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

Similarity:

Let Fₙ be the Fibonacci sequence defined by F₀=0, F₁=1, F n + 1 = F ₙ + F n - 1 ( n ≥ 1 ) . It is well known that F p - ( 5 / p ) ≡ 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 ∑ k ≡ r ( m o d 10 ) 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

Similarity:

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

Similarity:

For each m ≥ 2 , we consider the m -bonacci numbers defined by F k = 2 k for 0 ≤ k ≤ m - 1 and F k = F k - 1 + F k - 2 + ⋯ + F k - m for k ≥ 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...