On a Problem of Erdos and Graham

Date

2020-06-01

Author

Yildiz, Burak
Gurel, Erhan

Metadata

Show full item record

This work is licensed under a Creative Commons Attribution-NonCommercial-NoDerivatives 4.0 International License.

Item Usage Stats

466
views

0
downloads

An old conjecture of Erdos and Graham states that only finitely many integer squares could be obtained from product of disjoint blocks of consecutive integers of length greater than or equal to four. It is known by counterexamples that the conjecture is false for product of disjoint blocks of four and five consecutive integers. In this paper, we present new algorithms generating new polynomial parametrizations that extend the polynomial parametrization given by Bennett and Luijk (Indag Math (N.S.) 23(1-2):123-127, 2012). Moreover, we produce the first examples of integer squares obtained from product of disjoint blocks of consecutive integers such that each block has length six or seven.

Subject Keywords

General Mathematics

URI

https://hdl.handle.net/11511/64849

Journal

BULLETIN OF THE BRAZILIAN MATHEMATICAL SOCIETY

DOI

https://doi.org/10.1007/s00574-019-00158-9

Collections

Natural Sciences and Mathematics, Article

Suggestions

OpenMETU
Core

On a Fitting length conjecture without the coprimeness condition Ercan, Gülin (Springer Science and Business Media LLC, 2012-08-01) Let A be a finite nilpotent group acting fixed point freely by automorphisms on the finite solvable group G. It is conjectured that the Fitting length of G is bounded by the number of primes dividing the order of A, counted with multiplicities. The main result of this paper shows that the conjecture is true in the case where A is cyclic of order p (n) q, for prime numbers p and q coprime to 6 and G has abelian Sylow 2-subgroups.
A note on a theorem of Dwyer and Wilkerson Öztürk, Semra (Springer Science and Business Media LLC, 2001-01-03) We prove a version of Theorem 2.3 in [1] for the non-elementary abelian group Z(2) x Z(2n), n greater than or equal to 2. Roughly, we describe the equivariant cohomology of (union of) fixed point sets as the unstable part of the equivariant cohomology of the space localized with respect to suitable elements of the cohomology ring of Z(2) x Z(2n).
PERTURBATIONS OF NONASSOCIATIVE BANACH ALGEBRAS Dosi, Anar (Rocky Mountain Mathematics Consortium, 2009-01-01) In this note we prove that if either 21 is a Banach-Jordan algebra or a Banach-Lie algebra then all perturbations of the multiplication in 21 give algebras topologically isomorphic with 21 whenever certain small-dimension cohomology groups associated with 21 are vanishing.
Noncomplex smooth 4-manifolds with Lefschetz fibrations Korkmaz, Mustafa (2001-01-01) For every integer g ≥ 2 there exist infinitely many pairwise nonhomeomorphic smooth 4-manifolds admitting genus-g Lefschetz fibration over S2 but not carrying any complex structure. This extends a recent result of Ozbagci and Stipsicz.
A classification of equivariant principal bundles over nonsingular toric varieties Biswas, Indranil; Dey, Arijit; Poddar, Mainak (World Scientific Pub Co Pte Lt, 2016-12-01) We classify holomorphic as well as algebraic torus equivariant principal G-bundles over a nonsingular toric variety X, where G is a complex linear algebraic group. It is shown that any such bundle over an affine, nonsingular toric variety admits a trivialization in equivariant sense. We also obtain some splitting results.

Citation Formats

B. Yildiz and E. Gurel, “On a Problem of Erdos and Graham,” BULLETIN OF THE BRAZILIAN MATHEMATICAL SOCIETY, pp. 397–415, 2020, Accessed: 00, 2020. [Online]. Available: https://hdl.handle.net/11511/64849.