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Konstantine Zelator

Publications and source records attributed to Konstantine Zelator.

34 records · Page 2Linked to original sources
arXiv2009

An ancient Egyptian problem:the diophantine equation 4/n=1/x+1/y+1/z, n>or=2

From the Rhind Papyrus and other extant sources, we know that the ancient Egyptians were very iterested in expressing a given fraction into a sum of unit fractions, that is fractions whose numerators are equal to 1. One of the problems that has come down to us in the last 60 years, is known as the Erdos- Strauss conjecture which states that for each positive integer n>1; the fraction 4/n can be decomposed into a sum of three distinct unit fractions. Since 1950, a numberof partial results have been achieved, see references [1]- [8]; and also [10] and[11]. In this work we contribute four theorems. In Theorem 2, we prove that if the fraction 4/n is not equal to a sum of three distinct unit fractions, then each prime divisor of n; must be congruent to 1 modulo24. Moreover, if n contains a divisor noncongruent to 1mod24; then such a decomposition does exist and it is explicitly given.

math.GM↗
arXiv2009

A cornucopia of pythagorean triangles

Consider two circles, externally tangential,and with integer radii R1, R2; and with R1>R2.The two circles have three tangent lines in common, one of them being T1T2. If M is the midpoint of T1T2, and K the point of intersection of the lines C1C2 and T1T2;then 16 right triangles are formed(C1 and C2 are the two circle centers), see Figure 1.In Section 6 of this paper, we find the precice form the two integers R1 and R2 must have, in order that the sixteen aforementioned right triangles be Pythagorean.

math.HO↗
arXiv2009

Pythagorean Triangles with Repeated Digits-Repeated Bases

In 1998, in the winter issue of the journal Mathematics and Computer education (see [1]), Monte Zerger posed the following problem. He had noticed the Pythagorean triple (216,630,666);(216)^2+(630)^2=(666)^2. Note that 216=6^3 and 666 is the hypotenuse length. The question was then, whether there existed a digit d and a positive integer k(other than the above); such that d^k is the leglength of a Pythagorean triangle whose hypotenuse length has exactly k digits, each being equal to d. In 1999, F.Luca and P.Bruckman, answered the above question in the negative. In 2001, K.Zelator(see [2]), took this question further and showed that no Pythagorean triangle exists such that one leg has length d^k, while the other leglegth has exactly k digits in its decimal expansion, with each digit bein equal to d. In this work, we explore the above phenomenon from the point of view of number bases other than 10. We prove five Theorems. As an example, in Theorem 1, we prove that there exists no Pythagorean triangle with one of its leglengths being d^k, while the other leglength having exactly k digits in its base 4 expansion; and with digit being equal to d.

math.GM↗
arXiv2009

Integral points on hyperbolas: A special case

The subject matter of this work is integral points on conics described by the general equation, ax^2+bxy+cy^2+dx+ey+f=0 (1) where the six coefficients are integers satisfying the conditions, b^2-4ac=k^2, with a and c being nonzero and k a positive integer. It is well known the when b^2-4ac>0, equation (1) describes either a hyperbola on the plane or a pair of two straight lines(the degenerate case). The key integer is the number, I=k^2(d^2-4af)-(2ae-bd)^2. In Section 2, we show via a straightforward algebraic method that equation (1) can be put in the form, g(x,y)h(x,y)=I, where g(x,y) and h(x,y) are linear polynomials in x and y with integer coefficients. Thus, when I is not zero, equation (1) has only finitely many integer solutions (x,y). The process of finding these solutions is outlined in Section 3. In Section 4,we give a detailed numerical example. In Section 5, we offer some observations and remarks. In Section 6, we discuss the case b=d=e=0. In Section 7 we discuss the case a=1=k, and we show that when I=2^n, n>or=2, the above hyperbola has exactly 2(n-1) integral points, all explicitly given. In Section 10, we prove that these points are distinct. In Section 8 we discuss the case I=0, the hyperbola in (1) has either infinitely many integral points: or no integral points. Finally in Section 9, we discuss the special case d=e=f=0.

math.GM↗
arXiv2009

Distributive properties of the rationals

If o and * are two binary operations in a number system, then three elements a,b,c in that number system are said to satisfy the distributive property of the operation o over the operation * if, ao(b*c)= (aob)*(aoc) Now, suppose that the number system is the rationals,and the operations o and * are among the four usual operations of addition, multiplication, subtraction, and division. If we allow for o and * to be the same operation, then there are precisely 16 combinations with the operation o being one of the four usual operations in Q; and likewise for the operation *. Two of these 16 combinations are when o is the multiplication operationand * being the addition operation; and when is o is multiplication and * is subtraction. For these two combinations, the above stated distributive property is universally satisfied; that is, for ane three rational numbers a,b,and c. In this work, we examine the other fourteen combinations, to find out when the distributive property is satisfied. Of these 14 combinations or cases, eleven are easy/straightforward, in that almost always, one of the three rational numbers a, b, c; must be zero or 1. The remaining three cases or combinations are much more complicated, and number theory is involved.

math.GM↗
arXiv2009

A family of diophantine equations of the form x^4 +2nx^2y^2+my^4=z^2 with no solutions in (Z+)^3

In this work, we prove the following result(Theorem 1): Suppose that n is a positive integer, p an odd prime, and such that either n is congruent to 0 modulo4 and p congruent to 3 modulo8; or alternatively, n is congruent to 2 modulo4 and p congruent to 7 modulo 8. In addition to the above, assume that one of the following holds: Either (i) n^2-p>0 and the positive integer is a prime, Or (ii) n^2-p<0 and the positive integer N=-m=-(n^2-p) is a prime. Then the diophantine equation x^4 +2nx^2y^2+my^4=z^2 has no positive integer solutions. The method of proof is elementary in that it only uses congruence arguments, the method of (infinite) descent as originally applied by P.Fermat, and the general solution in positive inteagers to the 3-variable diophantine equation x^2+ly^2=z^2, l a positive integer. We offer 53 numerical examples in the form of two tableson pages 9 and 10; and historical commentary on pages 10-12.

math.NT↗
arXiv2009

Exploring Progressions: A Collection of Problems

The material of this work is aimed at mathematics educators, as well as math specialists with a keen interest in progressions. In this paper, we study the subject of arithmetic, geometric, mixed, and harmonic progressions or sequences. Some of the material found in Sections 2,3,4,and 5, can also be found in standard precalculus texts. For example, refer to the books in references 1 and 2. A substantial portion of the material in those sections, cannot be found in such books. In Section 6, we present 21 problems with detailed solutions. These are interesting, unusual problems not commonly found in mathematics texts, and most of them are quite challenging. In Section 7, we offer a collection of unsolved problems.

math.GM↗
arXiv2009

A Non-Existence Property of Pythagorean Triangles with a 3-D Application

After the introduction, in section 2 we state the well known parametric formulas that describe the entire family of Pythagorean triples. In section 3, we list four well known results from number theory, used later in the paper. in section 2, we prove two propositions. Proposition 1 says that the diophantine equation z^2=x^4+4y^4, has no solutions in positive integers x,y, and z. We then use Proposition 1, to prove the insolvability of a certain four-variable diophantine system, which is Proposition 2. In Section 5, we present some examplesof pairs of Pythagorean triangles with a common hypotenuse. In Section 6, we use Proposition 2 to prove Theorem 1: There exists no pair of Pythagorean triangles such that the leg of largest length in the first triangle is the hypotenuse of the second triangle; and in addition, with the leg of smallest length in the first triangle; having the same length as a leg in the second triangle. in Section 7, we present information on Pythagorean boxes, and we describe how to generate infinitely many Pythagorean boxes with a face diagonal of integral length. Finally, in Section 8, we apply Proposition 2 to prove Theorem 2: There exists no Pythagorean box with a pair of congruent (or opposite) faces being squares. And with the four diagonals of equal length in another pair of opposite faces, also having integral length.

math.GM↗
arXiv2008

The Seventeen Elements of Pythagorean Triangles

This is an exhaustive study of the seventeen elements of Pythagorean triangles, from the point of view of when such an element is an irrational number, a rational number, or an integer. For each of these 17 elements,precice conditions for their integrality,rationality, or irrationality; are given. These 17 elements are:The radius of the incircle, the radius of the circumscribed circle,the three radii of the three exterior circles tangential to the three lines containing the triangles'sides, the lengths of the three heights, the lenghts of the three internal angle bisectors, the lengthe of the three external angle bisectors, and the lengths of the three medians.

math.GM↗
arXiv2008

A simple method for generating rational triangles

In the early part of the paper, various geometrical formulas are derived. Then, at some point in the paper, the concept of a Pythagorean rational is introduced. A Pythagorean rational is a rational number which is the ratio of two integers which are the leglengths of a Pythagorean triangle. Using the idea of Pythagorean rationals, we generate two families of rational triangles. We define a rational triangle to be a triangle with rational sidelengths and area.

math.GM↗
arXiv2008

Lattice points on the plane ax+by+cz=d and the diophantine system ax+by+cz=d ex+fy+gz=h

The subject matter of this work are the linear, three variable diophantine equation ax+by+cz=d (1), and the diophantine system ax+by+cz=d (2) ex+fy+gz=h with the coefficients a,b,c,d,e,f,g,h being integers. Introductory number theory books, typically contain only a brief outline of how to solve equation (1). Even less or no material is offered on the system (2). The purpose of this work is to fill this gap. After some preliminary, introductory material, which includes the general solution of the two variable linear diophantine equation ax+by=c(material which we use later in the paper); we present a complete and detailed analysis of equation (1). We determine the precise conditions that the coefficients a,b,c,d must satisfy in order for integer solutions to exist. We then derive a two-parameter, parametric description of The solution set. The solution set of (1), if not empty, consists of all integer triples (x,y,z) that satisfy (1). Geometrically, this is interpreted as the set of all lattice points in 3-D space which lie on the plane with equation (1). Similarly, we offer an exhaustive analysis of the system (2) by determining the precise conditions that the coeffients must satisfy in order for integer solutions to exist. We offer seven examples with detailed solutions.

math.GM↗
arXiv2008

Heron isosceles with integral external radii

Each triangle has three exterior or external circles tangential to the three straight lines containing the three sides of the triangle.Among the preliminaries in this paper, is deriving formulas for the radii of the three exterior circles in terms of the triangle's sidelengths. After that we focus on Heron triangles. Heron triangles are known in the literature as triangles with integer sidelengths and integral area.Pythagorean triangles are examples of Heron triangles. In the first part of the paper, we parametrically describe all Heron isosceles triangles. In the second part, we parametrically(in terms of three independent parameters) describe the subfamily which consists of all Heron isosceles triangles which also have integral external radii. We provide numerical examples and tables.

math.GM↗
arXiv2008

On an area property of the sum cotA+cotB+cotC in a triangle

Given a triangle ABC, a new triangle A'B'C' can be formed as follows: Draw the perpendicular to the line AB at the pointA; then the perpendicular to the line BC at B, and lastly the perpendicular to the line CA at C.the two triangles ABC and A'B'C' are always similar. In Postulate1 we prove that the ratio E'/E is equal to (cotA+cotB+cotC)^2, which is the main result in this work.Here E' and E stand for the areas of the triangles A'B'C' and ABC respectively. In Postulate 2, we show that the above ratio has minimum value 3, which is attained when ABC and A'B'C' are equilateral triangles.In Postulate 3, we show that if we consider only those pairs of triangles (ABC,A'B'C'), with both ABC and A'B'C' being right triangles, then the minimum value of the above ratio of areas, is 4.

math.GM↗
arXiv2008

The equation asinx+bcosx=c and a family of cyclic Heron quadrilaterals

In the beginning of this paper, we present the general solution to the trigonometric equation asinx+bcosx=c. After that, we focus on the case when a^2+b^2=c^2. In this case, the general solution is expressed in terms of the acute angle theta which satisfies tan(theta)=a/b+c .From the right trianle with leglengths a and b, and hypotenuse length c, we construct a cyclic quadrilateral within which the angle theta is illustrated.Then we examine the case when a,b,and c are integers;and we derive or construct a family of cyclic Heron quadrilaterals. These are quadrilaterals with integer sidelengths or edges, integer diagonal lengths, and integer area.Also these quadrilaterals have their four vertices lying on a circle. The said family of quadrilaterals, is a three parameter infinite family.

math.GM↗
arXiv2008

When is sinx+cosx+tanx+cotx+secx+cscx an integer ?

In this paper, we investigate the one-variable equation, sinx+cosx+tanx+cotx+secx+cscx=n, where n is an integer. We prove that if n lies between(inclusively) -1 and 6; then the above equation has no real number solutions. While if n is greater than or equal to 7; or less than or equal to -2, then the said equation has a nonempty solution set which we describe.

math.GM↗