Sums of consecutive squares which result in a square (Q815028)
From MaRDI portal
 | This is the item page for this Wikibase entity, intended for internal use and editing purposes. Please use this page instead for the normal view: Sums of consecutive squares which result in a square |
scientific article; zbMATH DE number 5004598
| Language | Label | Description | Also known as |
|---|---|---|---|
| English | Sums of consecutive squares which result in a square |
scientific article; zbMATH DE number 5004598 |
Statements
Sums of consecutive squares which result in a square (English)
0 references
8 February 2006
0 references
The authors consider the Diophantine equation \( \sum_{j=1}^{n-1}(x+j)^2= y^2. \) They observe that for \(n=3k^2-1\) and \(n=(6k\pm 1)^2\) the equation has at least one integral solution. Moreover if \(n=3k^2-1\) is not a perfect square, then the Diophantine equation has infinitely many solutions. The authors also find necessary (but not sufficient) conditions for the existence of integral solutions. In the case of rational solutions they can prove also sufficient conditions, by using Hilbert symbols.
0 references
quadratic Diophantine equations
0 references
sums of squares
0 references
