Teoria liczb w twórczości Eulera

• Autorzy: Narkiewicz W.
• Języki publikacji: PL

Słowa kluczowe

PL

Euler; biografia naukowa; teoria liczb

• Wydawca: Polskie Towarzystwo Matematyczne
• Czasopismo: Roczniki Polskiego Towarzystwa Matematycznego. Seria 2: Wiadomości Matematyczne
• Rocznik: 2007
• Tom: [Z] 43
• Strony: 87--98
• Opis fizyczny: bibliogr. 58 poz.

Twórcy

autor

Narkiewicz W.

• Instytut Matematyczny Uniwersytet Wrocławski,

Bibliografia

• [Be] Bergmann, G., Űber Eulers Beweis des grossen Fermatschen Satzes für den Exponenten 3, Math. Ann., 164, 1966, 159-175.
• [Ch1] Chen, J.R., On the representation of a large even integer as the sum of a prime and the product of at most two primes, Sci. Sinica, 16, 1973, 157-176.
• [Che2] -, The exceptional set of Goldbach numbers, III, Chinese Quart. J. Math., 4, 1989, 1-15.
• [C] Chowla, S., An extension of Heilbronn's class-number theorem, Quart. J. Math., Oxford ser., 5, 1934, 304-307.
• [CB] Chowla, S., Briggs, W.E., On discriminants of binary quadratic forms with a single class in each genus, Canad. J. Math., 6, 1954, 463-470.
• [Co] Corput, J.G. van der, Sur l'hypothése de Goldbach pour presque tous les nombres pairs, a, 2, 1937, 266-290.
• [DERZ] Deshouillers, J.-M., Effinger, G., te Riele, H., Zinoviev, D., A complete Vinogradov 3-primes theorem under the Riemann hypothesis, Electron. Res. Announc. Amer. Math. Soc., 3, 1997, 99-104.
• [DHTN] Dickson, L.E., History of Number Theory, Washington, 1919-1923; reprinty: Stechert 1934, Chelsea 1952.
• [D] Dirichlet, P.G.L., Beweis des Satzes, dass jede unbegrenzte arithmetische Progression, deren erstes Glied und Differenz ganze Zahlen ohne gemeinschaftlichen Factor sind, unendlich viele Primzahlen enthält. Abh. Kgl. Preuß. Akad. Wiss. Berlin, 1837, 45-81; Werke, I, 313--342, Berlin 1889.
• [Du] Dujella, A., There are only finitely many Diophantine quintuples, J. Reine Angew. Math., 566, 2004, 183-214.
• [El] Elkies, N.D., On A4 + B4 + C4 = D4, Math. Comp., 51, 1988, 825-835.
• [E1] Euler, L., Vollständige Einleitung in die Algebra, St. Petersburg 1770; Opera omnia, I1.
• [E2] -, Observationis de theoremate quodam Fermatiano aliisque ad numeros primos spectantibus, Comment. Acad. Sci. Petropol., 6, 1738, 103-107; Opera Omnia, I2, 1-5.
• [E3] -, De solutione problematum diophanteorum per numeros integros, Comment. Acad. Sci. Petropol., 6, 1738, 175-188; Opera Omnia, I2, 6-17.
• [E4] -, Solutio problematis arithmetici de inveniendo numero, qui per datos numeros divisus relinquat data residua, Comment. Acad. Sci. Petropol., 7, 1740, 46-66; Opera Omnia, I2, 18-32.
• [E5] -, Theorematum quorundam ad numeros primos spectantium demonstratio, Comment. Acad. Sci. Petropol., 8, 1741, 141-146; Opera Omnia, I2, 33-37.
• [E6] -, Theorematum quorundam arithmeticorum demonstrationes, Comment. Acad. Sci. Petropol., 10, 1747, 125-146; Opera omnia , I2, 38-58.
• [E7] -, De numeris, qui sunt aggregata duorum quadratorum, Novi Comment. Acad. Sci. Petropol., 4, 1758, 3-40; Opera Omnia, I2, 295-327.
• [E8] -, Demonstratio theorematis Fermatiani omnem numerorum primum formae 4n + 1 esse summum duorum quadratum, Novi Comment. Acad. Sci. Petropol., 5, 1760, 3-13; Opera Omnia, I2, 328-337.
• [E9] -, Theoremata circa residua ex divisione potestatum relicta, Novi Comment. Acad. Sci. Petropol., 7 , 1761, 49-82 ; Opera Omnia, I2, 493-518.
• [E10] -, Theoremata arithmetica nova methodo demonstrata, Novi Comment. Acad. Sci. Petropol., 8, 1763, 74-104; Opera Omnia, I2, 531-555.
• [E11] -, De numeris primis valde magnis, Novi Comment. Acad. Sci. Petropol., 9, 1764, 99-153, Opera Omnia, I3, 1-45.
• [E12] -, De usu novi algorithmi in problemato Pelliano solvendo, Novi Comment. Acad. Sci. Petropol., 11, 1767, 29-66; Opera Omnia, I3, 73-111.
• [E13] -, Quomodo numeri praemagni sint explorandi utrum sint primi necne, Novi Comment. Acad. Sci. Petropol., 13, 1769, 67-88; Opera Omnia, I3, 112-130.
• [E14] -, De partitione numerorum in partes tam numero quam species datas, Novi Comment. Acad. Sci. Petropol., 14, 1770, 168-187; Opera omnia, I3, 131-147.
• [E15] -, Observationes circa bina biquadrata quorum summam in duam alia biquadrata resolvere liceat, Novi Comment. Acad. Sci. Petropol., 17, 1773, 64-69; Opera omnia, I3, 211-217.
• [E16] -, Novae demonstrationes circa resolutionem numerorum in quadrata, Nova Acta Eruditorum 1773, 193-211; Opera omnia, I3, 218-239.
• [E17] -, Demonstrationes circa residua ex divisione potestatum per numeros primos resultantia, Novi Comment. Acad. Sci. Petropol., 18, 1774, 85-135, Opera Omnia, I3, 240-281.
• [E18] -, Extrait d'une lettre de M.Euler le pére à M.Bernoulli concernant la mémoire imprimé parmi ceux de 1771 p.318, Nouv.Mém. Acad. Berlin, 1772/1774, 35-36; Opera Omnia, I3, 335-337.
• [E19] -, Observationes circa divisionem quadratorum per numeros primos, Opuscula varii argumenti, 1, 1783, 64-84; Opera Omnia, I3, 497-512.
• [E20] -, Extrait d'une lettre de M. Euler à M. Beguelin, Nouv.Mém. Acad. Sci. Berlin, 1776-1779, 337-339; Opera omnia , I3, 418-420.
• [E21] -, Evolutio producti infiniti (1 - x)(1 - xx)(1 - x3)(1 - x4)(1 - x5) in seriem simplicem, Acta Acad. Sci. Petropol., 1780/1783, 47-55; Opera omnia, I3, 472-479.
• [E22] -, Miscellanea analytica, Opuscula analytica, 1, 1783, 329-344; Opera Omnia, I4, 91-104.
• [E23] -, Speculationes circa quasdam insignes proprietates numerorum, Acta Acad. Sci. Petropol., 4, 1784, 18-30; Opera Omnia, I4, 105-115.
• [E24] -, Introductio in analysin infinitorum, Petropoli 1748; Opera Omnia, I8, I9.
• [E25] -, De summis serierum reciprocarum, Comment. Acad. Sci. Petropol., 7, 1740, 123-134; Opera Omnia, I14, 73-86.
• [E26] - Variae observationes circa series infinitas, Comment. Acad. Sci. Petropol., 9, 1744, 160-188; Opera Omnia, I14, 216-244.
• [E27] -, De seriebus quibusdam considerationes, Comment. Acad. Sci. Petropol., 12, 53-96, 1750; Opera omnia, I14, 407-462.
• [E28] -, Remarques sur un beau rapport entre les series des puissances tant directes que reciproques, Mémoires Acad. Sci. Berlin, 17, 1768, 83-106; Opera omnia, I15, 70-90.
• [E29] -, Tractatus de numerorum doctrina capita sedecim, quae supersunt, Commentationes arithmeticae, 2, 1849, 503-575, Opera omnia, I5, 182-283.
• [E30] -, Illustratio paradoxi circa progressionem numerorum idoneorum sive congruorum, Novi Acta Acad. Sci. Petropol., 15, 1806, 29-32; Opera omnia, I4, 395-398.
• [F] Fuss, P.-H., Correspondance mathématique et physique de quelques célèbres géomètres du XV IIIéme siécle, St.Pétersbourg 1843; reprint: Johnson 1968.
• [G] Gauss, C.F., Disquisitiones arithmeticae, Gottingae 1801.
• [Gi] Gibbs, P., Some rational Diophantine sextuples, Glasnik mat., 41, 2006, 195-203.
• [Gr] Grube, F., Ueber einige Euler'sche Sätze aus der Theorie der quadratischen Formen, Zeitschr. Math. Phys., (5) 19, 1874, 492-519.
• [H] Heegner, H., Diophantische Analysis und Modulfunktionen, Math. Z., 56, 1952, 227-253.
• [J] Jacobi, C.G.J., Fundamenta nova theoriae functionum ellipticarum, Regiomontani 1829; Gesammelte Werke, 1, 49-239, Berlin 1881; reprint: Chelsea 1969.
• [La] Lagrange, J.L., Démonstration d'un théorème d'arithmétique, Oeuvres, 3, 189-201, Paris 1869.
• [Le] Legendre, A.M., Essai sur la théorie des nombres, Paris 1798; 2 wyd. 1808, 3 wyd. (pod tytułem Théorie des Nombres), 1830.
• [LP] Lander, L.J., Parkin, T.R., Counterexample to Euler's conjecture on sums of like powers, Bull. Amer. Math. Soc., 72, 1966, str. 1079.
• [Ra] Rabinowitsch, G., Eindeutigkeit der Zerlegung in Primfaktoren in quadratischen Zahlkörpern, J. Reine Angew. Math., 142, 1913, 153-164.
• [Rie] Riemann, B., Ueber die Anzahl der Primzahlen unter einer gegebener Größe, Monatsber. Kgl. Preuß. Akad. Wiss. Berlin, 1860, 671-680.
• [S] Stark, H.M., A complete determination of the complex quadratic fields of class-number one, Michigan J. Math., 14, 1967, 1-27.
• [W] Weil, A., Number Theory, An approach through history, Birkhäuser 1983.
• [WC] Wang, T., Chen, J.-R., On odd Goldbach problem under general Riemann, hypothesis, Science in China, A, 36, 1993, 682-691.
• [Wi] Winogradow, I.M., Predstavlenie neqetnogo qisla v teorii qi-sel, Dokl. Akad. Nauk SSSR, 15, 1937, 291-294.
• [WJ] Winters, E., Ju škevič, A.P., (wydawcy), Leonhard Euler und Christian Goldbach: Briefwechsel 1729-1764 , Akademie-Verlag, 1965.
• [Z] Zinoview, D., On Vinogradov's constant in Goldbach's ternary problem, J. Number Theory, 65, 1997, 334-358.