117,137 research outputs found
Sums and differences of four k-th powers
We prove an upper bound for the number of representations of a positive
integer as the sum of four -th powers of integers of size at most ,
using a new version of the Determinant method developed by Heath-Brown, along
with recent results by Salberger on the density of integral points on affine
surfaces. More generally we consider representations by any integral diagonal
form. The upper bound has the form , whereas earlier
versions of the Determinant method would produce an exponent for of order
in this case. Furthermore, we prove that the number of
representations of a positive integer as a sum of four -th powers of
non-negative integers is at most for
, improving upon bounds by Wisdom.Comment: 18 pages. Mistake corrected in the statement of Theorem 1.2. To
appear in Monatsh. Mat
An Extreme Family of Generalized Frobenius Numbers
We study a generalization of the \emph{Frobenius problem}: given positive
relatively prime integers, what is the largest integer that cannot be
represented as a nonnegative integral linear combination of these parameters?
More generally, what is the largest integer that has exactly such
representations? We illustrate a family of parameters, based on a recent paper
by Tripathi, whose generalized Frobenius numbers
exhibit unnatural jumps; namely, $g_0, \ g_1, \ g_k, \ g_{\binom{k+1}{k-1}}, \
g_{\binom{k+2}{k-1}}, ...g_{\binom{k+j}{k-1}}\binom{k+j+1}{k-}$
representations. Along the way, we introduce a variation of a generalized
Frobenius number and prove some basic results about it.Comment: 5 pages, to appear in Integers: the Electronic Journal of
Combinatorial Number Theor
Bounds on generalized Frobenius numbers
Let and let be relatively prime integers.
The Frobenius number of this -tuple is defined to be the largest positive
integer that has no representation as where
are non-negative integers. More generally, the -Frobenius
number is defined to be the largest positive integer that has precisely
distinct representations like this. We use techniques from the Geometry of
Numbers to give upper and lower bounds on the -Frobenius number for any
nonnegative integer .Comment: We include an appendix with an erratum and addendum to the published
version of this paper: two inaccuracies in the statement of Theorem 2.2 are
corrected and additional bounds on s-Frobenius numbers are derive
Continuants and some decompositions into squares
In 1855 H. J. S. Smith proved Fermat's two-square using the notion of
palindromic continuants. In his paper, Smith constructed a proper
representation of a prime number as a sum of two squares, given a solution
of , and vice versa. In this paper, we extend the use of
continuants to proper representations by sums of two squares in rings of
polynomials on fields of characteristic different from 2. New deterministic
algorithms for finding the corresponding proper representations are presented.
Our approach will provide a new constructive proof of the four-square theorem
and new proofs for other representations of integers by quaternary quadratic
forms.Comment: 21 page
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