Abstract
In this chapter we give a variety of results, a good indication of the wide search for solutions of Fermat’s problem. There are reformulations into equivalent problems, also a number of consequences of the truth of Fermat’s theorem as well as statements which follow from the assumption that the theorem is false for some exponent.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Preview
Unable to display preview. Download preview PDF.
Bibliography
Euler, L., Theorematum qourundam arithmeticorum demonstrationes, Comm. Acad. Sci. Petrop., 10 (1738), 125–146; reprinted in Opera Omnia, Commentationes Arithmeticae, Vol. I, pp. 38–58, Teubner, Leipzig, 1915.
Lebesgue, V.A., Sur un théorème de Fermat, J. Math. Pures Appl., 5 (1840), 184–185.
Liouville, J., Sur l’èquation Z2n-Y2n = 2Xn, J. Math. Pures Appl., 5 (1840), 360.
Terquem, O., Theoremes sur les puissances des nombres, Nouv. Ann. Math., 5 (1846), 70–78.
Perrin, R., Sur l’équation indéterminee x 3 + y 3 = z 3, Bull. Soc. Math. France, 13 (1885), 194–197.
Bendz, T.R., Öfver Diophantiska Ekvationen xn + yn = zn, Almqvist & Wiksells Boktryckeri, Uppsala, 1901, 35 pp.
Hurwitz, A., Über die diophantische Gleichung x 3 y + y 3 z + z 3 x = 0, Math. Ann., 65 (1908), 428–430; reprinted in Math. Werke, Vol. II, pp. 427–429, Birkhauser, Basel, 1963.
Lind, B., Einige zahlentheoretische Sätze, Arch. Math. Phys., (3), 15 (1909), 368–369.
Pocklington, H.C., Some diophantine impossibilities, Proc. Cambridge Philos. Soc, 17 (1913), 108–121.
Siegel, C.L., Über einige Anwendungen diophantischer Approximationen, Abh. Preuss. Akad. Wiss. Berlin, Phys. Math. Kl., 1(1929), 57 pp.
Fueter, R., Über kubische diophantische Gleichungen, Comment. Math. Helv., 2 (1930), 69–89.
Kapferer, H., Über die diophantischen Gleichungen \( Z^3 - Y^2 = 3^3 \cdot 2{\mathbf{ }}^\lambda \cdot X^{\lambda + 2} \) und deren Abhängigkeit von der Fermatschen Vermutung, Heidelberger Akad., Math. Naturwiss. Klasse, Abh., 2 (1933), 32–37.
Krasner, M., Sur le théorème de Fermat, C. R. Acad. Sci. Paris, 208 (1939), 1468–1471.
Pérez-Cacho, L., El último teorema de Fermat y los números de Mersenne, Rev. Real Acad. Cienc. Exact. Fiis. y Natur., Madrid, 40 (1946), 39–57.
Pérez-Cacho, L., Sobre algunas cuestiones de la teoria de números, Rev. Mat. Hisp.-Amer., (4), 18 (1958), 10–27 and 113–124.
Christilles, W.E., A note concerning Fermat’s conjecture, Amer. Math. Monthly, 74 (1967), 292–294.
Rivoire, P., Dernier Théorème de Fermat et Groupes de Classes dans Certains Corps Quadratiques Imaginaires, Thèse, Université Clermont-Ferrand, 1968, 59 pp.; reprinted in Ann. Sci. Univ. Clermont-Ferrand II, 68 (1979), 1–35.
Hellegouarch, Y., Courbes Elliptiques et Équation de Fermat, Thesis, Univiversité Besangon, 1972.
Yahya, Q.A.M.M., On general proof of Fermat’s last theorem, Portugal. Math., 32 (1973), 157–170.
Gandhi, J.M. and Stuff, M., Comments on certain results about Fermat’s last theorem, Notices Amer. Math. Soc., 22 (1975), A-502.
Chowla, S., L-series and elliptic curves, Part 4, On Fermat’s last theorem. Number Theory Day, Springer Lecture Notes, No. 626, pp. 19–24, Springer-Verlag, New York, 1978.
Vrănceanu, G., Une interprëtation gëomëtrique du théoreme de Fermat, Rev. Roumaine Math. Pures Appl., 24 (1979), 1137–1140.
Inkeri, K., On certain equivalent statements for Fermat’s last theorem — with requisite corrections, Ann. Univ. Turkuenis, Ser. AI, 186 (1984) 12–22; reprinted in Collected Papers of Kustaa Inkeri (editor, P. Ribenboim), Queen’s Papers in Pure and Applied Mathematics, Vol. 91, Kingston, Ontario, 1992.
Frey, G., Elliptic curves and solutions of A-B = C, in: Sém. Th. Nombres, Paris, 1985–1986 (editor, C. Goldstein), Progress in Math., Birkhauser, Boston, 1986, pp. 39–51.
Frey, G., Links between elliptic curves and solutions of A-B = C, J. Indian Math. Soc., 51 (1987), 117–145.
Quine, W.V., Fermat’s last theorem in combinatorial form, Amer. Math. Monthly, 95 (1989), 626.
Ribenboim, P., Catalan’s Conjecture, Academic Press, Boston, 1994.
Bibliography
Fermat, P. de, Lettre à Frenicle (Février 1657) et Second Défi aux Mathematiciens, Oeuvres, Vol. II, pp. 333–335, Gauthier-Villars, Paris, 1894.
Lagrange, J.L., Solution d’un probléme d’arithmétique, Miscellanea Taurinensia, 4 (1766–1769); reprinted in Oeuvres, Vol. I, pp. 671–731, Gauthier-Villars, Paris, 1867.
Jonquières, E. de, Sur une question d’algèbre qui a des liens avec le dernier theoreme de Fermat, C. R. Acad. Sci. Paris, 120 (1895), 1139–1143.
Bini, U., Sopra alcune congruenze, Period. Mat., (3), 22 (1907), 180–183.
Bôcher, M., Introduction to Higher Algebra, Macmillan, New York, 1907.
Bottari, A., Soluzioni intere in progressione aritmetica appartinenti a equazione indeterminate del tipo \( \sum\nolimits_{v = 1}^r x _v^n = \mathop x\nolimits_{r + i}^n \) Period. Mat., (3), 22 (1907), 156–158.
Mirimanoff, D., Sur les congruences du troisieme degre, Enseign. Math., 9 (1907), 381–384.
Cailler, R., Sur les congruences du troisieme degre, Enseign. Math., 10 (1908), 474–487.
Cattaneo, P., Osservazioni sopra due articoli del Signor Amerigo Bottari, Period. Mat., (3), 23 (1908), 218–220.
Mirimanoff, D., Sur le dernier théoreme de Fermat, Enseign. Math., 11 (1909), 49–51.
Goldziher, H., Hatványszamok Telbontdsa hatványszamok összegere, Közepiskolai Math. Lapok, 21 (1913), 177–184.
Niewiadomski, R., Question 4205, L’Interm. Math., 20 (1913), 98–100.
Métrod, G., Sur la question 4205 de Niewiadomski, L’Interm. Math., 20 (1913), 215–216.
Pollaczek, F., Über den großen Fermat’schen Satz, Sitzungsber. Akad. Wiss., Wien, Abt. Ha, 126 (1917), 45–59.
Pérez-Cacho, L., Una proposición sobre el indicador, Rev. Mat. Hisp.-Amer., (2), 3 (1928), 273–275
Bussi, C, Sull’ultimo teorema di Fermat, Boll. Un. Mat. Ital., 11 (1932), 267–269.
Skolem, T., Zwei Sätze über kubische Kongruenzen, Det Kongel. Norske Vidensk. Selskab Forhandlinger, Trondhejm, 10 (1937), no. 24, 89–92.
Skolem, T., Die Anzahl der Wurzeln der Kongruenz x3 + ax + b ≡ 0 (mod p) für die verschiedenen Paare a, b, Det Kongel. Norske Vidensk. Selskab Forhandlinger, Trondhejm, 14 (1941), no. 43, 161–164.
Bussi, C, Osservazione sull’ultimo teorema di Fermat, Boll. Un. Mat. Ital., (2), 5 (1943), 42–43.
Pierre, C, Remarques arithmetiques en connexion avec le dernier théorème de Fermat, C. R. Acad. Sci. Paris, 218 (1944), 23–25.
Inkeri, K., Untersuchungen über die Fermatsche Vermutung, Ann. Acad. Sci. Fenn., Ser. Al, Nr. 33, 1946, 60 pp.
Kapferer, H., Über ein Kriterium zur Fermatschen Vermutung, Comment. Math. Helv., 23 (1949), 64–75.
Mihaljinec, M., Prilog Fermatovu problemu (Une contribution au probleme de Fermat), Hrvatsko Prirodoslovno Drustvo, Glas. Mat. Fiz. Astr. Ser. II, 7 (1952), 12–18.
Pérez-Cacho, L., Sobre algunas cuestiones de la teoria de números, Rev. Mat. Hisp.-Amer., (4), 18 (1958), 10–27 and 113–124.
Pignataro, S., Una osservazione sull’ultimo teorema di Fermat, Rend. Accad. Sci. Fis. Mat. Napoli, (4), 30 (1963), 281–286.
Rameswar Rao, D., Some theorems on Fermat’s last theorem, Math. Student, 37 (1969), 208–210.
Swistak, J.M., A note on Fermat’s last theorem, Amer. Math. Monthly, 76 (1969), 173–174.
Ribenboim, P., Algebraic Numbers, Wiley-Interscience, New York, 1972.
Schaumberger, N., Question 572, Math. Mag., 46 (1973), 168.
Klamkin, M.S., Solution of the question 572, Math. Mag., 47 (1974), 177–178.
Kiss, P. and Phong, Bui Minh, Divisibility properties in second order recurrences, Publ. Math. Debrecen, 26 (1979), 187–197.
Ribenboim, P., 13 Lectures on Fermat’s Last Theorem, Springer-Verlag, New York, 1979.
Kiss, P., Connection between second-order recurrences and Fermat’s last theorem, Period. Math. Hungar., 11 (1980), 151–157.
Ribenboim, P., The Classical Theory of Algebraic Numbers, Springer-Verlag, New York, 1999.
Rights and permissions
Copyright information
© 1999 Springer-Verlag New York, Inc.
About this chapter
Cite this chapter
(1999). Reformulations, Consequences, and Criteria. In: Fermat’s Last Theorem for Amateurs. Springer, New York, NY. https://doi.org/10.1007/978-0-387-21692-8_9
Download citation
DOI: https://doi.org/10.1007/978-0-387-21692-8_9
Publisher Name: Springer, New York, NY
Print ISBN: 978-0-387-98508-4
Online ISBN: 978-0-387-21692-8
eBook Packages: Springer Book Archive