Generalized fruit Diophantine equation and super elliptic curves

In this article, we are interested in finding rational points on certain superelliptic curves.

💡 Research Summary

The paper investigates a family of Diophantine equations, dubbed “fruit Diophantine equations,” of the form

  a·x^{d₁} – y^{d₂} – z² + xyz – b = 0  (2.1)

where d₁ ≥ 3 is odd, d₂ ≥ 2 is even, and the coefficients satisfy the congruence conditions a ≡ 1 (mod 12) and b = 2^{d₁}a – 3·d₁, together with coprimality hypotheses (3 | d₁, 2 | d₂, gcd(d₁,d₂)=1). The authors first prove that (2.1) admits no integer solution. The proof proceeds by fixing x = α and treating the parity of α separately.

If α is even, they write d₂ = 2s and rewrite the equation as a quadratic form

 Y² – β² – k²Z² = k²(aα^{d₁} – b)

with β = α², k = y^{s} – 1, Y = y^{2s‑1} – αz², Z = z. Reducing modulo 4 shows that Y² ≡ 3 (or Y² ± Z² ≡ 3) which is impossible in ℤ/4ℤ, yielding a contradiction.

If α is odd, they set α = 2n + 1 and consider three residue classes modulo 12: α ≡ 1, 5, 9. For each class they reduce the resulting quadratic form modulo 3 or modulo a suitably chosen prime p ≡ 5, 7 (mod 12). In every sub‑case they obtain an impossible congruence such as Y² ≡ –1 (mod 3) or Y² ≡ 3 (mod p) where 3 is a non‑quadratic residue. Hence (2.1) cannot have any integer solution. Corollaries 2.2 and 2.3 refine the result, showing that even when x is forced to be even or x ≡ 5 (mod 12), no solutions exist.

Having established arithmetic rigidity, the paper turns to a geometric reinterpretation. It introduces super‑elliptic curves

 C*: y^{n} = f(x)  (n ≥ 3, f∈ℤ