EUDML

In this note we prove that the equation $(\binom{k}{2}) - 1 = q^{n} + 1$ , $q \geq 2, n \geq 3$ , has only finitely many positive integer solutions $(k, q, n)$ . Moreover, all solutions $(k, q, n)$ satisfy $k 10^{10^{182}}$ , $q 10^{10^{165}}$ and $n 2 \cdot 10^{17}$ .

A note on primes p with $σ (p^{m}) = z^{n}$

Maohua Le — 1991

Colloquium Mathematicae

A note on Jeśmanowicz' conjecture

Maohua Le — 1996

Colloquium Mathematicae

Lower bounds for the solutions in the second case of Fermat's equation with prime power exponents

Maohua Le — 1993

Colloquium Mathematicae

The solvability of the diophantine equation $D_{1} x^{2} - D_{2} y^{4} = 1$

Maohua Le — 1995

Colloquium Mathematicae

A note on the integer solutions ofhyperelliptic equations

Maohua Le — 1995

Colloquium Mathematicae

A note on the diophantine equation $x ² + b^{y} = c^{z}$

Maohua Le — 1995

Acta Arithmetica

On the diophantine equation D₁x⁴ -D₂y² = 1

Maohua Le — 1996

Acta Arithmetica

On the diophantine equation $(x^{m} + 1) (x^{n} + 1) = y ²$

Maohua Le — 1997

Acta Arithmetica

1. Introduction. Let ℤ, ℕ, ℚ be the sets of integers, positive integers and rational numbers respectively. In [7], Ribenboim proved that the equation (1) $(x^{m} + 1) (x^{n} + 1) = y ²$ , x,y,m,n ∈ ℕ, x > 1, n > m ≥ 1, has no solution (x,y,m,n) with 2|x and (1) has only finitely many solutions (x,y,m,n) with 2∤x. Moreover, all solutions of (1) with 2∤x satisfy max(x,m,n) < C, where C is an effectively computable constant. In this paper we completely determine all solutions of (1) as follows. Theorem. Equation (1)...

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On the generalized Ramanujan-Nagell equation $x^{2} - D = p^{n}$

On the diophantine equation $D ₁ x ² + D ₂ = 2^{n + 2}$

A note on perfect powers of the form $x^{m - 1} + . . . + x + 1$

The diophantine equation $x^{2} + D^{m} = p^{n}$

A note on the diophantine equation $(x^{m} - 1) / (x - 1) = y^{n}$

On the number of solutions of the generalized Ramanujan-Nagell equation $x ² - D = 2^{n + 2}$

Upper bounds for class numbers of real quadratic fields

A note on the diophantine equation $(\binom{k}{2}) - 1 = q^{n} + 1$

A note on primes p with $σ (p^{m}) = z^{n}$

A note on Jeśmanowicz' conjecture

Lower bounds for the solutions in the second case of Fermat's equation with prime power exponents

The solvability of the diophantine equation $D_{1} x^{2} - D_{2} y^{4} = 1$

A note on the integer solutions ofhyperelliptic equations

A note on the diophantine equation $x ² + b^{y} = c^{z}$

On the diophantine equation D₁x⁴ -D₂y² = 1

On the diophantine equation $(x^{m} + 1) (x^{n} + 1) = y ²$

A note on the number of solutions of the generalized Ramanujan-Nagell equation $x ² - D = k^{n}$

A correction to the paper "Upper bounds for class numbers of real quadratic fields" (Acta Arith. 68 (1994), 141-144)

On Terai's conjecture concerning Pythagorean numbers

A note on Jeśmanowicz' conjecture concerning primitive Pythagorean triplets