On ternary Diophantine equations of signature $ p,p,2 $ over number fields

Key Findings & Data:We prove that there is a constant $B_K$ depending only on $K$ such that for any prime exponent $p>B_K$ the Fermat type equation $x^p+y^p=z^2$ with $x,y,z\in O_K$ does not have certain type of solutions.

💡

Conclusion & Impact:Our main tools in the proof are modularity, level lowering, and image of inertia comparisons.

🔬Design:Research Article

📍Scope:Comprehensive Sample

⚙️Variables:Fermat equation · generalized Fermat equation · S-units

Abstract

Let $K$ be a totally real number field with narrow class number one and $O_K$ be its ring of integers. We prove that there is a constant $B_K$ depending only on $K$ such that for any prime exponent $p>B_K$ the Fermat type equation $x^p+y^p=z^2$ with $x,y,z\in O_K$ does not have certain type of solutions. Our main tools in the proof are modularity, level lowering, and image of inertia comparisons.

Keywords

Fermat equation generalized Fermat equation S-units modularity

🔖

Cite This Article

IŞIK, E., KARA, Y., KARAKURT, E. Ö. (2020). On ternary Diophantine equations of signature $ p,p,2 $ over number fields. Turkish Journal of Mathematics, Cilt: 44 - Sayı: 4, 1197–1211. https://ajindex.com/articles/185520

References (2)

Alvarado A, Koutsianas A, Malmskog B, Rasmussen C, Vincent C et al. A robust implementation for solving the S -unit equation and several applications. arXiv:1903.00977 [2] Anni S, Siksek S. Modular elliptic curves over real abelian fields and the generalized Fermat equation x 2+y 2m = z p. Algebra & Number Theory 2016; 6 (10): 1147-1172. [3] Bennett MA. The equation x 2n + y 2n = z 5. Journal de théorie des nombres de Bordeaux 2006; 2 (18): 315-321. [4] Bennett MA, Chen I. Multi-Frey Q-curves and the Diophantine equation a 2 + b 6 = c n. Algebra & Number Theory 2012; 4 (6): 707-730. [5] Bennett MA, Chen I, Dahmen SR, Yazdani S. Generalized Fermat equations: a miscellany. International Journal of Number Theory 2015; 1 (11): 1-28. [6] Bennett MA, Ellenberg JS, Ng NC. The diophantine equation A 4 + 2δB2 = C n. International Journal of Number Theory 2010; 2 (6): 311-338. [7] Bennet M, Mihăilescu P, Siksek S. The generalized Fermat equation, in Open problems in mathematics 2016; Springer, pp. 173-205. [8] Bennett MA, Skinner CM. Ternary Diophantine equations via Galois representations and modular forms. Canadian Journal of Mathematics 2004; 1 (56): 23-54. [9] Billerey N, Chen I, Dieulefait L, Freitas N. A multi-Frey approach to Fermat equations of signature(r, r, p).Transactions of American Mathematical Society 2019; 12 (371): 8651-8677. [10] Billerey N, Chen I, Dieulefait L, Freitas N. A result on the equation x p + y p = z r using Frey abelian varieties. Proceedings of the American Mathematical Society 2017; 10 (145), pp.4111-4117. [11] Billerey N, Dieulefait L. Solving Fermat-type equations x 5 + x 5 = dzp. Mathematics of Computation 2010; 269 (79), pp.535-544. [12] Blasius D. Elliptic curves. Hilbert modular forms and the Hodge conjecture, In: Hida H, Ramakrishnan D, Shahidi F (editors). Contributions to Automorphic Forms, Geometry, and Number Theory. Baltimore, MD, USA: Johns Hopkins University Press, pp. 83-103. [13] Brown D. Primitive Integral Solutions to x 2 + y 3 = z 10, International Mathematics Research Notices IMRN 2012: 423-436. [14] Bruin N. The Diophantine Equations x 2 ± y 4 = z 6 and x 2 + y 8 = z 3. Compositio Mathematica 1999; 3 (118): 305-321. [15] Bruin N. On powers as sums of two cubes. In: International Algorithmic Number Theory Symposium 2000;
July, Springer, Berlin, Heidelberg, pp. 169-184. [16] Bruin N. The primitive solutions to x 3 + y 9 = z 2. J. Number Theory 2005; 1 (111): 179-189. [17] Chen I. On the equation a 2