On the distribution of rank two $\tau$-congruent numbers

It is known that a positive integer $n$ is the area of a right triangle with rational sides if and only if the elliptic curve $E^{(n)}: y^{2} = x(x^{2}-n^{2})$ has a rational point of order different than 2. A generalization of this result states that a positive integer $n$ is the area of a triangle with rational sides if and only if there is a nonzero rational number $\tau$ such that the elliptic curve $E^{(n)}_{\tau}: y^{2} = x(x-n\tau)(n+n\tau^{-1})$ has a rational point of order different than 2. Such $n$ are called $\tau$-congruent numbers. It is shown that for a given integer $m>1$, each congruence class modulo $m$ contains infinitely many distinct $\tau$-congruent numbers.

Number theory is the part of mathematics concerned with the mysterious and hidden properties of the integers and rational numbers (by a rational number, we mean the ratio of two integers). The congruent number problem, the written history of which can be traced back at least a millennium, is the oldest unsolved major problem in number theory, and perhaps in the whole of mathematics. We say that a right-angled triangle is “rational” if all its sides have rational length. A positive integer N is said to be “congruent” if it is the area of a rational right-angled triangle. If we multiply any congruent number N by the square of an integer, we again get a congruent number, and so it suffices to consider only those integers N that are square-free (meaning not divisible by the square of an integer >1). The congruent number problem is simply the question of deciding which square-free positive integers are, or are not, congruent numbers. Long ago, it was realized that an integer N ≥ 1 is congruent if and only if there exists a point (x, y) on the elliptic curve y2 = x3 − N2x, with rational coordinates x, y and with y ≠ 0. Until the 17th century, mathematicians made numerical tables of congruent numbers by using ingenuity to write down the corresponding rational right-angled triangles. For example, the integers 5, 6, and 7 were all known to be congruent, as they are the areas of the right-angled triangles, whose sides lengths are given respectively by [40/6, 9/6, 41/6], [3, 4, 5], and [288/60, 175/60, 337/60]. The first important theoretical result about congruent numbers was established by Fermat, who proved in the 17th century that 1 is not a congruent number. As explained in more …

A positive integer N is called a $$\theta $$ -congruent number if there is a $${\theta }$$ -triangle (a, b, c) with rational sides for which the angle between a and b is equal to $$\theta $$ and its area is $$N \sqrt{r^2-s^2}$$ , where $$\theta \in (0, \pi )$$ , $$\cos (\theta )=s/r$$ , and $$0 \le |s|<r$$ are coprime integers. It is attributed to Fujiwara (Number Theory, de Gruyter, pp 235–241, 1997) that N is a $${\theta }$$ -congruent number if and only if the elliptic curve $$E_N^{\theta }: y^2=x (x+(r+s)N)(x-(r-s)N)$$ has a point of order greater than 2 in its group of rational points. Moreover, a natural number $$N\ne 1,2,3,6$$ is a $${\theta }$$ -congruent number if and only if rank of $$E_N^{\theta }({{\mathbb {Q}}})$$ is greater than zero. In this paper, we answer positively to a question concerning with the existence of methods to create new rational $${\theta }$$ -triangle for a $${\theta }$$ -congruent number N from given ones by generalizing the Fermat’s algorithm, which produces new rational right triangles for congruent numbers from a given one, for any angle $${\theta }$$ satisfying the above conditions. We show that this generalization is analogous to the duplication formula in $$E_N^{\theta }({{\mathbb {Q}}})$$ . Then, based on the addition of two distinct points in $$E_N^{\theta }({{\mathbb {Q}}})$$ , we provide a way to find new rational $${\theta }$$ -triangles for the $${\theta }$$ -congruent number N using given two distinct ones. Finally, we give an alternative proof for the Fujiwara’s Theorem 2.2 and one side of Theorem 2.3. In particular, we provide a list of all torsion points in $$E_N^{\theta }({{\mathbb {Q}}})$$ with corresponding rational $${\theta }$$ -triangles.

A positive integer is called a congruent number if it is the area of a rightangled triangle, all of whose sides have rational length. The problem of determining which positive integers are congruent is buried in antiquity (see Chapter 9 of Dickson [6]), with it long being known that the numbers 5, 6, and 7 are congruent. Fermat proved that 1 is not a congruent number, and similar arguments show that also 2 and 3 are not congruent numbers. No algorithm has ever been proven for infallibly deciding whether a given integer n ≥ 1 is congruent. The reason for this is that it can easily be seen that an integer n ≥ 1 is congruent if and only if there exists a point (x, y), with x and y rational numbers and y = 0, on the elliptic curve ny2 = x3− x. Moreover, assuming n to be square free, a classical calculation of root numbers shows that the complex L-function of this curve has zero of odd order at the center of its critical strip precisely when n lies in one of the residue classes of 5, 6, or 7 modulo 8. Thus, in particular, the unproven conjecture of Birch and Swinnerton-Dyer predicts that every positive integer lying in the residue classes of 5, 6, and 7 modulo 8 should be a congruent number. The aim of this paper is to prove the following partial results in this direction.

We explore congruent numbers through a unified approach combining geometric constructions, elliptic curve theory, and arithmetic progressions of squares. Leveraging recent advances in modular forms and computational techniques, we construct infinite explicit families of congruent numbers parameterized by Pell-type equations and related Diophantine conditions. We complement this with a detailed statistical analysis of the residue classes, rank distributions, and root numbers associated with these families, providing empirical insights that deepen understanding of intricate conjectures in the arithmetic of elliptic curves.

On a problem of Diophantus for rationals

The congruent number problem. A congruent numberis a rational number qthat is the area of a right triangle, all of whose sides have rational length. We observe that if the triangle has sides a, b, and c, and if sis a rational number, then s2qis also a congruent number whose associated triangle has sides sa, sb,and sc.So it is enough to ask which square free integers nare congruent numbers. If we take cto be the length of the hypotenuse, then we are looking for square free integers nsuch that there are rational numbers a, b, csatisfying A simple algebraic calculation shows that the positive solutions to the simultaneous equations (25.1.1) are in one-to-one correspondence with the positive solutions to the equation Thus nis a congruent number if and only if (25.1.2) has a solution in positive rational numbers xand y.

The notion of $\theta $ -congruent numbers generalises the classical congruent number problem. A positive integer n is $\theta $ -congruent if it is the area of a rational triangle with an angle $\theta $ whose cosine is rational. Das and Saikia [‘On $\theta $ -congruent numbers over real number fields’, Bull. Aust. Math. Soc. 103 (2) (2021), 218–229] established criteria for numbers to be $\theta $ -congruent over certain real number fields and concluded their work by posing four open questions regarding the relationship between $\theta $ -congruent and properly $\theta $ -congruent numbers. In this work, we provide complete answers to those questions. Indeed, we remove a technical assumption from their result on fields with degrees coprime to six, provide a definitive answer for real cubic fields without congruence restrictions, extend the analysis to fields of degree six and examine the exceptional cases $n=1, 2, 3$ and $6$ .

$ \theta $ -congruent numbers are defined by extending congruent numbers. It has been known that a natural number n is $ \theta $ -congruent number if and only if the corresponding elliptic curve has positive rational rank. Using a criterion of Birch and modular parametrizations, we construct a non-trivial point on some elliptic curves by studying Heegner points on the modular curves $ X_0(24) $ and $ X_0(48) $ .

A Heron quadrilateral is a cyclic quadrilateral whose area and side lengths are rational. In this work, we establish a correspondence between Heron quadrilaterals and a family of elliptic curves of the form y 2 = x 3 + αx 2 - n 2 x. This correspondence generalizes the notions of Goins and Maddox who established a similar connection between Heron triangles and elliptic curves. We further study this family of elliptic curves, looking at their torsion groups and ranks. We also explore their connection with congruent numbers, which are the α = 0 case. Congruent numbers are positive integers which are the area of a right triangle with rational side lengths.

Let $l$ be the prime $3$, $5$ or $7$, and let $m_{1}$,~$m_{2}$, $n_{1}$ and $n_{2}$ be non-zero rational numbers. We construct an infinite family of pairs of distinct quadratic fields $\mathbb{Q}(\sqrt{m_{1}D+n_{1}})$ and $\mathbb{Q}(\sqrt{m_{2}D+n_{2}})$ with $D\in\mathbb{Q}$ such that both class numbers are divisible by $l$, using rational points on an elliptic curve with positive Mordell-Weil rank to parametrize such quadratic fields.

Let m O01U be a square-free positive integer. We say that a positive integer n is a congruent number over QO AAAA p U if it is the area of a right triangle with three sides in QO AAAA p U. We put Ka QO AAAA p U. We prove that if m0 2, then n is a congruent number over K if and only if EnOKU has a positive rank, where EnOKU denotes the group of K-rational points on the elliptic curve En defined by y 2 a x 3 ˇ n 2 x. Moreover, we classify right triangles with area n and three sides in K.

A positive integer $A$ is called a \emph{congruent number} if $A$ is the area of a right-angled triangle with three rational sides. Equivalently, $A$ is a \emph{congruent number} if and only if the congruent number curve $y^2 = x^3 − A^2 x$ has a rational point $(x, y) \in {\mathbb{Q}}^2$ with $y \ne 0$. Using a theorem of Fermat, we give an elementary proof for the fact that congruent number curves do not contain rational points of finite order.

A polynomial in one variable is an expression of the form a n x n + a n−1 x n−1 + ⋯ +a 0, where x is a variable*, where n is a nonnegative integer, and where a n , a n-1, ⋯ , a 0 are numbers. (Here, and in the remainder of the book, “number” will mean “rational number” unless otherwise stated.) The a’s are called the coefficients of the polynomial. The leading term of a polynomial is the term a i x i with a i ≠ 0 for which i is as large as possible. Normally one assumes a n ≠ 0, so a n x n is the leading term. The leading coefficient of a polynomial is the coefficient of the leading term; the degree is the exponent of x in the leading term. A polynomial of degree 0 is simply a nonzero rational number. The polynomial 0 has no leading term; it is considered to have degree −∞ in order to make the degree of the product of two polynomials equal to the sum of their degrees in all cases. For a similar reason, the leading coefficient of the polynomial 0 is considered to be 0. A polynomial is called monic if its leading coefficient is 1.

Families of non-congruent numbers with odd prime factors of the form 8k + 3