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Representations of Numbers as Sums and Differences of Unlike Powers

Enrico Jabara — 2010

Bollettino dell'Unione Matematica Italiana

In this paper we prove that every n ∈ 𝐙 can be written as n = ϵ 1 ⁢ x 1 2 + ϵ 2 ⁢ x 2 3 + ϵ 3 ⁢ x 3 4 + ϵ 4 ⁢ x 4 5 and as n = ϵ 1 ⁢ x 1 3 + ϵ 2 ⁢ x 2 4 + ϵ 3 ⁢ x 3 5 + ϵ 4 ⁢ x 4 6 + ϵ 5 ⁢ x 5 7 + ϵ 6 ⁢ x 6 8 + ϵ 7 ⁢ x 7 9 + ϵ 8 ⁢ x 8 10 with x i ∈ 𝐙 and ϵ i ∈ { - 1 , 1 } . We also prove some other results on numbers expressible as sums or differences of unlike powers.

A note on a class of factorized p -groups

Enrico Jabara — 2005

Czechoslovak Mathematical Journal

In this note we study finite p -groups G = A B admitting a factorization by an Abelian subgroup A and a subgroup B . As a consequence of our results we prove that if B contains an Abelian subgroup of index p n - 1 then G has derived length at most 2 n .

Every 2 -group with all subgroups normal-by-finite is locally finite

Enrico Jabara — 2018

Czechoslovak Mathematical Journal

A group G has all of its subgroups normal-by-finite if H / H G is finite for all subgroups H of G . The Tarski-groups provide examples of p -groups ( p a “large” prime) of nonlocally finite groups in which every subgroup is normal-by-finite. The aim of this paper is to prove that a 2 -group with every subgroup normal-by-finite is locally finite. We also prove that if | H / H G | ≤ 2 for every subgroup H of G , then G contains an Abelian subgroup of index at most 8 .

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