1,720,969 research outputs found
Random Diophantine inequalities of additive type
No full textAbstractUsing the Davenport–Heilbronn circle method, we show that for almost all additive Diophantine inequalities of degree k in more than 2k variables the expected asymptotic formula for the density of solutions holds true
The density of twins of k-free numbers
No full textFor k >= 2, we consider the number A(k)(Z) of positive integers n <= Z such that both n and n + 1 are k-free. We prove an asymptotic formula A(k)(Z) = c(k)Z + O(Z(14/9k+is an element of)), where the error term improves upon previously known estimates. The main tool used is the approximative determinant method of Heath-Brown
Random Thue and Fermat equations
Get PDFWe consider Thue equations of the form axk+byk=1, and assuming the truth of the abc-conjecture, we show that almost all locally soluble Thue equations of degree at least three violate the Hasse principle. A similar conclusion holds true for Fermat equations axk+byk+czk=0 of degree at least six
On the gaps between values of binary quadratic forms
Get PDFAbstractAmong the values of a binary quadratic form, there are many twins of fixed distance. This is shown in quantitative form. For quadratic forms of discriminant −4 or 8 a corresponding result is obtained for triplets.</jats:p
Rational lines on cubic hypersurfaces
Get PDFWe show that any smooth projective cubic hypersurface of dimension at least 29 over the rationals contains a rational line. A variation of our methods provides a similar result over p-adic fields. In both cases, we improve on previous results due to the second author and Wooley.We include an appendix in which we highlight some slight modifications to a recent result of Papanikolopoulos and Siksek. It follows that the set of rational points on smooth projective cubic hypersurfaces of dimension at least 29 is generated via secant and tangent constructions from just a single point
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