Pythagorean Triangles Research Articles

Observation on the Ternary Quadratic Diophantine Equation X<SUP>2</SUP>= 2<SUP>2k+1</SUP> (3z<SUP>2</SUP> - Y<SUP>2</SUP>)

Objectives: Finding the integral solutions of ternary quadratic equation and the relation between parameters/solutions. The ternary quadratic Diophantine equation is studied for it's non-trivial distinct integral solutions. Findings: Employing the integral solutions of the equation under consideration infinitely many Pythagorean triangles are obtained where each of which satisfies the relation Hypotenuse-4(area/Perimeter) is a Nasty number

Trisecting Angles in Pythagorean Triangles

Suppose that a2 + b2 = c2, where a, b, and c are relatively prime positive integers, and consider the right triangle T with sides a, b, and c. We prove that both of the acute angles in T can be trisected with a compass and unmarked straightedge if and only if c is a perfect cube.

A Connection between Hardy-Ramanujan Number and Special Pythagorean Triangles

Special Pythagorean Triangles are obtained in relation with the Hardy-Ramanujan Number 1729. Some special cases are also discussed. A few interesting results are obtained.

On the mean value of a kind of zeta functions

Let $d_{\alpha ,\beta }(n)=\sum _{ n=kl,\,\alpha l< k\leq \beta l}1$ be the number of ways of factoring $n$ into two almost equal integers. For fixed rational numbers $\alpha >0$ and $\beta >0$, we consider a zeta function of the type $\zeta _{\alpha ,\be

A rational sine and cosine of the angles of a triangle

The article deals with triangles whose sines and cosines of the angles are the rational numbers. We prove that these triangles are similar to Pythagorean or Heronian triangles, and vice versa.

Two extreme Diophantine problems concerning the perimeter of Pythagorean triangles

Two extreme Diophantine problems concerning the perimeter of Pythagorean triangles

The mathematics in the structures of Stonehenge

The development of ancient civilizations and their achievements in sciences such as mathematics and astronomy are well researched for script-using civilizations. On the basis of oral tradition and mnemonic artifacts illiterate ancient civilizations were able to attain an adequate level of knowledge. The Neolithic and Bronze Age earthworks and circles are such mnemonic artifacts. Explanatory models are given for the shape of the stone formations and the ditch of Stonehenge reflecting the circular and specific non-circular shapes of these structures. The basic mathematical concepts are Pythagorean triangles, thus adopting the construction procedures of the Neolithic circular ditches of Central Europe in the fifth Millennium bc and later earthworks and stone circles in Britain and Brittany. This knowledge was extended with new elliptical concepts. Approximations for the values of π and the square root of 2 are encoded in the henge. All constructions were performed using a standardized “Babylonian” metrology that shows a remarkable consistency and comprehensible development over some 14 centuries.

Assessing Outcomes in Facial Reanimation: Evaluation and Validation of the SMILE System for Measuring Lip Excursion During Smiling

Assessing Outcomes in Facial Reanimation: Evaluation and Validation of the SMILE System for Measuring Lip Excursion During Smiling

On the distribution of primitive Pythagorean triangles

On the distribution of primitive Pythagorean triangles

Affine linear sieve, expanders, and sum-product

Let $\mathcal{O}$ be an orbit in ℤ n of a finitely generated subgroup Λ of GL n (ℤ) whose Zariski closure Zcl(Λ) is suitably large (e.g. isomorphic to SL2). We develop a Brun combinatorial sieve for estimating the number of points on $\mathcal{O}$ at which a fixed integral polynomial is prime or has few prime factors, and discuss applications to classical problems, including Pythagorean triangles and integral Apollonian packings. A fundamental role is played by the expansion property of the “congruence graphs” that we associate with $\mathcal{O}$ . This expansion property is established when Zcl(Λ)=SL2, using crucially sum-product theorem in ℤ/qℤ for q square-free.

Nesting similar Pythagorean triangles

Nesting similar Pythagorean triangles

On the Number of Primitive Pythagorean Triangles with a given Inradius

On the Number of Primitive Pythagorean Triangles with a given Inradius

89.01 Perfect, and almost perfect, Pythagorean triangles

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88.48 Escribed and inscribed circles of Pythagorean triangles

88.48 Escribed and inscribed circles of Pythagorean triangles

Geometry of Venice

Having observed that ancient Venice belfries are located in such a way that they generate many Pythagorean triangles, having a great number of vertices in common, it has been decided to test the null hypothesis of random location by statistical and probabilistic methods. A simple index, called Pythagorean Ratio, is proposed, for checking which triangles are to be considered as Pythagorean. Then, a Monte Carlo simulation is performed, generating samples of "random belfries" in the historical kernel of Venice; a Poisson model seems to fit very well the number X of Pythagorean triangles. Combining this number with the number of connections, the null hypothesis is rejected. Adding a further belfry (S.Simeon Grande) to the original group of belfries, the significance becomes even higher.

Historical Mathematical Blunders: The Case of Barbaro's Cannonballs

The negative solution represents a new right triangle whose EGS triangle has integer sides; but the positive solution (1768, 2415) does not. However, the positive solutions are important. It is not difficult to show that, starting with a positive solution, Aubry's formulas always produce both a positive and a negative solution thus assuring an infinite number of negative primitive solutions and so an infinite number of incongruent Pythagorean triangles whose EGS triangle is of the same type.

Inscribed circles of Pythagorean triangles

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Pythagorean triangles with legs less than n

We obtain asymptotic estimates for the number of Pythagorean triples ( a, b, c) such that a<n, b<n . These estimates (considering the triple ( a, b, c) different from ( b, a, c)) is (4π −2 log(1+ 2 ))n+ O( n ) in the case of primitive triples, and (4π −2 log(1+ 2 ))n log n+ O(n) in the case of general triples. Furthermore, we derive, by a self-contained elementary argument, a version of the first formula which is weaker only by a log-factor. Also, we tabulate the number of primitive Pythagorean triples with both legs less than n, for selected values of n⩽1 000 000 000 , showing the excellent precision obtained.

On the number of primitive Pythagorean triangles

On the number of primitive Pythagorean triangles

Pythagorean triangles and arithmetical progression

Pythagorean triangles and arithmetical progression