PROOF OF THE IMPOSSIBILITY OF THE PERFECT CUBOID EXISTENCE

H.  Gevorgyan

doi:10.31618/ESSA.2782-1994.2021.1.70.70

Автор(и)

H. Gevorgyan

DOI:

https://doi.org/10.31618/ESSA.2782-1994.2021.1.70.70

Ключові слова:

Euler parallelepideds, perfect cuboid, spatial diagonal, cube doubling problem, irrationality

Анотація

The problem of finding, among the Euler parallelepipeds, one with an integer spatial diagonal, called the perfect cuboid problem, is one of the unsolved mathematical problems from the section of number theory. This article provides mathematical proof of the impossibility of the existence of the perfect cuboide among all possible Euler parallelepipeds. A mathematical justification for an equivalence of the problem of doubling a cube and the problem of constructing a perfect cuboid is also given.

Посилання

Halcke P. Deliciae mathematicae oder mathematisches Sinnen-Confect // N. Sauer, Hamburg, Germany, 1719.

Saunderson N. Elements of algebra // Vol. 2, Cambridge Univ. Press, Cambridge, 1740.

Euler L. Vollstandige Anleitung zur Algebra // Kayserliche Akademie der Wissenschaften, St. Petersburg, 1771.

Pocklington H. C. Some Diophantine impossibilities // Proc. Cambridge Phil. Soc., Vol. 17, 1912, P. 108–121.

Kraitchik M. On certain rational cuboids // Scripta Math., Vol. 11, 1945, P. 317–326.

Sokolowsky B. D., VanHooft A. G., Volkert R. M., Reiter C. A. An infinite family of perfect parallelepipeds // Math. Comp. 83(2014), No. 289, P. 2441 – 2454.

Wyss W. On Perfect Cuboids // arxiv.org/abs/1506.02215v2 [math.NT] 27 Jun 2015.

Sawyer J. F., Reiter C. A. Perfect parallelopipeds exist // Math. Comp. 80(2011), No. 274, P. 1037 – 1040.

Sharipov R. A. Asimptoticheskij podhod k zadache o sovershennom kuboide // Ufimskij matematicheskij zhurnal. T. 7, N3 (2015), S. 100 – 113.

Wantzel P. L. Recherches sur les moyens de reconnaitre si un Probleme de Geometrie peut se resoudre avec la regle et le compas. Journal de Mathematiques Pures et Appliquees, 1837, Vol.1, Issue 2, PP. 366 – 372.