Big pleasure

A deserving scientist of such a name, especially a mathematician, experiences in his work the same impression as an artist; his pleasure is so big and of the same nature.

Jules Henri Poincaré

INFINITUM. Mathematical appointments

Although many people do not understand it, the mathematician experiences a magnificent sensation realizing his work. As there says Poincaré, a big pleasure; and, as says Isa, a subidón.

What do you think?

Sums of fractions and 2010

I leave the problem of this week to you, in this case related to the year in which we are:

To demonstrate that for any one verifies positive real numbers that

Luck.

The Nagel line

This article is a collaboration sent for fede to gaussianos (25 pound) gmail (point) com.

Brief biographical Nagel critique

Christian Heinrich von Nagel

Christian Heinrich von Nagel, German geometer, was born on February 28, 1803 in Stuttgart, Germany, and German died on October 27, 1882 in also Ulm.

In 1821 Nagel began to study Theology, finishing his studies in 1825. But during these four years his interests also went towards the mathematics and the physics.

So much it was so he became a teacher of mathematics of secondary in the German city of Tübingen. But the thing did not stay there. In 1826 Nagel confers a doctorate thanks to his triangulis rectangulis work on ex-algebraic aequatione construendis (On triangles rectangles construibles from an algebraic equation). Later, in 1830, Nagel moves to Ulm where it is employed at the Gymnasium (school of secondary preparatory for top studies) of this locality.

His principal contribution to the mathematics is framed in the geometry of the triangle. In this article we are going to see, between other things, two constructions related to the triangle that take his name: the point of Nagel and the line of Nagel.

Introduction

As the distance of the baricentro to an apex is the double of the distance of to the average point of the opposite side, the homotecia with center and reason-1/2 transforms the triangle, antimedial or anticomplementary of, in the triangle, and this one in his medial or complementary triangle.

The applet GeoGebra-Java could not have executed.

In geometry of the triangle it is called sometimes a complement of a point to his image in the homotecia and anticomplement of to his image in the homotecia

The point, a point, his complement, and his anticomplement they are aligned, and placed so that it is the average point of and.

If in the figure we place the point in the circuncentro of, the point is the circuncentro of the antimedial triangle (that is the ortocentro of), the point is the circuncentro of the medial triangle (that is to say the center of the circle of 9 points), and the line is the line of Euler of the triangle.

On the other hand if we place the point in the uncenter of, the point is the uncenter of the antimedial triangle, the point the uncenter of the medial triangle, and the line is the line that in Wolfram MathWorld they have decided to call somewhat arbitrarily a Nagel line, for the fact that the uncenter of the antimedial triangle is the Nagel point, as we will demonstrate next.

On the other hand the Spieker point is for definition the uncenter of the medial triangle, and of the previous remarks one concludes that the uncenter, the baricentro, the point of Spieker and the point of Nagel are aligned, it is the average point of the segment and.

The Nagel point

We call ceviana of Nagel to the line, in the figure, which joins an apex with the point of touch of the exwritten circumference opposite to the apex with the opposite side.

The point, to the being the intersection of the common tangent one to the circumferences written and exwritten put up to with the line that joins the centers of these circumferences is a center of a homotecia that transforms the circumference exinscribed in the written circumference. This homotecia takes the radio to the radio, parallel to and therefore perpendicularly to.

Therefore the ceviana happens for the point diametrically put up in the circle inscribed to the point of touch of this circle with the side put up to.

Since we saw in the post on the circles tritangentes and therefore if it is the average point of.
Since also, it proves that the parallel lines and they are.

And as the homotecia, which transforms the triangle into his antimedial one, transforms the line into a line that passes for parallel to, that is to say in the Nagel ceviana $AE$, turns out that the Nagel cevianas meet in a point, the point of Nagel and this point is the uncenter of the antimedial triangle.

Of the post on circles tritangentes we conclude also that the Nagel cevianas bisect the perimeter of the triangle, that is to say two parts of the perimeter of the triangle placed to one and another side of every ceviana of Nagel they have equal length.

The Spieker point

The Spieker point is the center of the circle inscribed in the medial triangle, or I circulate of Spieker, and it has some quite interesting properties.

If in the figure it is the average point of and we prolong the side up to so that, and it is the average point of, and they are parallel and.

Camo $BE$ is perpendicular to $AH$, and this line is the exterior bisectriz of, $A_1F$ it is parallel to the interior bisectriz of, and therefore it is a bisectriz of the medial triangle.

Therefore the lines that join the average point of every side with the Spieker point, that is to say the bisectrices of the medial triangle, bisect the perimeter of the triangle, like the cevianas of Nagel.

If, the segments are respectively equal to the segments, and the average points of these equal segments are placed at the same distance of the straight line.
Then the center of gravity of a mass distributed uniformly by the perimeter of the triangle is in the line $A_1F$. Since also it is in other bisectrices of the medial triangle, it turns out that the Spieker point is the center of gravity of the perimeter of the triangle.

The average point of equidistant es of the points of touch of the exwritten circumferences put up to and with the side, and therefore it is in the radical axis of these circumferences.
As the radical axis is perpendicular to the line that joins the centers, which it is the exterior bisectriz of the angle in, it turns out that the radical axis of two exwritten circumferences is the bisectriz of the medial triangle, and therefore the Spieker point is the radical center of three exwritten circumferences, that is to say the tangent ones from the Spieker point have the same length to the exwritten circumferences.

The circumferences of Jenkins of sound three tangent circumferences internally to an exwritten circumference and outwardly to others two.

Three Jenkins circumferences are cut in the Spieker point, since the investment with regard to the orthogonal circle to three exwritten circumferences, which center is the Spieker point, transforms the sides of the triangle into the Jenkins circumferences.

And also if the Spieker point is on the circumference inscribed in, three circumferences of Jenkins are tangent to a straight line perpendicular to the line of Nagel, and in another case the center of the tangent circumference to three Jenkins circumferences is in the Nagel line, because this circumference is inverse of the written circumference.

Certainly the latter point is not, seems to me, in ETC.: Will it be new? According to Geogebra his first coordinate trilineal for (6,9,13) is 166.495. and it is not on the search page of ETC.

The following figure tries to illustrate the previous properties.

The applet GeoGebra-Java could not have executed.

Sources used for the biographical critique:

• Personal page of Clark Kimberling, of the University of Evansville, from which also I have extracted the image of Nagel that illustrates the beginning of the article.
• Christian Heinrich von Nagel in the English Wikipedia.

Difficult to communicate

In company of friends, the writers can discuss on his books, the economists on the state of the economy, the lawyers his last suits and the businessmen his last procurement, but the mathematicians cannot speak on his mathematics by no means. And the deeper is his work, the less understandable it is.

Alfred Adler

INFINITUM. Mathematical appointments

I agree with Adler. For a mathematician it is very complicated to explain to someone who should not be much put in the matter what is what it does. It is sure that some of you you have been in a situation like that one day. The comments are the best way of counting your experiences.

Torrent Gossip Girl S03E14 The Lady Vanished free

Numeri idonei

This article has been promoted to appear in the front of Wiggle me. If you have liked and want to vote for it enter this linkage and bundle click in Wiggle it.

Introduction

Since we have already commented one day, Leonhard Euler is the most prolific mathematician of the history. We can find his name in almost all the branches of the mathematics, from algebra up to complex analysis, happening for geometry and topology. But the more it investigates one in his works the more it is surprised. However much we think that we know the Euler works it always appears by surprise with a new topic that us was foreign. This itself is what it has spent to me a few days ago. And, how not, am I going to tell it to you.

Numeri idonei

In a letter directed to the Swiss physicist Nicolas Béguelin, Euler he was commenting on the following thing:

All the suppressed numbers of only one form prime in sound or cousins' doubles where and they are prime between themselves. I have observed that other similar expressions of the form enjoy the same property giving to the letter suitable values.

This is, any number that can express itself of the only form how, for and relative cousins, it is prime or the double of a cousin. In particular, any odd number that could express itself of the only form in the previous sense is prime.

But there is still more. Not only it serves an expression of the type, but certain values exist of such that an expression of the type fulfills the same property. To these values of es to which there are called they numeri idonei (suitable numbers or suitable numbers in Spanish and suitable numbers or idoneal numbers in English).

At least this was the initial definition of suitable number. But this way of defining this type of numbers presents some problems. For example, it is a suitable number (we will see it further on) and for him it is fulfilled that:

it is the only representation of the number 9 as. But since we all know 9 it is not prime, although yes it is a potency of a cousin, since. Therefore we should say that it is a suitable number if any odd number that could express itself of the only form as it is prime or promotes of a cousin, but it is possible to play in tune a little more to eliminate this new possibility, this is, that the number is a potency of a prime number (in the first linkage of the sources you can see some of the conditions that we can add to him to the definition to avoid this).

Knowing a little the way of working of any Euler he can imagine that it did not remain there, that his investigations on this topic did not end in the establishment of the definition of this type of numbers. Knowing about his character investigator one he tends to think that it tried to study in depth more the matter. And having little information about his achievements it is not difficult to become convinced of that it did it, and very deeply. Since yes, this way it was. Euler prepared a list of suitable numbers. It is the following one:

In whole 65 numbers that Euler verified that they were suitable (in the sense commented previously). In fact it investigated more: it used it is ready to construct prime numbers even of eight numbers.

Come to this point the most logical thing is that we rise the following question to ourselves: is the set of suitable numbers infinite? The answer is not. In 1934, the mathematician Sarvadaman Chowla demonstrated that the set of suitable numbers is finite.

Knowing this us another question arises: are there more suitable numbers apart from the found ones by Euler? Unfortunately for this question there is no answer yet, although yes information is had. Specifically it is known that like much one more suitable number exists, apart from that they are in the list. And that if the latter suitable number in fact exists, must be major than 100000000.

Major prime number met the suitable numbers

We have mentioned before Euler that it used these numbers to find prime numbers relatively grades (up to eight numbers). The biggest prime number that Euler found with this téctica was. To demonstrate that this number of eight numbers is prime it would be necessary to verify that the only solution of the equation

The only one is 26

Introduction

It already does enough time we comment on a curious property of the number 26. Specifically it is this one:

The number 26 is the only natural number that is placed between a square () and a bucket ().

Apparently it was Fermat who demonstrated the above mentioned result, but in the post where we realized of this characteristic of 26 no test of this fact was happening. It was Juanbuffer who was contributing in a comment a pdf with a demonstration of the same one (that if I do not remember badly was not in Spanish). Unfortunately it seems that it is already not possible to gain access to the above mentioned document (at least I cannot). For this motive I started looking … and I have found it. My admired Carlos Ivorra is who has provided the above mentioned test to me. Very well, in fact I do not know if it is his, but it appears in one of the books in format pdf that it has available in his web: Theory of Numbers.

In this article you are going to be able to see this demonstration.

The unicidad of 26

In fact the demonstration that I am going to present to you of the fact of which the only natural number is 26 with the mentioned property previously is relatively elementary. The interesting of the test is that it leaves of the set of the natural numbers to demonstrate a characteristic in. The fact to rest on a set major that to demonstrate something in him is a quite useful, made argument of which many mathematicians took advantage when they became convinced of the potency of the above mentioned argument.

Centrémonos in the topic. We are going to do the demonstration in (the entire numbers). Then the statement of the result to demonstrate the following one:

Theorem:

The only entire solutions of the equation

they are.

Demonstration:

A simple glance to the equation says to us that it cannot be an even number. If out we would have that also it would be a pair. The contradiction would be in the fact that the right part of the equality would be divisible between 8, but the left part would not be not even divisible between 4. Therefore it has to be an odd number.

We leave now of for penetrating into the ring. We think that the previous equation in this ring his expression can happen factorizada of the following way:

We consider in this ring the following norm:

It is simple to verify that the above mentioned norm is multiplicativa, this is, that it is positive for any element different from zero of, that zero is zero for the element and that the norm of a product of two elements of es the product of the norms of saying elements.

Let's suppose now that they fulfill the initial equation and let's take the elements and of. Any element that is their common divisor two must divide also to his sum, and to his difference. Taking norms in this situation we would have the following thing:

Therefore. The only pairs of values that fulfill this are them following:

With the first two possibilities we obtain the elements and-1$ of, that are units of this ring. In other cases we obtain the elements and, all of them with norm pair (2 ó 4), therefore they cannot divide to, whose norm () is odd.

With this we come to the following thing: and they are prime between themselves.

Now, we had the initial equation factorizada of the following form:

Joining these two facts we have that the product of two elements of which they are prime between themselves is equal to a bucket. It forces that each of these elements is he itself a bucket. In particular:

Let's develop now the right part of the latter equality:

Equaling coefficients of of the initial and final expressions we come to the following equality:

A simple analysis of the values of and it takes us that the only possible values are and (let's remember that and there are entire numbers). For obtenemos that and this implies that. And for obtenemos and therefore, that is the looked result.

Do you know any other demonstration of this fact? The comments are yours.

Let's calculate the highest common factor

In view of the title of the post there is quite clear the subject-matter of the problem of this week: truth? There it goes:

He calculates the following highest common factor:

Luck.