Nagell–Ljunggren equation

TheoremThe Nagell–Ljunggren cube equation for \(n \equiv 5 \pmod 6\)#●ready to prove

Let \(n\) be a natural number with \(n \equiv 5 \pmod 6\). Then there are no integers \(x\) and \(y\) with \(x \equiv 2, 3, 5\) or \(6 \pmod 9\) such that \(x^n - 1 = (x - 1) y^3\). Since such an \(x\) is not \(1\), this says that \((x^n - 1)/(x - 1) = y^3\) has no such integer solution.

public theorem_wanted nagell_ljunggren_cube_no_solution_of_mod_six_eq_five (n : ℕ)
    (hn : n % 6 = 5) (x y : ℤ) (hx : x % 9 = 2 ∨ x % 9 = 3 ∨ x % 9 = 5 ∨ x % 9 = 6) :
    x ^ n - 1 ≠ (x - 1) * y ^ 3
Sources
  • Tran Duc Son, A Note on the Nagell–Ljunggren Equation (xⁿ - 1)/(x - 1) = y³, INTEGERS 26 (2026), Article A104, Theorem 2.

TheoremThe Nagell–Ljunggren cube equation for \(x \equiv 2, 3, 5, 6 \pmod 9\)#●ready to prove

Let \(n \ge 3\) be a natural number. Then there are no integers \(x\) and \(y\) with \(x \equiv 2, 3, 5\) or \(6 \pmod 9\) such that \(x^n - 1 = (x - 1) y^3\). Since such an \(x\) is not \(1\), this says that \((x^n - 1)/(x - 1) = y^3\) has no such integer solution.

The bound \(n \ge 3\) is the classical range \(n > 2\) of the Nagell–Ljunggren equation; for \(n = 1\) every \(x\) gives the solution \(y = 1\).

public theorem_wanted nagell_ljunggren_cube_no_solution (n : ℕ) (hn : 3 ≤ n) (x y : ℤ)
    (hx : x % 9 = 2 ∨ x % 9 = 3 ∨ x % 9 = 5 ∨ x % 9 = 6) :
    x ^ n - 1 ≠ (x - 1) * y ^ 3
Sources
  • Tran Duc Son, A Note on the Nagell–Ljunggren Equation (xⁿ - 1)/(x - 1) = y³, INTEGERS 26 (2026), Article A104, Corollary 1.

TheoremLjunggren's theorem on the Nagell–Ljunggren equation#●ready to prove

Let \(x\) and \(y\) be integers and \(n\) and \(q\) natural numbers with \(|x| > 1\), \(y > 1\), \(n > 2\) and \(q \ge 2\), and suppose that \(x^n - 1 = (x - 1) y^q\), that is, \((x^n - 1)/(x - 1) = y^q\) (as \(x \ne 1\)). If \(q = 2\), or \(3 \mid n\), or \(4 \mid n\), or \(q = 3\) and \(n \not\equiv 5 \pmod 6\), then \((x, y, n, q)\) is one of \((3, 11, 5, 2)\), \((7, 20, 4, 2)\), \((18, 7, 3, 3)\) and \((-19, 7, 3, 3)\).

public theorem_wanted nagell_ljunggren_solutions_of_conditions (x y : ℤ) (n q : ℕ)
    (hx : 1 < x.natAbs) (hy : 1 < y) (hn : 2 < n) (hq : 2 ≤ q)
    (heq : x ^ n - 1 = (x - 1) * y ^ q)
    (hcond : q = 2 ∨ 3 ∣ n ∨ 4 ∣ n ∨ (q = 3 ∧ n % 6 ≠ 5)) :
    (x, y, n, q) = (3, 11, 5, 2) ∨ (x, y, n, q) = (7, 20, 4, 2) ∨ (x, y, n, q) = (18, 7, 3, 3) ∨
      (x, y, n, q) = (-19, 7, 3, 3)
Sources
  • Tran Duc Son, A Note on the Nagell–Ljunggren Equation (xⁿ - 1)/(x - 1) = y³, INTEGERS 26 (2026), Article A104, Theorem 1, quoting Ljunggren.

TheoremStructure of a further solution of the Nagell–Ljunggren equation#●ready to prove

Let \(x, y, n, q\) be natural numbers with \(x \ge 2\), \(y \ge 2\), \(n \ge 3\) and \(q \ge 2\) such that \((x^n - 1)/(x - 1) = y^q\), and suppose that \((x, y, n, q)\) is none of the known solutions \((3, 11, 5, 2)\), \((7, 20, 4, 2)\) and \((18, 7, 3, 3)\). Then the least prime divisor of \(n\) is at least \(29\), \(n\) has at most four prime factors counted with multiplicity, \(n\) is prime if \(q = 3\), and \(n = q\) if \(q \mid n\).

The source attributes the result to Bugeaud and Mihăilescu and to its references [2], [4].

public theorem_wanted nagell_ljunggren_exponent_structure (x y n q : ℕ) (hx : 2 ≤ x)
    (hy : 2 ≤ y) (hn : 3 ≤ n) (hq : 2 ≤ q) (h : (x ^ n - 1) / (x - 1) = y ^ q)
    (hnew : (x, y, n, q) ≠ (3, 11, 5, 2) ∧ (x, y, n, q) ≠ (7, 20, 4, 2) ∧
      (x, y, n, q) ≠ (18, 7, 3, 3)) :
    29 ≤ n.minFac ∧ ArithmeticFunction.cardFactors n ≤ 4 ∧ (q = 3 → n.Prime) ∧
      (q ∣ n → n = q)
Sources
  • Tran Duc Son, On the Nagell--Ljunggren Equation \frac{x^{5}-1}{x-1}=y^{3}, INTEGERS 26 (2026), Article A44, Theorem 2.

TheoremThe Nagell–Ljunggren cube equation when \(5 \mid n\)#●ready to prove

There are no natural numbers \(x \ge 2\), \(y \ge 2\) and \(n \ge 3\) with \(5 \mid n\) such that \((x^n - 1)/(x - 1) = y^3\).

public theorem_wanted nagell_ljunggren_cube_of_five_dvd (x y n : ℕ) (hx : 2 ≤ x) (hy : 2 ≤ y)
    (hn : 3 ≤ n) (h5 : 5 ∣ n) : (x ^ n - 1) / (x - 1) ≠ y ^ 3
Sources
  • Tran Duc Son, On the Nagell--Ljunggren Equation \frac{x^{5}-1}{x-1}=y^{3}, INTEGERS 26 (2026), Article A44, Corollary 1.
5 ready to prove

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