This paper seeks to showcase how a new approach can breathe new life into research within the traditional domain of Pythagorean triples, introducing innovative applications to invigorate the field. ...This serves not only as an exemplar but also as a wellspring of inspiration for students at both school and university levels. The demonstrations will underscore that, with fundamental mathematical concepts and unencumbered by intricate calculations, one can unveil novel results and applications with ease. The new results and applications, along with those found in the Preliminary Results section, show how the field of Pythagorean triples is still interesting and stimulating to study, despite the centuries that have elapsed.
Quadratic forms and their Berggren trees Cha, Byungchul; Nguyen, Emily; Tauber, Brandon
Journal of number theory,
April 2018, 2018-04-00, Letnik:
185
Journal Article
Recenzirano
Odprti dostop
An old result of Berggren's says that there exist three 3×3 matrices N1,N2,N3 with the following remarkable property: Start with (3,4,5) or (4,3,5) and multiply N1,N2, or N3 by it in any order any ...number of times. This yields another primitive Pythagorean triple (x,y,z), that is, a triple of positive integers without common factor satisfying x2+y2−z2=0. Furthermore, every primitive Pythagorean triple can be obtained uniquely this way. In other words, all primitive Pythagorean triples can be given a tree-like structure with each edge representing a multiplication by Nj. In this paper, we present a geometric algorithm for producing such trees that is applicable to any integral quadratic form. Although this algorithm does not always yield a tree, we find a few other trees arising from different quadratic forms.
This paper proposes a fractal modification of tropical algebra for noise removal and optimal control. Fractal addition, fractal multiplication, and fractal dot product are defined and explained in a ...fractal space. A fractal tropical polynomials function can model a coastline at any scale, providing a new tool for fractal analysis of any irregular curve. An example is given to solve an optimal control subject to a fractal tropical polynomials function. This paper provides a new window for tropical algebra, combining fractal geometry and algebra.
In 1956, Jeśmanowicz conjectured that the exponential Diophantine equation (m2−n2)x+(2mn)y=(m2+n2)z has only the positive integer solution (x,y,z)=(2,2,2), where m and n are positive integers with ...m>n, gcd(m,n)=1 and m≢n(mod2). We show that if n=2, then Jeśmanowicz' conjecture is true. This is the first result that if n=2, then the conjecture is true without any assumption on m.
The data in this article was obtained from the algebraic and statistical analysis of the first 331 primitive Pythagorean triples. The ordered sample is a subset of the larger Pythagorean triples. A ...primitive Pythagorean triple consists of three integers a, b and c such that; a2+b2=c2. A primitive Pythagorean triple is one which the greatest common divisor (gcd), that is; gcd(a,b,c)=1 or a, b and c are coprime, and pairwise coprime. The dataset describe the various algebraic and statistical manipulations of the integers a, b and c that constitute the primitive Pythagorean triples. The correlation between the integers at each analysis was included. The data analysis of the non-normal nature of the integers was also included in this article. The data is open to criticism, adaptation and detailed extended analysis.
In 1956 L. Jeśmanowicz conjectured, for any primitive Pythagorean triple (a,b,c) satisfying a2+b2=c2, that the equation ax+by=cz has the unique solution (x,y,z)=(2,2,2) in positive integers x, y and ...z. This is a famous unsolved problem on Pythagorean numbers. In this paper we broadly extend many of classical well-known results on the conjecture. As a corollary we can verify that the conjecture is true if a−b=±1.
In this note we study the existence of integer solutions of the Diophantine equationz2=f(x)2±g(y)2 for certain polynomials f,g∈Zx of degree ≥3. In particular, for given k∈N we prove that for all ...a,b∈Z satisfying the condition a2+b2≠0, the above Diophantine equation, with f(x)=xk(x+a), g(x)=xk(x+b) and any choice of the sign, has infinitely many solutions in integers x,y,z such that f(x)g(y)≠0. Moreover, we prove that for f(x)=x3 and g(x)=x(x+1)(x+2) the system of Diophantine equationsz12=f(x)2+g(y)2,z22=f(x)2+g(y+1)2 has infinitely many solutions in positive integers x,y,z1,z2 with gcd(x,y)=1. Similar result is proved for the systemz12=f(x)2+g(y)2,z22=f(x+1)2+g(y)2. We also present some experimental results concerning the construction of polynomials with rational coefficients for which the Diophantine equation z2=f(x)2±g(y)2 has infinitely many solutions in integers.
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