Nu-configurations in tiling the square
Authors:
Andrew Bremner and Richard K. Guy
Journal:
Math. Comp. 59 (1992), 195-202
MSC:
Primary 11D25; Secondary 11G05, 11Y50, 52C20
DOI:
https://doi.org/10.1090/S0025-5718-1992-1134716-2
MathSciNet review:
1134716
Full-text PDF Free Access
Abstract | References | Similar Articles | Additional Information
Abstract: In order to tile the unit square with rational triangles, at least four triangles are needed. There are four candidate configurations: one is conjectured not to exist; two others are dealt with elsewhere; the fourth is the "nu-configuration," corresponding to rational points on a quartic surface in affine 3-space. This surface is examined via a pencil of elliptic curves. One rank-3 curve is treated in detail, and rational points are given on 772 curves of the pencil. Within the range of the search there are roughly equal numbers of odd and even rank and those of rank 2 or more seem to be at least 0.45 times as numerous as those of rank 0. Symmetrical solutions correspond to rational points on a curve of rank 1, which exhibits an almost periodic behavior.
- Andrew Bremner and Richard K. Guy, The delta-lambda configurations in tiling the square, J. Number Theory 32 (1989), no. 3, 263–280. MR 1006593, DOI https://doi.org/10.1016/0022-314X%2889%2990083-8
- Joe P. Buhler, Benedict H. Gross, and Don B. Zagier, On the conjecture of Birch and Swinnerton-Dyer for an elliptic curve of rank $3$, Math. Comp. 44 (1985), no. 170, 473–481. MR 777279, DOI https://doi.org/10.1090/S0025-5718-1985-0777279-X
- J. W. S. Cassels, Diophantine equations with special reference to elliptic curves, J. London Math. Soc. 41 (1966), 193–291. MR 199150, DOI https://doi.org/10.1112/jlms/s1-41.1.193
- David A. Cox and Steven Zucker, Intersection numbers of sections of elliptic surfaces, Invent. Math. 53 (1979), no. 1, 1–44. MR 538682, DOI https://doi.org/10.1007/BF01403189
- G. Faltings, Endlichkeitssätze für abelsche Varietäten über Zahlkörpern, Invent. Math. 73 (1983), no. 3, 349–366 (German). MR 718935, DOI https://doi.org/10.1007/BF01388432
- Richard K. Guy, Tiling the square with rational triangles, Number theory and applications (Banff, AB, 1988) NATO Adv. Sci. Inst. Ser. C Math. Phys. Sci., vol. 265, Kluwer Acad. Publ., Dordrecht, 1989, pp. 45–101. MR 1123070
- B. Mazur, Modular curves and the Eisenstein ideal, Inst. Hautes Études Sci. Publ. Math. 47 (1977), 33–186 (1978). With an appendix by Mazur and M. Rapoport. MR 488287
- B. Mazur, Rational isogenies of prime degree (with an appendix by D. Goldfeld), Invent. Math. 44 (1978), no. 2, 129–162. MR 482230, DOI https://doi.org/10.1007/BF01390348
- Joseph H. Silverman, Computing heights on elliptic curves, Math. Comp. 51 (1988), no. 183, 339–358. MR 942161, DOI https://doi.org/10.1090/S0025-5718-1988-0942161-4
- Lawrence C. Washington, Number fields and elliptic curves, Number theory and applications (Banff, AB, 1988) NATO Adv. Sci. Inst. Ser. C Math. Phys. Sci., vol. 265, Kluwer Acad. Publ., Dordrecht, 1989, pp. 245–278. MR 1123077 A. Weil, Sur les courbes algébriques et les variétés qui s’en déduisent, Hermann, Paris, 1948.
- Don Zagier, Large integral points on elliptic curves, Math. Comp. 48 (1987), no. 177, 425–436. MR 866125, DOI https://doi.org/10.1090/S0025-5718-1987-0866125-3
Retrieve articles in Mathematics of Computation with MSC: 11D25, 11G05, 11Y50, 52C20
Retrieve articles in all journals with MSC: 11D25, 11G05, 11Y50, 52C20
Additional Information
Article copyright:
© Copyright 1992
American Mathematical Society