On the Diophantine Equation x₁³-x₂²x₁+1=0 over ℚ(√(2))

2026/07/22 by Pinki Khatun
Mathematics · #math.GM

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arxiv created 2026/07/22 · arxiv updated 2026/07/30

Abstract

I investigate the Diophantine equation x₁³-x₂²x₁+1=0 where x₁∈ℚ(√(2)) and x₂∈ℤ[√(2)]. Using the arithmetic of the quadratic integer ring ℤ[√(2)], together with norm arguments, divisibility properties, and the explicit description of its unit group, I prove that the equation has exactly two solutions, namely (x₁,x₂)=(-1,0)~ and ~(1,√(2)) As an application, I consider the family of elliptic curves C_m:Y²=X³-m²X+1,~ m∈ℤ[√(2)], and deduce that, for every m≠0,√(2) the Mordell--Weil group C_m(ℚ(√(2))) contains no rational point of order two.

Citations

• A criterion to rule out torsion groups for elliptic curves over number fields

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