On the Carlitz problem on the number of solutions to some special equations over finite fields

Ioulia N. Baoulina

Displaying similar documents to “On the Carlitz problem on the number of solutions to some special equations over finite fields”

Fibonacci numbers and Fermat's last theorem

Zhi-Wei Sun (1992)

Acta Arithmetica

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Let Fₙ be the Fibonacci sequence defined by F₀=0, F₁=1, $F_{n + 1} = F ₙ + F_{n - 1} (n \geq 1)$ . It is well known that $F_{p - (5 / p)} \equiv 0 (m o d p)$ for any odd prime p, where (-) denotes the Legendre symbol. In 1960 D. D. Wall [13] asked whether $p ² | F_{p - (5 / p)}$ is always impossible; up to now this is still open. In this paper the sum $\sum_{k \equiv r (m o d 10)} (\binom{n}{k})$ is expressed in terms of Fibonacci numbers. As applications we obtain a new formula for the Fibonacci quotient $F_{p - (5 / p)} / p$ and a criterion for the relation $p | F_{(p - 1) / 4}$ (if p ≡ 1 (mod 4), where p ≠ 5 is an odd prime. We also prove that the affirmative...

A new lower bound for ${∥ (3 / 2)}^{k} ∥$

Wadim Zudilin (2007)

Journal de Théorie des Nombres de Bordeaux

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We prove that, for all integers $k$ exceeding some effectively computable number $K$ , the distance from ${(3 / 2)}^{k}$ to the nearest integer is greater than $0 . 5803^{k}$ .

On the Number of Partitions of an Integer in the $m$ -bonacci Base

Marcia Edson, Luca Q. Zamboni (2006)

Annales de l’institut Fourier

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For each $m \geq 2,$ we consider the $m$ -bonacci numbers defined by $F_{k} = 2^{k}$ for $0 \leq k \leq m - 1$ and $F_{k} = F_{k - 1} + F_{k - 2} + \dots + F_{k - m}$ for $k \geq m .$ When $m = 2,$ these are the usual Fibonacci numbers. Every positive integer $n$ may be expressed as a sum of distinct $m$ -bonacci numbers in one or more different ways. Let $R_{m} (n)$ be the number of partitions of $n$ as a sum of distinct $m$ -bonacci numbers. Using a theorem of Fine and Wilf, we obtain a formula for $R_{m} (n)$ involving sums of binomial coefficients modulo $2 .$ In addition we show that this formula may be used to determine the...

Jacobsthal numbers and alternating sign matrices.

Frey, Darrin D., Sellers, James A. (2000)

Journal of Integer Sequences [electronic only]

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Inverse series relations, formal power series and Blodgett-Gessel's type binomial identities.

Chu, Wenchang (1992)

Séminaire Lotharingien de Combinatoire [electronic only]

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On some formula in connected cocommutative Hopf algebras over a field of characteristic 0

Piotr Wiśniewski (2000)

Colloquium Mathematicae

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Let H be a cocommutative connected Hopf algebra, where K is a field of characteristic zero. Let $H^{+} = K e r$ and $h^{+} = h - (h)$ for $h \in H$ . We prove that $d_{h} = \sum_{r = 1}^{\infty} ({(- 1)}^{r + 1} / r) \sum h_{1}^{+} . . . h_{r}^{+}$ is primitive, where $\sum h_{1} \otimes . . . \otimes h_{r} = Δ_{r - 1} (h)$ .

Displaying similar documents to “On the Carlitz problem on the number of solutions to some special equations over finite fields”

Fibonacci numbers and Fermat's last theorem

A new lower bound for ∥ ( 3 / 2 ) k ∥

On the Number of Partitions of an Integer in the m -bonacci Base

Jacobsthal numbers and alternating sign matrices.

Inverse series relations, formal power series and Blodgett-Gessel's type binomial identities.

On some formula in connected cocommutative Hopf algebras over a field of characteristic 0

A new lower bound for ${∥ (3 / 2)}^{k} ∥$

On the Number of Partitions of an Integer in the $m$ -bonacci Base