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The Josephus problem

Lorenz Halbeisen, Norbert Hungerbühler (1997)

Journal de théorie des nombres de Bordeaux

We give explicit non-recursive formulas to compute the Josephus-numbers j ( n , 2 , i ) and j ( n , 3 , i ) and explicit upper and lower bounds for j ( n , k , i ) (where k ≥ 4 ) which differ by 2 k - 2 (for k = 4 the bounds are even better). Furthermore we present a new fast algorithm to calculate j ( n , k , i ) which is based upon the mentioned bounds.

The positivity problem for fourth order linear recurrence sequences is decidable

Pinthira Tangsupphathawat, Narong Punnim, Vichian Laohakosol (2012)

Colloquium Mathematicae

The problem whether each element of a sequence satisfying a fourth order linear recurrence with integer coefficients is nonnegative, referred to as the Positivity Problem for fourth order linear recurrence sequence, is shown to be decidable.

The terms of the form 7kx² in the generalized Lucas sequence with parameters P and Q

Olcay Karaatlı (2016)

Acta Arithmetica

Let Vₙ(P,Q) denote the generalized Lucas sequence with parameters P and Q. For all odd relatively prime values of P and Q such that P² + 4Q > 0, we determine all indices n such that Vₙ(P,Q) = 7kx² when k|P. As an application, we determine all indices n such that the equation Vₙ = 21x² has solutions.

Towards Bauer's theorem for linear recurrence sequences

Mariusz Skałba (2003)

Colloquium Mathematicae

Consider a recurrence sequence ( x k ) k ∈ ℤ of integers satisfying x k + n = a n - 1 x k + n - 1 + . . . + a ₁ x k + 1 + a ₀ x k , where a ₀ , a ₁ , . . . , a n - 1 ∈ ℤ are fixed and a₀ ∈ -1,1. Assume that x k > 0 for all sufficiently large k. If there exists k₀∈ ℤ such that x k ₀ < 0 then for each negative integer -D there exist infinitely many rational primes q such that q | x k for some k ∈ ℕ and (-D/q) = -1.

Currently displaying 321 – 340 of 351