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Criteria for testing Wall's question

Jiří Klaška (2008)

Czechoslovak Mathematical Journal

In this paper we find certain equivalent formulations of Wall's question and derive two interesting criteria that can be used to resolve this question for particular primes.

Groups with large Noether bound

Kálmán Cziszter, Mátyás Domokos (2014)

Annales de l’institut Fourier

The finite groups having an indecomposable polynomial invariant of degree at least half the order of the group are classified. It turns out that –apart from four sporadic exceptions– these are exactly the groups with a cyclic subgroup of index at most two.

Iterated digit sums, recursions and primality

Larry Ericksen (2006)

Acta Mathematica Universitatis Ostraviensis

We examine the congruences and iterate the digit sums of integer sequences. We generate recursive number sequences from triple and quintuple product identities. And we use second order recursions to determine the primality of special number systems.

Linear recurrence sequences without zeros

Artūras Dubickas, Aivaras Novikas (2014)

Czechoslovak Mathematical Journal

Let a d - 1 , , a 0 , where d and a 0 0 , and let X = ( x n ) n = 1 be a sequence of integers given by the linear recurrence x n + d = a d - 1 x n + d - 1 + + a 0 x n for n = 1 , 2 , 3 , . We show that there are a prime number p and d integers x 1 , , x d such that no element of the sequence X = ( x n ) n = 1 defined by the above linear recurrence is divisible by p . Furthermore, for any nonnegative integer s there is a prime number p 3 and d integers x 1 , , x d such that every element of the sequence X = ( x n ) n = 1 defined as above modulo p belongs to the set { s + 1 , s + 2 , , p - s - 1 } .

Mod p structure of alternating and non-alternating multiple harmonic sums

Jianqiang Zhao (2011)

Journal de Théorie des Nombres de Bordeaux

The well-known Wolstenholme’s Theorem says that for every prime p > 3 the ( p - 1 ) -st partial sum of the harmonic series is congruent to 0 modulo p 2 . If one replaces the harmonic series by k 1 1 / n k for k even, then the modulus has to be changed from p 2 to just p . One may consider generalizations of this to multiple harmonic sums (MHS) and alternating multiple harmonic sums (AMHS) which are partial sums of multiple zeta value series and the alternating Euler sums, respectively. A lot of results along this direction...

Multigeometric sequences and Cantorvals

Artur Bartoszewicz, Małgorzata Filipczak, Emilia Szymonik (2014)

Open Mathematics

For a sequence x ∈ l 10, one can consider the achievement set E(x) of all subsums of series Σn=1∞ x(n). It is known that E(x) has one of the following structures: a finite union of closed intervals, a set homeomorphic to the Cantor set, a set homeomorphic to the set T of subsums of Σn=1∞ x(n) where c(2n − 1) = 3/4n and c(2n) = 2/4n (Cantorval). Based on ideas of Jones and Velleman [Jones R., Achievement sets of sequences, Amer. Math. Monthly, 2011, 118(6), 508–521] and Guthrie and Nymann [Guthrie...

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