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Displaying 141 – 160 of 350

Multiplicative character sums for nonlinear recurring sequences

Harald Niederreiter, Arne Winterhof (2004)

Acta Arithmetica

Multiplicative relations on binary recurrences

Florian Luca, Volker Ziegler (2013)

Acta Arithmetica

Given a binary recurrence ${u_{n}}_{n \geq 0}$ , we consider the Diophantine equation $u_{n_{1}}^{x_{1}} \dots u_{n_{L}}^{x_{L}} = 1$ with nonnegative integer unknowns $n_{1}, . . ., n_{L}$ , where $n_{i} \neq n_{j}$ for 1 ≤ i < j ≤ L, $m a x | x_{i} | : 1 \leq i \leq L \leq K$ , and K is a fixed parameter. We show that the above equation has only finitely many solutions and the largest one can be explicitly bounded. We demonstrate the strength of our method by completely solving a particular Diophantine equation of the above form.

$n$ -color partitions with weighted differences equal to minus two.

Agarwal, A.K., Balasubrananian, R. (1997)

International Journal of Mathematics and Mathematical Sciences

Neither $\prod_{k = 1}^{n} (4 k 2 + 1)$ nor $\prod_{k = 1}^{n} (2 k (k - 1) + 1)$ is a perfect square.

Fang, Jin-Hui (2009)

Integers

New formulae for $K_{i} (z)$ function.

Petojević, Aleksandar (2005)

Novi Sad Journal of Mathematics

Noncirculant Toeplitz matrices all of whose powers are Toeplitz

Kent Griffin, Jeffrey L. Stuart, Michael J. Tsatsomeros (2008)

Czechoslovak Mathematical Journal

Let $a$ , $b$ and $c$ be fixed complex numbers. Let $M_{n} (a, b, c)$ be the $n \times n$ Toeplitz matrix all of whose entries above the diagonal are $a$ , all of whose entries below the diagonal are $b$ , and all of whose entries on the diagonal are $c$ . For $1 \leq k \leq n$ , each $k \times k$ principal minor of $M_{n} (a, b, c)$ has the same value. We find explicit and recursive formulae for the principal minors and the characteristic polynomial of $M_{n} (a, b, c)$ . We also show that all complex polynomials in $M_{n} (a, b, c)$ are Toeplitz matrices. In particular, the inverse of $M_{n} (a, b, c)$ is a Toeplitz matrix when...