Exponential sums and the distribution of inversive congruential pseudorandom numbers with prime-power modulus

Harald Niederreiter; Igor E. Shparlinski

Displaying similar documents to “Exponential sums and the distribution of inversive congruential pseudorandom numbers with prime-power modulus”

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

Ioulia N. Baoulina (2011)

Journal de Théorie des Nombres de Bordeaux

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We consider an equation of the type $a_{1}^{} x_{1}^{2} + \dots + a_{n}^{} x_{n}^{2} = b x_{1} \dots x_{n}$ over the finite field $𝔽_{q} = 𝔽_{p^{s}}$ . Carlitz obtained formulas for the number of solutions to this equation when $n = 3$ and when $n = 4$ and $q \equiv 3 (mod 4)$ . In our earlier papers, we found formulas for the number of solutions when $d = gcd (n - 2, (q - 1) / 2) = 1$ or $2$ or $4$ ; and when $d > 1$ and $- 1$ is a power of $p$ modulo $2 d$ . In this paper, we obtain formulas for the number of solutions when $d = 2^{t}$ , $t \geq 3$ , $p \equiv 3 or 5 (mod 8)$ or $p \equiv 9 (mod 16)$ . For general case, we derive lower bounds for the number of solutions.

Solving the multidimensional difference biharmonic equation by the Monte Carlo method.

Mikhajlov, G.A., Lukinov, V.L. (2001)

Sibirskij Matematicheskij Zhurnal

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A note on the integer solutions ofhyperelliptic equations

Maohua Le (1995)

Colloquium Mathematicae

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Distribution of the linear flow length in a honeycomb in the small-scatterer limit.

Boca, Florin P. (2010)

The New York Journal of Mathematics [electronic only]

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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...