment:: Text parallel in griechisch und deutsch abgedruckt.

Document type:: Multivolume work

Structure type:: Section

Section

Title:: 17. [Sokr. Nun wenn es Dir so genehm ist, [...]]

Document type:: Multivolume work

Structure type:: Section

Leonardi Euleri Commentationes Arithmeticae Collectae
Leonardi Euleri Commentationes Arithmeticae Collectae (Tomus 1)
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Title page
Title page
Title page
PROOEMIUM
ELOGE DE LÉONARD EULER LU A L'ACADÉMIE IMPÉRIALE DES SCIENCES DE St.-PÉTERSBOURG. le 23 Octobre 1783, par NICOLAS FUSS.
INDEX SYSTÉMATIQUE ET RAISONNÉ DES MÉMOIRES ARITHMÉTIQUES DE LÉONARD EULER, CONTENUS DANS LES DEUX VOLUMES DE CETTE COLLECTION. PAR MM. V. BOUNIKOWSKY et P. TCHÉBYCHEW.
INDEX SYSTÉMATIQUE DES MÉMOIRES DE L. EULER RELATIFS A LA THÉORIE DES NOMBRES.
SCRIPTA ACADEMICA ET COLLECTIONES in quibus singulae hujus libri dissertationes primum editae fuerunt, adjecto, pro quovis tomo, anno emissionis.
SUPPLEMENTA PROOEMII.
CONSPECTUS TOMI PRIORIS.
CORRIGENDA ET EMENDANDA IN TOMO I.
Title page
I. Observationes de theoremate quodam Fermatiano, aliisque ad numeros primos spectantibus. (Comment. VI. 1732 - 33. p. 103.)
II. De solutione problematum Diophanteorum per numeros Integros. (Comment. VI. 1732 - 33. p. I75.)
III. Solutio problematis arithmetici de inveniendo numero, qui per datos numeros divisus, relinquat data residua. (Comment. VII. 1734 - 35. p. 46.)
IV. Theorematum quorundam ad numeros primos spectantium demonstratio. (Comment. VIII. 1736. p. 141.)
V. Theorematum quorundam arithmeticorum demonstrationes. (Comment. X. 1738, p. 125.)
VI. Theoremata circa divisores numerorum in hac forma paa ± qbb contentorum. (Comment. XIV. 1744 - 46. p. 151.)
VII. Theoremata circa divisores numerorum. (N. Comment. I. 1747 - 48, p. 20. Exhib, 1748 Oct. 17.)
VIII. Solutio problematis difficillimi a Fermatio propositi. (N. Comment. II. 1749 p. 49. Exhib. 1748. Oct. 17.)
IX. De partitione numerorum. (N. Comment, III. 1750 - 51. p. 125. Exhib. 1750. .lan. 26.)
X. De numeris amicabilibus. (Opuscula varii argum. II. 1750. p. 23.)
XI. Observatio de summis divisorum. (N. Comment. V. 1754 - 55. p. 59. Exhib. 1752. Apr. 6.)
XII. De numeris. qui sunt aggregata duorum quadratorum. (N. Comment. IV. 1752 - 53. p. 3.)
XIII. Specimen de usu observationum in mathesi pura. (N. Comment. VI. 1756 - 57. p. 185. Exhib. 1754. Sept. 30.)
XIV. Solutio generalis quorundam problematum Diophabteorum, quae vulgo nonnisi solutiones speciales admittere videntur. (N. Comment. VI. 1756 - 57. p. 155. Exhib. 1754. Sept. 30.)
XV. Demonstratio theorematis Fermatiani, omnem numerum primum formae 4n + 1 esse summam duorum quadratorum. (N. Comment. V. 1754 - 55. p. 3.)
XVI. Demonstratio theorematis circa ordinem in summis divisorum observatum. (N. Comment. V. 1754 - 55. p. 75.)
XVII. Solutio problematis de investigatione trium numerorum, quorum tam summa, quam productum, nec non summa productorum ex binis, sint numeri quadrati. (N. Comment. VIII. 1760 - 61. p. 64. Exhib. 1756. Mart. 8.)
XVIII. De problematibus indeterminatis, quae videntur plus quam determinata. (N. Comment. VI. 1756 - 57. p. 85.)
XIX. Theoremata circa residua ex divisione potestatum relicta. (N. Comment. VII. 1758 - 59. p. 49.)
XX. Theoremata arithmetica etova methodo demonstrata. (N. Comment. VIII. 1760 - 61. p. 74. Exhib. 1759 Oct. 15.)
XXI. Supplementum quorundam theorematum arithmeticorum, quae in nonnullis demonstrationibus supponuntur. (N. Comment. VIII. 1760 - 61. p. 105. Exhib. 1759. Oct. 15.)
XXII. De resolutione formularum quadraticarum indeterminatarum per numeros integros. (N. Comment. IX. 1762 - 63. p. 3. Exhib. 1759 Oct. 15.)
XXIII. De usu novi algorithmi in problemate Pelliano solvendo. (N. Comment. XI. 1765 p. 28. Exhib. 1759 Oct. 15.)
XXIV. Solution d'une question curieuse qui ne paraît soumise à aucune analyse. (Mémoires de Berlin 1759. p. 310.)
XXV. De numeris primis valde magnis. (N. Comment. IX. 1762 - 63 p. 99. Exhib. 1760. Dec. 1.)
XXVI. Quomodo numeri pracmagnl sint explorandi, utrum sint primi, nec ne. (N. Comment. XIII. 1768 p. 67. Exhib. 1765. Dec. 19.)
XXVII. De partitione numerorum In partes tam numero quam specie datas. (N. Comment. XIV. I. 1769. p. 168. Exhib. 1768 Aug. 18.)
XXVIII. De inventione qnotcunque mediarum proportionalium citra radicum extractionem. (N. Comment. XIV. I. 1769 p. 188. Exhib. 1768. Aug. 18.)
XXIX. Solutio problematis, quo duo quaeruntur numeri, quorum productum tam summa, quam differentia eorum, sive auctum sive minutum, fiat quadratum. (N. Comment. XV, 1770. p. 29. Exhib. 1770 Mart. 5.)
XXX. Problema algebraicum ob affectiones prorsus singulares memorabile. (N. Comment. XV. 1770. p. 75. Exhib. 1770 Mart. 5.)
XXXI. Solutio quorundam problematum Diophanteorum. (N. Comment. XX, 1775. p. 48. Exhib. 1771 Jul. 4.)
XXXII. Problematls cujusdam Diophantei evolutio. (N. Comment. XVII. 1772. p. 24. Exhib. 1772 Jan. 13.)
XXXIII. Observationes circa bina biquadrata, quorum summam in duo alia biquadrata resolvere liceat. (N. Comment. XVII. 1772. p. 64. Exhib. 1772 Jan. 13.)
XXXIV. Observationes circa divisionem quadratorum per numeros primos. (Op. anal. I. p. 64. Exhib. 1772?)
XXXV. Disquisitio accuratior circa residua ex divisione quadratorum altiorumque potestatum per numeros primos relicta. (Op. anal. I. 121. Exhib. 1772. Maji 18.)
XXXVI. Solutio problematis de inveniendo triangulo in quo rectae ex singulis angulis latera opposita bisecantes sint rationales. (N. Comment. XVIII. 1773. p. 171. Exhib. 1772 Aug. 24.)
XXXVII. Demonstrationes circa residua ex divisione potestatum per numeros primos resultantia. (N. Comment. XVIII. 1773. p. 85. Exhib. 1772 Maj. 18.)
XXXVIII. Novac demonstrationes circa resolutionem numerorum in quadrata. (Acta Erudit. Lips. 1773 p. 193. Acta Petrop. I. II. 1775 p. 48. Exhib. 1772. Sept. 21.)
XXXIX. Resolutio aequationis Ax² + 2Bxy + Cy² + 2Dx + 2Ey + F = 0 per muneros tam rationales, quam integros. (N. Comment. XVIII. 1773. p. 185. Exhib. 1772 Nov. 19.)
XL. De criteriis aequationis [...]xx + gyy = hzz, utrum ea resolutionem admittat, nec ne? (Op. anal. I. 211. Exhib. 1772. Dec. 7.)
XLI. De resolutione Irrationalium per fractiones continuas, ubi simul пovа quacdam et singularis species minimi exponitur. (N. Comment. XVIII. 1773 p. 218. Exhib. 1772 Dec. 10).
ADDITAMENTUM AD ANNUM 1772. Extrait d'une lettre à M. Bernoulli, concernant le mémoire imprimé parmi ceux de 1771 p. 318. (Mémoires de Berlin 1772 Hist. p. 35.)
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178
L. EULERI OPERA ARITHMETICA. 1754.
Hlota. Proprietatum, quas hic circa numeros formae 2aa-\-bb eorumque divisores observavimus,
aliae ita sunt comparatae, ut earum veritas facile ostendi possit, aliae autem majorem demonstrationis
apparatum requirunt, aliae vero denique profundissimae indaginis sunt judicandae, cum summa
sollertia ad eas demonstrandas sit opus. Ad primum genus referendae sunt observationes prima,
secunda, tertia, quarta et pars prior sextae; ad genus secundum autem pertinent observationes quinta,
pars posterior sextae, et septima, quae eo redit. Profundissimae autem indaginis est observatio
octava. Proprietates autem istae similes sunt iis, quas circa summas duorum quadratorum proposui;
quarum veritatem cum feliciter eruerim, operam dabo, ut etiam bas proprietates observatas simili
modo demonstrationibus confirmem. Incipiam ergo ab observationibus facillimis.
1. Theorema 1. Si numerus N fuerit numerus formae 2aa 66, tum quoque
ejus duplum 2N erit numerus ejusdem formae.
Demonstratio. Sit enim N = 2mm -j- nn, erit 2N= hmm -|- 2nn, ponatur 2m = k, fietque
2N=kk-\-2nn, sicque >2N erit quoque numerus formae 2«a-)-66. Q. E. D.
2. Coroll. 1. Ac si N fuerit pluribus modis numerus formae 2aa-\-bb, totidem quoque
modis ejus duplum 2 N erit numerus formae 2 aa ~j- 66.
3. Coroll. 2. Constat ergo veritas observationis secundae, simulque ratio perspicitur, cur
numerorum, qui inter numeros formae 2aa-\-bb supra expositos bis occurrunt, eorum quoque
dupla ibidem bis reperiantur.
k. Theorema 2. Si numerus par 2N fuerit numerus formae 2aa -j- 66, tum quo
que ejus semissis N erit numerus ejusdem formae.
Demonstratio. Posito 2N = 2mm -\- nn, quo 2minnn sit numerus par, quoniam pars
2mm jam est par, necesse est, ut altera pars nn sit quoque numerus par, ¡deoque et ejus radix n.
Ponatur ergo n = 2k, fietque 2iV = 2mm -|- hkk, unde per 2 dividendo oritur N= mm -\- 2kk,
ita ut quoque semissis N sit in forma 2aa -|- 66 contentus. Q. E. D.
5. Coroll. 1. Hinc etiam evidens est, si numerus propositus par 2N fuerit pluribus modis
numerus formae 2aa-\-bb, totidem quoque modis ejus semissem A fore numerum ejusdem formae.
6. Coroll. 2. Si ergo numerus N fuerit unico modo numerus formae 2aa + 66, tum etiam
ejus duplum 2 N unico modo erit numerus formae 2 aa^-bh; si enim pluribus modis esset hujus
formae, totidem quoque modis ejus semissis N foret ejusdem formae contra hypothesin.
7. Coroll. 3. Hinc autem porro duplicando numeri kN, SN, 16N etc. omnes unico tantum
modo in forma 2aa -f- 66 continebuntur, siquidem numerus simplex N unico modo in ista forma
reperiatur.
8. Coroll. 4. Quod vero hic de unico modo resolutionis in formam 2aa-\-bb est dictum,
patet quoque ad duos pluresve modos. Ex qualibet enim resolutione numeri N in formam 2aa-\-bb 1
sponte nascitur resolutio numeri, sive dupli, sive dimidii, sicque observationem tertiam demonstra
tam dedimus.
9. Theorema 3. Si habeantur duo numeri M et N formae 2aa -|- 66, erit quoque
eorum productum MN numerus ejusdem formae.

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