Sums of cubes of polynomials

TheoremIntegers of the form \(p^3 + q^3\) or \(2(p^6 - q^6)\) as sums of four polynomial cubes#●ready to prove

Let \(p\) and \(q\) be integers and let \(N = p^3 + q^3\) or \(N = 2(p^6 - q^6)\). Then there are polynomials \(f, g, h, k \in \mathbb{Z}[x]\), each of degree at least \(1\), such that \(f^3 + g^3 + h^3 + k^3 = N\) as an identity in \(\mathbb{Z}[x]\).

theorem_wanted int_sum_four_polynomial_cubes_of_cube_form
    (N p q : ℤ)
    (hN : N = p ^ 3 + q ^ 3 ∨ N = 2 * (p ^ 6 - q ^ 6)) :
    ∃ f g h k : Polynomial ℤ,
      1 ≤ f.natDegree ∧ 1 ≤ g.natDegree ∧ 1 ≤ h.natDegree ∧
        1 ≤ k.natDegree ∧
        f ^ 3 + g ^ 3 + h ^ 3 + k ^ 3 = Polynomial.C N

Sources

• Ajai Choudhry, "Expressing an Integer as a Sum of Cubes of Polynomials", Theorem 1.

TheoremEvery integer is a sum of five cubes of polynomials#●ready to prove

For every integer \(N\) there are polynomials \(f, g, h, k, l \in \mathbb{Z}[x]\), each of degree at least \(1\), such that \(f^3 + g^3 + h^3 + k^3 + l^3 = N\) as an identity in \(\mathbb{Z}[x]\).

theorem_wanted int_eq_sum_five_polynomial_cubes (N : ℤ) :
    ∃ f g h k l : Polynomial ℤ,
      1 ≤ f.natDegree ∧ 1 ≤ g.natDegree ∧ 1 ≤ h.natDegree ∧
        1 ≤ k.natDegree ∧ 1 ≤ l.natDegree ∧
        f ^ 3 + g ^ 3 + h ^ 3 + k ^ 3 + l ^ 3 = Polynomial.C N

Sources

• Ajai Choudhry, "Expressing an Integer as a Sum of Cubes of Polynomials", Theorem 2.

2 ready to prove

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