Arithmetic progressions represented by binary forms

TheoremRational representations of an arithmetic progression scale to integral ones#●ready to prove

Let \(a, b, c\) be integers, \(n\) a natural number and \(u_0, d\) rational numbers. If for every natural number \(i < n\) there are rational numbers \(x, y\) with \(ax^2 + bxy + cy^2 = u_0 + id\), then there is a nonzero integer \(k\) such that for every natural number \(i < n\) there are integers \(x, y\) with \(ax^2 + bxy + cy^2 = k^2(u_0 + id)\).

The nonzero condition on \(k\) is the paper's common-denominator choice; it also excludes the trivial zero scaling.

theorem_wanted exists_int_representations_of_rat_arithmetic_progression
    (a b c : ℤ) (n : ℕ) (u0 d : ℚ)
    (hrep : ∀ i ∈ Finset.range n, ∃ x y : ℚ,
      (a : ℚ) * x ^ 2 + (b : ℚ) * x * y + (c : ℚ) * y ^ 2 =
        u0 + i * d) :
    ∃ k : ℤ, k ≠ 0 ∧ ∀ i ∈ Finset.range n, ∃ x y : ℤ,
      (a * x ^ 2 + b * x * y + c * y ^ 2 : ℚ) =
        (k : ℚ) ^ 2 * (u0 + i * d)

Sources

• Ajai Choudhry, "Arithmetic Progressions Represented by Binary Quadratic Forms", Lemma 1.

TheoremNine-term arithmetic progressions represented by \(px^2 + qy^2\)#●ready to prove

Let \(p, q, g, h\) be integers such that \(-4pq\) is not a perfect square, and put

\[ u_1 = 4p^5g^{10} - 47p^4qg^8h^2 - 156p^3q^2g^6h^4 + 2052p^2q^3g^4h^6 + 4833pq^4g^2h^8 + 324q^5h^{10} \]

and

\[ d = 12pqg^2h^2(pg^2 - 3qh^2)(pg^2 + 6qh^2)(2pg^2 + 3qh^2). \]

Then for every natural number \(i < 9\) there are integers \(x, y\) with \(u_1 + id = px^2 + qy^2\).

theorem_wanted exists_representation_nine_term_diagonal_quadratic
    (p q g h : ℤ) (hirr : ¬IsSquare (-4 * p * q)) :
    ∀ i : ℕ, i < 9 → ∃ x y : ℤ,
      (4 * p ^ 5 * g ^ 10 - 47 * p ^ 4 * q * g ^ 8 * h ^ 2 -
          156 * p ^ 3 * q ^ 2 * g ^ 6 * h ^ 4 +
          2052 * p ^ 2 * q ^ 3 * g ^ 4 * h ^ 6 +
          4833 * p * q ^ 4 * g ^ 2 * h ^ 8 + 324 * q ^ 5 * h ^ 10) +
        i * (12 * p * q * g ^ 2 * h ^ 2 *
          (p * g ^ 2 - 3 * q * h ^ 2) * (p * g ^ 2 + 6 * q * h ^ 2) *
          (2 * p * g ^ 2 + 3 * q * h ^ 2)) =
        p * x ^ 2 + q * y ^ 2

Sources

• Ajai Choudhry, "Arithmetic Progressions Represented by Binary Quadratic Forms", Theorem 1, relations (3.1).

TheoremBinary quadratic forms with non-square discriminant represent nine-term progressions#●ready to prove

Let \(a, b, c\) be integers such that the discriminant \(b^2 - 4ac\) is not a perfect square. Then there are integers \(u_0\) and \(d \ne 0\) such that for every natural number \(i < 9\) there are integers \(x, y\) with \(u_0 + id = ax^2 + bxy + cy^2\).

The condition \(d \ne 0\) excludes constant progressions.

theorem_wanted exists_nine_term_arithmetic_progression_binary_quadratic
    (a b c : ℤ) (hirr : ¬IsSquare (b ^ 2 - 4 * a * c)) :
    ∃ u0 d : ℤ, d ≠ 0 ∧ ∀ i : ℕ, i < 9 → ∃ x y : ℤ,
      u0 + i * d = a * x ^ 2 + b * x * y + c * y ^ 2

Sources

• Ajai Choudhry, "Arithmetic Progressions Represented by Binary Quadratic Forms", Theorem 2.

3 ready to prove

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