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Primality testing: recap
Dear IIT Students: Address to IIT Hyderabad by present President of India, Shri Ram Nath Kovind
Pre RMO Practice question: 2018: How long does it take for a news to go viral in a city? And, a cyclist vs horseman
Problem 1:
Some one arrives in a city with very interesting news and within 10 minutes tells it to two others. Each of these tells the news within 10 minutes to two others(who have not heard it yet), and so on. How long will it take before everyone in the city has heard the news if the city has three million inhabitants?
Problem 2:
A cyclist and a horseman have a race in a stadium. The course is five laps long. They spend the same time on the first lap. The cyclist travels each succeeding lap 1.1 times more slowly than he does the preceding one. On each lap the horseman spends d minutes more than he spent on the preceding lap. They each arrive at the finish line at the same time. Which of them spends the greater amount of time on the fifth lap and how much greater is this amount of time?
I hope you enjoy “mathematizing” every where you see…
Good luck for the Pre RMO in Aug 2018!
Nalin Pithwa.
A leaf out of Paul Erdos’ biography: My Brain is Open: by Bruce Schechter
Reference:
My Brain is Open: The Mathematical Journeys of Paul Erdos by Bruce Schechter, a TouchStone Book, Published by Simon and Schuster, New York.
Amazon India link:
Chapter One: Traveling.
The call might come at midnight or an hour before dawn — mathematicians are oddly unable to handle the arithmetic of time zones. Typically, a thickly accented voice on the other end of the line would abruptly begin:
“I am calling from Berlin. I want to speak to Erdos.”
“He’s not here, yet.”
“Where is he?”
“I don’t know.”
“Why don’t you know!” Click!
Neither are mathematicians always observant of social graces.
For more than sixty years mathematicians around the world have been roused from their abstract dreams by such calls, the first of the many disruptions that constituted a visit from Paul Erdos. The frequency of the calls would increase over the next several days and would culminate with a summons to the airport, where Erdos himself would appear, a short, frail man in a shapeless old suit, clutching two small suitcases that contained all of his worldly possessions. Stepping off the plane he would announce to the welcoming group of mathematicians, “My brain is open!”
Paul Erdos’ brain, when open, was one of the wonders of the world, an Ali Baba’s cave, glittering with mathematical treasures, gems of the most intricate cut and surpassing beauty. Unlike Ali Baba’s cave, which was hidden behind a huge stone in a remote desert, Erdos and his brain were in perpetual motion. He moved between mathematical meetings, universities, and corporate think tanks, logging hundreds of thousands of miles. “Another roof, another proof,” as he liked to say. “Want to meet Erdos?” mathematicians would ask. “Just stay here and wait. He’ll show up.” Along the way, in borrowed offices, guest bedrooms, and airplane cabins, Erdos wrote in excess of 1600 papers, books and articles, more than any other mathematician who ever lived. Among them are some of the great classics of the twentieth century, papers that opened up entire new fields and became the obsession and inspiration of generations of mathematicians.
The meaning of life, Erdos often said, was to prove and conjecture. Proof and conjecture are the tools with which mathematicians explore the Platonic universe of pure form, a universe that to many of them is as real as the universe in which they must reluctantly make their homes and livings, and far more beautiful. “If numbers aren’t beautiful, I don’t know what is,” Erdos frequently remarked. And although, like all mathematicians, he was forced to make his home in the temporal world, he rejected worldly encumbrances. He had no place on earth called home, nothing resembling a conventional yearround, ninetofive job, and no family in the usual sense of the word. He arranged his life with only one purpose, to spend areas many hours a day as possible engaged in the essential, lifeaffirming business of proof and conjecture.
For Erdos, the mathematics that consumed most of his waking hours was not a solitary pursuit but a social activity, a movable feast. One of the greatest mathematical discoveries of the twentieth century was the simple equation that two heads are better than one. Ever since Archimedes traced his circles in sand, mathematicians, for the most part, have laboured alone — that is, until some forgotten soul realized that mathematics could be done anywhere. Only paper and pencil were needed, and those were not strictly essential. A tablecloth would do in a pinch, or the mathematician could carry his equations in his head, like a chessmaster playing blindfolded. Strong coffee, and in Erdos’ case even more powerful stimulants, helped too. Mathematicians began to frequent the coffeehouses of Budapest, Prague, and Paris, which led to the quip often attributed to Erdos:”A mathematician is a machine for turning coffee into theorems.” Increasingly, mathematical papers became the work of two, three, or more collaborators. That radical transformation of how mathematics is created is the result of many factors, not the least of which was the infectious example set by Erdos.
Erdos had more collaborators than most people have acquaintances. He wrote papers with more than 450 collaborators — the exact number is still not known, since Erdos participated in the creation of new mathematics until the last day of his life, and his collaborators are expected to continue writing and publishing for years. The briefest encounter could lead to a publication — for scores of young mathematicians a publication that could become the cornerstone of their life’s work. He would work with anyone who could keep up with him, the famous or the unknown. Having been a child prodigy himself, he was particularly interested in meeting and helping to develop the talents of young mathematicians. Many of the world’s leading mathematicians owe their careers to an early meeting with Erdos.
Krishna Alladi, who is now a mathematician at the University of Florida, Gainesville, is one of the many young mathematicians whom Erdos helped. In 1974, when Alladi was an undergraduate in Madras, India, he began an independent investigation of a certain number theoretic function. His teachers could not help Alladi with his problem, nor could his father, who was a theoretical physicist and head of Madras Institute of Mathematics. Alladi’s father told some of his knowledgeable friends about his son’s difficulty, and they suggested that he write to Erdos.
Because Erdos was constantly on the move, Alladii sent a letter to the Hungarian Academy of Sciences. In an astonishingly short time, Alladi heard from Erdos, who said he would soon be lecturing in Calcutta. Could Alladi come there to meet him? Unfortunately, Alladi had examinations and could not attend, so he sent his father in his place to present the results of his research. After his father’s talk, Alladi recounts, “Erdos walked up to him and told him in very polite terms that he was not interested in the father but in the son.” Determined to meet with the promising young mathematician, Erdos, who was bound for Australia, rerouted his trip to stop briefly in Madras, which lies about 860 miles south of Calcutta.
Alladi was astonished that a great mathematician should change his plans to visit a student. He was nervous when he met Erdos at the airport, but that soon passed. “He talked to me as if he had known me since childhood,” Alladi recalls. The first thing Erdos asked was, “Do you know my poem about Madras?” And then he recited:
This the city of Madras
The home of the curry and the dhal,
Where Iyers speak only to Iyengars
And Iyengars speak only to God.
The Iyers and Iyengars are two Brahmin sects. The Iyers worship Shiva the Destroyer but will also worship in the temples of the Iyengars, who worship only Lord Vishnu, the Protector. Erdos explained that this was his variation on the poem about Boston and the pecking order among the Lowells, the Cabots, and God. Having put Alladi at ease, Erdos launched into a discussion of mathematics. Erdos was so impressed with Alladi, who was applying to graduate schools in the United States, that he wrote a letter on his behalf. Within a month, Alladi received the Chancellor’s Fellowship at the University of California, Los Angeles.
A celebrated magazine article about Erdos was called, “The Man Who Loved Only Numbers.” While it is true that Erdos loved numbers, he loved much more. He loved to talk about history, politics, and almost any other subject. He loved to take long walks and to climb towers, no matter how dismal the prospective view, he loved to play pingpong, chess, and Go, he loved to perform silly tricks to amuse children and to make sly jokes and thumb his nose at authority. But, most of all, Erdos loved those who loved numbers, mathematicians. He showed that love by opening his pocket as well as his mind. Having no permanent job, Erdos also had little money, but whatever he had was at the service of others. If he heard of a graduate student who needed money to continue his studies, he would sent a cheque. Whenever he lectured in Madras, he would send his fee to the needy widow of the great Indian mathematician Srinivasa Ramanujan; he had never met Ramanujan or his wife, but the beauty of Ramanujan’s equations had inspired Erdos as a young mathematician. In 1984, he won the prestigious Wolf prize, which came with a cash reward of $at 50000, easily the most money Erdos had ever received at one time. He gave $30000 to endow a postdoctoral fellowship in the name of his parents at the Technion in Haifa, Israel, and used the remainder to help relatives, graduate students, and colleagues:”I kept only $720,” Erdos recalled.
In the years before the internet, there was Paul Erdos. He carried a shopping bag crammed with latest papers, and his brain was stuffed with the latest gossip as well as an amazing database of the world of mathematics. He knew everybody: what they were interested in; what they had conjectured, proved, or were in the midst of proving; their phone numbers; the names and ages of their wives, children, pets; and, much more. He could tell off the top of his head on which page in which obscure Russian journal a theorem similar to the one you were working on was proved in 1922. When he met a mathematician in Warsaw, say, he would immediately take up the conversation where they had left it two years earlier. During the iciest years of the Cold War Erdos’s fame allowed him freely to cross the Iron Curtain, so that he became vital link between the East and the West.
In 1938, with Europe on the brink of war, Erdos fled to the United States and embarked on his mathematical journeys. This book is the story of those adventures. Because they took Erdos everywhere mathematics is done, this is also the story of the world of mathematics, a world virtually unknown to outsiders.. Today perhaps the only mathematician most people can name is Theodore Kacznyski. The names of Karl Friedrich Gauss, Bernhard Riemann, Georg Cantor and Leonhard Euler, who are to Mathematics what Shakespeare is to literature and Mozart to music, are virtually unknown outside of the worlds of math and science.
For all frequent flier miles Erdos collected, his true voyages were journeys of the mind. Erdos carefully constructed his life to allow himself as much time as possible for those inward journeys, so a true biography of Erdos should spend almost as much time in the Platonic realm of mathematics as in the real world. For a layman this may seem to be a forbidding prospect. Fortunately, many of the ideas that fascinated Erdos can be easily grasped by anyone with a modest recollection of high school mathematics. The proofs and conjectures that made Erdos famous are, of course, far more difficult to follow, but that should not be of much concern to the reader. As Ralph Boas wrote, “Only professional mathematicians learn anything from proofs. Other people learn from explanations.” Just as it is not necessary to understand how Glenn Gould fingers a difficult passage to be dazzled by his performance of thee “Goldberg Variations,” one does not have to understand the details of Erdos’s elegant proofs to appreciate the beauty of mathematics. And, it is the nature of Erdos’s work that while his proofs are difficult, the questions he asks can be quite easy to understand. Erdos often offered money for the solution to problems he proposed. Some of those problems are enough for readers of this book to understand — and, perhaps, even solve. Those who decide to try should be warned that, as Erdos has pointed out, when the number of hours it takes to solve one of his problems is taken into account, the cash prizes rarely exceed minimum wage. The true prize is to share in the joy that Erdos knew so well, joy in understanding a page of the eternal book of mathematics.
— shared by Nalin Pithwa (to motivate his students and readers.)
Diophantus of Alexandria: some trivia, some tidbits
The name/word Diophantine equation honours the mathematician Diophantus, who initiated the study of such equations. Practically, nothing is known of Diophantus as an individual, save that he lived in Alexandria sometime around 250 A.D. The only positive evidence as to the date of his activity is that the Bishop of Laodicea, who began his episcopate in 270, dedicated a book on Egyptian computation to his friend Diophantus. Although Diophantus’ works were written in Greek and he displayed the Greek genius for theoretical abstraction, he was most likely a Hellenized Babynolian. The only personal particulars we have of his career come from the wording of an epigramproblem (apparently dating from the 4th century). His boyhood lasted 1/6 of his life; his beard grew after 1/12 more; after 1/7 more he married; and his son was born 5 years later, the son lived to half his father’s age and the father died 4 years after his son. If x was the age at which Diophantus died, these data lead to the equation:
with solution . Thus, he must have reached an age of 84, but in what year or even in what century is not certain.
The great work upon which the reputation of Diophantus rests is his Arithmetica, which may be described as the earliest treatise on algebra. Only six Books of the original thirteen have been preserved. It is in the Arithmetica that we find the first systematic use of mathematical notation, although the signs employed are of the nature of abbreviations for words rather than algebraic symbols in the sense with which we use them today. Special symbols are introduced to represent frequently occurring concepts, such as the unknown quantity in an equation and the different powers of the unknown up to the sixth power. Diophantus also had a symbol to express subtraction, and another for equality.
It is customary to apply the term Diophantine equation to any equation in one or more unknowns that is to be solved in the integers. The simplest type of Diophantine equation is the linear Diophantine equation in two variables:
where a, b, c are given integers and a, b are not both zero. A solution of this equation is a pair of integers that, when substituted in to the equation, satisfy it; that is, we ask that . Curiously enough, the linear equation does not appear in the extant works of Diophantus (the theory required for its solution is to be found in Euclid’s Elements), because he viewed it as trivial, most of his problems deal with finding squares or cubes with certain properties.
Reference:
Elementary Number Theory, David M. Burton, 6th edition, Tata McGraw Hill Edition.
More such interesting information about famous mathematical personalities is found in the classic, “Men of Mathematics by E. T. Bell”.
— Nalin Pithwa
Remembering Richard Feynman on his 100th birth centenary
Thanks to Hindustan Times and Prof. Arnab Bhattacharya, TIFR, Mumbai.
Coordinate geometry practice for IITJEE Maths: Ellipses
Problem 1:
Find the locus of the point of intersection of tangents to the ellipse , which are at right angles.
Solution I:
Any tangent to the ellipse is ….call this equation I.
Equation of the tangent perpendicular to this tangent is …call this equation II.
The locus of the intersection of tangent lines (i) and (ii) is obtained by eliminating m between these equations. So, squaring and adding them, we get
which is a circle with its centre at the centre of the ellipse and radius equal to the length of the line joining the ends of the major and minor axis. This circle is called the director circle of the ellipse.
Problem II:
A tangent to the ellipse meets the ellipse at P and Q. Prove that the tangents at P and Q of the ellipse at right angles.
Solution II:
Let the tangent at to the ellipse meet the ellipse at P and Q.
Let the tangents at P and Q to the second ellipse intersect at the point . Then, PQ is the chord of contact of the point with respect to ellipse two, and so its equation is
….call this “A”.
PQ is also the tangent at to the first ellipse and so the equation can be written as ….call this “B”.
Comparing “A” and “B”, we get
and
The locus of is , or , which is the director circle of the second ellipse.
Hence, the tangents at P and Q to the ellipse (ii) are at right angles (by the solution to the previous example).
Problem 3:
Let d be the perpendicular distance from the centre of the ellipse to the tangent drawn at a point P on the ellipse. If and are the two foci of the ellipse, then show that
Solution 3:
Equation of the tangent at the point on the given ellipse is . Thus,
We know
…call this equation I.
Also, , which in turn equals,
, that is,
that is,
Now, from I, we get ,
also,
Hence,
we will continue later,
Cheers,
Nalin Pithwa
B.S. in Mathematics: IIT Bombay program:
http://www.math.iitb.ac.in/Academics/bs_programme.php
Note that the admission is through IITJEE Advanced only.
–Nalin Pithwa.
Coordinate Geometry problems for IITJEE : equations of median, area of a triangle, and circles
Problem I:
If , and are the vertices of a triangle ABC, then prove that the equation of the median through A is given by:
Solution I:
If D is the midpoint of BC, its coordinates are
Therefore, equation of the median AD is , which in turn, implies that,
Now apply the row transformation to the previous determinant. So, we get
, using the sum property of determinants.
Hence, the proof.
Problem 2:
If is the area of the triangle with vertices , and is the area of the triangle with vertices , , and , and is the area of the triangle with vertices , , . Then, prove that there is no value of for which the areas of triangles, , and are in GP.
Solution 2:
We have , and
.
Applying the following column transformations to the above determinant, and , we get
and
so that .
Now, , and are in GP, if
, that is,
, where . But, for this value of , the vertices of the given triangles are not defined. Hence, , and and cannot be in GP for any value of .
Problem 3:
Two points P and Q are taken on the line joining the points and such that . Circles are drawn on AP, PQ, and QB as diameters. The locus of the point S, the sum of the squares of the length of the tangents from which to the three circles is equal to , is
(a)
(b)
(c)
(d) .
Ans. b.
Solution 3:
Since , the coordinates of P are and of Q are , equations of the circles on AP, PQ, and QB as diameters are respectively.
Please draw the diagram.
So, we get
So, if be any point of the locus, then .
So, the required locus of is .
More later,
Nalin Pithwa.