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Let k be a real quadratic field and let $_{k}$ and $_{k}^{\times}$ be the ring of integers and the group of units, respectively. A method of solving the Diophantine equation X³ = u+v ( $X \in_{k}$ , $u, v \in_{k}^{\times}$ ) is developed.

The diophantine equation X3 + Y3 = DZ3 and the conjectures of Birch and Swinnerton-Dyer.

N.M. Stephens (1968)

Journal für die reine und angewandte Mathematik

The Diophantine equation x⁴ - Dy² = 1, II

J. H. E. Cohn (1997)

Acta Arithmetica

The Diophantine equation xⁿ = Dy² + 1

J. H. E. Cohn (2003)

Acta Arithmetica

The Diophantine equation X(X + 1) = Y(Y... + 1).

J.H.E. Cohn (1991)

Mathematica Scandinavica

The Diophantine equation x(x+1)...(x+(m-1)) + r = yⁿ

Yu. F. Bilu, M. Kulkarni, B. Sury (2004)

Acta Arithmetica

The Diophantine equation $y^{2} = D x^{4} + 1$ , II

J. Cohn (1975)

Acta Arithmetica

The Diophantine equation y...=Dx...+1, III.

J.H.E. Cohn (1978)

Mathematica Scandinavica

The diophantine equations ${(x^{2} - c)}^{2} = (t^{2} \pm 2) y^{2} + 1$

J. Cohn (1976)

Acta Arithmetica

The diophantine equations $x^{2} - k = 2 \cdot T_{n} (\frac{b^{2} \pm 2}{2})$ .

Udrea, Gheorghe (1998)

Portugaliae Mathematica

The distribution of primes satisfying the condition $a^{p - 1} \equiv 1 (m o d p^{2})$

Leo Murata (1991)

Acta Arithmetica

The divisibility of $a^{n} - b^{n}$ by powers of $n$ .

Győry, Kálmán, Smyth, Chris (2010)

Integers

The divisor problem for binary cubic forms

Tim Browning (2011)

Journal de Théorie des Nombres de Bordeaux

We investigate the average order of the divisor function at values of binary cubic forms that are reducible over $ℚ$ and discuss some applications.

The equation $a x^{m} + b y^{m} = c x^{n} + d y^{n}$

T. Shorey (1982)

Acta Arithmetica

The equation ${(j + k + 1)}^{2} - 4 k = Q n^{2}$ and related dispersions.

Kimberling, Clark (2007)

Journal of Integer Sequences [electronic only]

The equation n₁n₂ = n₃n₄, the gcd-sum function and the mean values of certain character sums

Huaning Liu, Wenguang Zhai (2012)

Acta Arithmetica

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