In this article, we discuss whether a single congruent number $t$ can have two (or more) distinct triangles with the same hypotenuse. We also describe and carry out computational experimentation providing evidence that this does not occur.
We introduce a shifted convolution sum that is parametrized by the squarefree natural number $t$. The asymptotic growth of this series depends explicitly on whether or not $t$ is a emph{congruent number}, an integer that is the area of a rational right triangle. This series presents a new avenue of inquiry for The Congruent Number Problem.
We take an approach toward counting the number of n for which the curves E_n: y^2=x^3-n^2x have 2-Selmer groups of a given size. This question was also discussed in a pair of Invent. Math. papers by Roger Heath-Brown. We discuss the connection between computing the size of these Selmer groups and verifying cases of the Birch and Swinnerton-Dyer Conjecture. The key ingredient for the asymptotic formulae is the ``independence of the Legendre symbol evaluated at the prime divisors of an integer with exactly k prime factors.
In this paper, $p$ and $q$ are two different odd primes. First, We construct the congruent elliptic curves corresponding to $p$, $2p$, $pq$, and $2pq,$ then, in the cases of congruent numbers, we determine the rank of the corresponding congruent elliptic curves.
Let $mathbb{F}_q$ be a finite field of odd characteristic. We study Redei functions that induce permutations over $mathbb{P}^1(mathbb{F}_q)$ whose cycle decomposition contains only cycles of length $1$ and $j$, for an integer $jgeq 2$. When $j$ is $4$ or a prime number, we give necessary and sufficient conditions for a Redei permutation of this type to exist over $mathbb{P}^1(mathbb{F}_q)$, characterize Redei permutations consisting of $1$- and $j$-cycles, and determine their total number. We also present explicit formulas for Redei involutions based on the number of fixed points, and procedures to construct Redei permutations with a prescribed number of fixed points and $j$-cycles for $j in {3,4,5}$.
We study a curious class of partitions, the parts of which obey an exceedingly strict congruence condition we refer to as sequential congruence: the $m$th part is congruent to the $(m+1)$th part modulo $m$, with the smallest part congruent to zero modulo the length of the partition. It turns out these obscure-seeming objects are embedded in a natural way in partition theory. We show that sequentially congruent partitions with largest part $n$ are in bijection with the partitions of $n$. Moreover, we show sequentially congruent partitions induce a bijection between partitions of $n$ and partitions of length $n$ whose parts obey a strict frequency congruence condition -- the frequency (or multiplicity) of each part is divisible by that part -- and prove families of similar bijections, connecting with G. E. Andrewss theory of partition ideals.