We discover suprising connections between three seemingly different problems: finding right triangles with rational sides in a non-Euclidean geometry, finding three integers such that the difference of the squares of any two is a square, and the problem of finding rational points on an algebraic surface in algebraic geometry. We will also reinterpret Euler's work on the second problem with a modern point of view.
The method of generating Pythagorean triples is known for about 2000 years. Though the classical formulas produce all primitive triples they do not generate all possible triples, especially non-primitive triples. This paper presents a direct method to generate all possible triples both primitive and non-primitive for any given number. Then the technique is generalized to produce Pythagorean Quadruples and n-tuples. Our method utilizes the fact that the difference between lengths of the hypotenuse and one leg of a Pythagorean triangle can have only certain distinct values depending on the lengt…
The paper presents a systematic construction of primitive Pythagorean triples. The order of enumeration on the set of primitive Pythagorean triples is defined. The order is based on the representation of a primitive Pythagorean triple by dividing the side of the generating square into two groups of factors using gnomons. In paper is shown the inverse mapping of the elements of a primitive Pythagorean triple to the parameters of the partition of the side of the generating square, the representation of primitive Pythagorean triples as the sum of two connected gnomons, a one-to-one correspondence…
We define operations that give the set of all Pythagorean triples a structure of commutative monoid. In particular, we define these operations by using injections between integer triples and $3 \times 3$ matrices. Firstly, we completely characterize these injections that yield commutative monoids of integer triples. Secondly, we determine commutative monoids of Pythagorean triples characterizing some Pythagorean triple preserving matrices. Moreover, this study offers unexpectedly an original connection with groups over conics. Using this connection, we determine groups composed by Pythagorean …
In this article we consider numeric palindromes as a component of a pythagorean triple. We first show that there are infinitely many non-primitive pythagorean triples that contains (i) a single numeric palindrome as a component, (ii) two numeric palindromes as a component and (iii) three numeric palindromes as a component. We then focus on numeric palindromes in primitive pythagorean triples. Open problem related to the topic was also given.
In 1987, Orrin Frink introduced the concept of almost Pythagorean triples. He defined them as an ordered triple $(x,y,z)$ that satisfies the equation $x^2+y^2=z^2+1$ where $x,y$ and $z$ are positive integers. In his paper, he showed that there were infinitely many almost Pythagorean triples by giving a characterization which suggests a method on generating all of them. However, this method does not explicitly and readily give a particular almost Pythagorean triple. In this note, using basic algebraic operations, we extend his result by giving a characterization that explicitly and readily give…
We study the family of elliptic curves $y^2=x(x-a^2)(x-b^2)$ parametrized by Pythagorean triples $(a,b,c)$. We prove that for a generic triple the lower bound of the rank of the Mordell-Weil group over $\mathbb{Q}$ is 1, and for some explicitly given infinite family the rank is 2. To each family we attach an elliptic surface fibered over the projective line. We show that the lower bounds for the rank are optimal, in the sense that for each generic fiber of such an elliptic surface its corresponding Mordell-Weil group over the function field $\mathbb{Q}(t)$ has rank 1 or 2, respectively. In ord…
We explore primitive Pythagorean triples of special forms $(a,b,b+g)$ and $(a,a+f,c)$, with $g,f\in\mathbb{Z}^+$. For each $g$ and $f$, we provide a method to generate infinitely many such primitive triples. Lastly, for each $g$, we describe the asymptotic density of primitive $(a,b,b+g)$ triples within all primitive triples of the same parity.
We show how the multiplicative structure on Pythagorean triples defined by Beauregard--Suryanarayan arises from automorphisms of the hyperbolic plane, and how the structure theorem of this product follow naturally from this connection. We also show how the natural involution on Pythagorean triples is related to Pell's equation with the parameter 2, and prove a generalization of this phenomenon.
This paper is a preliminary expository paper that outlines the relationship between solutions to the Erdős-Straus conjecture for a given prime $p$ and their corresponding Pythagorean triples. This paper also uses Bézout Coefficients to find a relationship between the solutions to the conjecture and roots of a second degree polynomial.
It is well known that Pythagorean triples can be parametrized by two triples of polynomials with integer coefficients. We show that no single triple of polynomials with integer coefficients in any number of variables is sufficient, but that there exists a parametrization of Pythagorean triples by a single triple of integer-valued polynomials.
Given a right triangle and two inscribed squares, we show that the reciprocals of the hypotenuse and the sides of the squares satisfy an interesting Pythagorean equality. This gives new ways to obtain rational(integer)right triangles from a given one.
We study $N$-point rational distance sets ($\textrm{RDS}(N)$) on the parabola $y=x^2$. Previous approaches to the problem include efforts made using elliptic curves and diophantine chains, with successful analysis for $N\leq 4$. We extend the analysis for arbitrary $N$ by establishing a correspondence between $\textrm{RDS}(N)$s and Pythagorean triplets. Our main result gives sufficient and necessary conditions for the existence and nature of the $\textrm{RDS}(N)$s for arbitrary $N$. Our approach also leads to an efficient computational algorithm to construct new $\textrm{RDS}(N)$s, and we prov…
One unsolved mathematical problem remains the perfect cuboid problem. A perfect cuboid is a rectangular parallelepiped whose edges, face diagonals and space diagonal are all expressed as integers. No such cuboid has yet been discovered and its existence has also not been proven. This paper shows a proof of the non-existence of a perfect cuboid.
The general formulas for finding the quantity of all primitive and nonprimitive triples generated by the given number x have been proposed. Also the formulas for finding the complete quantity of the representations of the integers as a difference of two squares have been obtained.
Is it possible to color the naturals with finitely many colors so that no Pythagorean triple is monochromatic? This question is even open for two colors. A natural strategy is to show that some small nonbipartite triple systems cannot be realized as a family of Pythagorean triples. It suffices to consider partial triple systems (PTS's), and it is therefore natural to consider the Fano plane, the smallest nonbipartite PTS. We show that the Pythagorean triples do not contain any Fano plane. In fact, our main result is that a much larger family of "ordered" triple systems (viz. those with a certa…
The aim of this paper is to introduce a new family of elliptic curves in the form of $y^2=x(x-a^2)(x-b^2)$ that have positive ranks. We first generate a list of pythagorean triples $(a,b,c)$ and then construct this family of elliptic curves. It turn out that this new family have positive ranks and search for the upper bound for their ranks.
Let $p>5$ be a fixed prime. We obtain an asymptotic formula related to small solutions of quadratic congruences of the form $x_1^2+x_2^2\equiv x_3^2\bmod{p^n}$ where $\max\{|x_1|,|x_2|,|x_3|\}\le p^{νn}$ with $ν>1/2$.
For the ternary quadratic form Q(x) = x^2 + y^2 - z^2 and a non-zero Pythagorean triple x_0 in Z^3 lying on the cone Q(x) = 0, we consider an orbit O = x_0 Gamma of a finitely generated subgroup Gamma < SO_Q(Z) with critical exponent exceeding 1/2. We find infinitely many Pythagorean triples in O whose hypotenuse, area, and product of side lengths have few prime factors, where "few" is explicitly quantified. We also compute the asymptotic of the number of such Pythagorean triples of norm at most T, up to bounded constants.
It is well-known that pythagorean triples can be represented by points of the unit circle with rational coordinates. These points form an abelian group, and we describe its structure. This structural description yields, almost immediately, an enumeration of the normalized pythagorean triples with a given hypotenuse, and also to an effective method for producing all such triples. This effective method seems to be new. This paper is intended for the general mathematical audience, including undergraduate mathematics students, and therefore it contains plenty of background material, some history a…
Given relatively prime integers $a_1, \dotsc, a_n$, the Frobenius number $g(a_1, \dotsc, a_n)$ is defined as the largest integer which cannot be expressed as $x_1 a_1 + \dotsb + x_n a_n$ with $x_i$ nonnegative integers. In this article, we give the Frobenius number of primitive Pythagorean triples. That is, \[ g(m^2-n^2, 2mn, m^2+n^2) = (m-1)(m^2-n^2) + (m-1)(2mn) - (m^2 + n^2). \]
Let $m$ be a fixed square-free positive integer, then equivalence classes of solutions of Diophantine equation $x^2+m\cdot y^2=z^2$ form an infinitely generated abelian group under the operation induced by the complex multiplication. A basis of this group is constructed here using prime ideals and the ideal class group of the field $\mathbb Q (\sqrt{-m})$.
A Pythagorean triple is a triple of positive integers a, b, c $\in$ N${}^{+}$ satisfying a${}^2$ + b${}^2$ = c${}^2$. Is it true that, for any finite coloring of N${}^{+}$ , at least one Pythagorean triple must be monochromatic? In other words, is the Dio-phantine equation X${}^2$+ Y${}^2$ = Z${}^2$ regular? This problem, recently solved for 2-colorings by massive SAT computations [Heule et al., 2016], remains widely open for k-colorings with k $\ge$ 3. In this paper, we introduce morphic colorings of N + , which are special colorings in finite groups with partly multiplicative properties. We …
We study spherical tetrahedra with rational dihedral angles and rational volumes. Such tetrahedra occur in the Rational Simplex Conjecture by Cheeger and Simons, and we supply vast families, discovered by computational efforts, of positive examples that confirm this conjecture. As a by-product, we also obtain a classification of all spherical Pythagorean triples, previously found by Smith.
We address the question of the "partition regularity" of the Pythagorean equation a^2+b^2=c^2; in particular, can the natural numbers be assigned a 2-coloring, so that no Pythagorean triple (i.e., a solution to the equation) is monochromatic? We prove that the hypergraph of Pythagorean triples can contain no Steiner triple systems, a natural obstruction to 2-colorability. Then, after transforming the question into one about 3-CNF satisfiability and applying some reductions, a SAT solver is used to find a 2-coloring for {1,...,7664}. Work continues as we seek to improve the reductions and exten…
Two well-studied Diophantine equations are those of Pythagorean triples and elliptic curves; for the first, we have a parametrization through rational points on the unit circle, and for the second we have a structure theorem for the group of rational solutions. Recently Yekutieli discussed a connection between these two problems and described the group structure of Pythagorean triples and the number of triples for a given hypotenuse. We generalize these methods and results to Pell's equation. We find a similar group structure and count on the number of solutions for a given $z$ to $x^2 + Dy^2 …
Let $m$, $n$ be positive integers such that $m>n$, $\gcd(m,n)=1$ and $m \not\equiv n \bmod 2$. In 1956, L. Jeśmanowicz \cite{Jes} conjectured that the equation $(m^2 - n^2)^x + (2mn)^y = (m^2+n^2)^z$ has only the positive integer solution $(x,y,z) = (2,2,2)$. This problem is not yet solved. In this paper, combining a lower bound for linear forms in two logarithms due to M. Laurent \cite{Lau} with some elementary methods, we prove that if $mn \equiv 2 \bmod 4$ and $m > 30.8 n$, then Jeśmanowicz' conjecture is true.
We consider the triples of integer numbers that are solutions of the equation $x^2+qy^2=z^2$, where $q$ is a fixed, square-free arbitrary positive integer. The set of equivalence classes of these triples forms an abelian group under the operation coming from complex multiplication. We investigate the algebraic structure of this group and give a complete analysis when $q\in\{2,3,5,6\}$.
On the Burgess bound and Pythagorean triples involving primitive roots
School of Science, Nanjing University of Posts and Telecommunications（南京邮电大学理学院）；Department of Applied Mathematics, Nanjing Audit University（南京审计大学应用数学系）
We show that the space of Euclid's parameters for Pythagorean triples is endowed with a natural symplectic structure and that it emerges as a spinor space of the Clifford algebra $\mathbb{R}_{2,1}$, whose minimal version may be conceptualized as a 4-dimensional real algebra of "kwaternions." We observe that this makes Euclid's parameterization the earliest appearance of the concept of spinors. We present an analogue of the "magic correspondence" for the spinor representation of Minkowski space and show how the Hall matrices fit into the scheme. The latter obtain an interesting and perhaps unex…
We construct a piecewise onto 3-to-1 dynamical system on the positive quadrant of the unit circle, such that for rational points (which correspond to normalized Primitive Pythagorean Triples), the associated ternary expansion is finite, and is equal to the address of the PPT on Barning's ternary tree of PPTs, while irrational points have infinite expansions. The dynamical system is conjugate to a modified Euclidean algorithm. The invariant measure is identified, and the system is shown to be conservative and ergodic. We also show, based on a result of Aaronson and Denker, that the dynamical sy…
This article highlights interactions of diverse areas: the Heron formula for the area of a triangle, the Descartes circle equation, and right triangles with integer or rational sides. New and old results are synthesized. We show that every primitive Pythagorean triple (PPT), furnishes a Descartes quadruple of tangent circles with integral curvatures, and so generates an integral Apollonian packing (IAP) containing a rectangle of centers. Thus Pythagorean triples serve to generate a large number of in-equivalent integral packings.
Let $(a, b, c)$ be a primitive Pythagorean triple parameterized as $a=u^2-v^2,\ b=2uv,\ c=u^2+v^2$,\ where $u>v>0$ are co-prime and not of the same parity. In 1956, L. Je{ś}manowicz conjectured that for any positive integer $n$, the Diophantine equation $(an)^x+(bn)^y=(cn)^z$ has only the positive integer solution $(x,y,z)=(2,2,2)$. In this connection we call a positive integer solution $(x,y,z)\ne (2,2,2)$ with $n>1$ exceptional. In 1999 M.-H. Le gave necessary conditions for the existence of exceptional solutions which were refined recently by H. Yang and R.-Q. Fu. In this paper we give a un…
The paper found a geometric and algebraic interpretation of the parameters m and n from the formulas for obtaining primitive Pythagorean triples, which are solutions of the equation ${x^2+y^2=z^2}$, namely: ${x=m^2-n^2}$, ${y=2mn}$, ${z=m^2+n^2}$. The study was based on the process of building figurate numbers using gnomons. The paper discusses the process of building squares. The addition of the gnomon U to the original square leads to a larger square: ${x^2+U=z^2}$. The first stage of the investigation was the construction of the gnomon U, which is equal to the area of a square ${y^2}$. The …
Let $F=x^2+y^2-z^2$, $x_0 \in \mathbb{Z}^3$ primitive with $F(x_0)=0$, and $Γ\leq SO_F(\mathbb{Z})$ be a finitely generated thin subgroup. We consider the resulting thin orbits of Pythagorean triples $x_0 \cdot Γ$ - specifically which hypotenuses, areas, and products of all three coordinates arise. We produce infinitely many $R$-almost primes in these three cases whenever $Γ$ has exponent $δ_Γ>δ_0(R)$ for explicit $R$, $δ_0$.
Problem 2 at the 56th International Mathematical Olympiad (2015) asks for all triples (a,b,c) of positive integers for which ab-c, bc-a, and ca-b are all powers of 2. We show that this problem requires only a primitive form of arithmetic, going back to the Pythagoreans, which is the arithmetic of the even and the odd.
It has long been known that the set of primitive pythagorean triples can be enumerated by descending certain ternary trees. We unify these treatments by considering hyperbolic billiard tables in the Poincare disk model. Our tables have m>=3 ideal vertices, and are subject to the restriction that reflections in the table walls are induced by matrices in the triangle group PSU^\pm_{1,1}\Zbb[i]. The resulting billiard map \tilde B acts on the de Sitter space x_1^2+x_2^2-x_3^2=1, and has a natural factor B on the unit circle, the pythagorean triples appearing as the B-preimages of fixed points. We…
Two well-studied Diophantine equations are those of Pythagorean triples and elliptic curves, for the first we have a parametrization through rational points on the unit circle, and for the second we have a structure theorem for the group of rational solutions. Recently, Yekutieli discussed a connection between these two problems, and described the group structure of Pythagorean triples and the number of triples for a given hypotenuse. In arXiv:2112.03663 we generalized these methods and results to Pell's equation. We find a similar group structure and count on the number of solutions for a giv…
We prove that there exist infinitely many rationals a, b and c with the property that a^2-1, b^2-1, c^2-1, ab-1, ac-1 and bc-1 are all perfect squares. This provides a solution to a variant of the problem studied by Diophantus and Euler.
Let (a,b,c) be a primitive Pythagorean triple, i.e., a^{2}+b^{2}=c^{2} with gcd(a,b,c)=1, a even and b odd. Terai's conjecture says that the Diophantine equation x^{2}+b^{y}=c^{z} has only the positive integer solutions (x,y,z)=(a,2,2). In this study, we prove that Terai's conjecture is true when b is a product of two primes and c=5(mod8).
In this paper, we study the alternating Euler $T$-sums and $§$-sums, which are infinite series involving (alternating) odd harmonic numbers, and have similar forms and close relations to the Dirichlet beta functions. By using the method of residue computations, we establish the explicit formulas for the (alternating) linear and quadratic Euler $T$-sums and $§$-sums, from which, the parity theorems of Hoffman's double and triple $t$-values and Kaneko-Tsumura's double and triple $T$-values are further obtained. As supplements, we also show that the linear $T$-sums and $§$-sums are expressible in…
Andrews and Merca investigated a truncated version of Euler's pentagonal number theorem and showed that the coefficients of the truncated series are nonnegative. They also considered the truncated series arising from Jacobi's triple product identity, and they that its coefficients are nonnegative. This conjecture was posed by Guo and Zeng independently and confirmed by Mao and Yee using different approaches. In this paper, we provide a new combinatorial proof of their nonnegativity result related to Euler's pentagonal number theorem. Meanwhile, we find an analogous result for a truncated serie…
We say that $(a_1,...,a_k)$ is pairwise non-coprime if $\gcd(a_i,a_j) \ne 1$ for all $1 \le i <j \le k$. Let $a_1,a_2,a_3$ be positive integers less than $H$. We obtain an asymptotic formula for the number of $(a_1,a_2,a_3)$ that are pairwise non-coprime. The probability that a randomly chosen unbounded positive integer triple is pairwise non-coprime is approximately 17.4%. Let $φ(n)$ be the Euler totient function. We also give an upper bound on the error term in an asymptotic formula for $\sum_{n=1}^H (φ(n)/n)^m$ for $m \ge 2$ and as $H \rightarrow \infty$.
