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On ternary quadratic forms over the rational numbers

Amir Jafari, Farhood Rostamkhani (2022)

Czechoslovak Mathematical Journal

For a ternary quadratic form over the rational numbers, we characterize the set of rational numbers represented by that form over the rational numbers. Consequently, we reprove the classical fact that any positive definite integral ternary quadratic form must fail to represent infinitely many positive integers over the rational numbers. Our proof uses only the quadratic reciprocity law and the Hasse-Minkowski theorem, and is elementary.

On the Diophantine equation x 2 - k x y + y 2 - 2 n = 0

Refik Keskin, Zafer Şiar, Olcay Karaatlı (2013)

Czechoslovak Mathematical Journal

In this study, we determine when the Diophantine equation x 2 - k x y + y 2 - 2 n = 0 has an infinite number of positive integer solutions x and y for 0 n 10 . Moreover, we give all positive integer solutions of the same equation for 0 n 10 in terms of generalized Fibonacci sequence. Lastly, we formulate a conjecture related to the Diophantine equation x 2 - k x y + y 2 - 2 n = 0 .

On the diophantine equation x²+x+1 = yz

A. Schinzel (2015)

Colloquium Mathematicae

All solutions of the equation x²+x+1 = yz in non-negative integers x,y,z are given in terms of an arithmetic continued fraction.

On the matrix negative Pell equation

Aleksander Grytczuk, Izabela Kurzydło (2009)

Discussiones Mathematicae - General Algebra and Applications

Let N be a set of natural numbers and Z be a set of integers. Let M₂(Z) denotes the set of all 2x2 matrices with integer entries. We give necessary and suficient conditions for solvability of the matrix negative Pell equation (P) X² - dY² = -I with d ∈ N for nonsingular X,Y belonging to M₂(Z) and his generalization (Pn) i = 1 n X i - d i = 1 n Y ² i = - I with d ∈ N for nonsingular X i , Y i M ( Z ) , i=1,...,n.

On the powerful part of n 2 + 1

Jan-Christoph Puchta (2003)

Archivum Mathematicum

We show that n 2 + 1 is powerfull for O ( x 2 / 5 + ϵ ) integers n x at most, thus answering a question of P. Ribenboim.

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