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The largest known prime (found by GIMPS [Great Internet Mersenne Prime Search] in November 2003) is the 40th Mersenne prime: M 20996011 which has 6320430 decimal digits.
The largest known twin primes are 242206083 
238880 
1. They have 11713 digits and were announced by Indlekofer and Ja'rai in November, 1995.
The largest known factorial prime (prime of the form n! 1) is 3610! - 1. It is a number of 11277 digits and was announced by Caldwell in 1993.
The largest known primorial prime (prime of the form n # 1 where n # is the product of all primes 
n) is 24029# + 1. It is a number of 10387 digits and was announced by Caldwell in 1993.
Assignments:
1. Speak on the latest prime records trying to avoid peeping into the text.
Assignments for unit II:
1. Give English equivalents of the following words and expressions:
совершенное число, десятичный знак, дружественные числа, простое число, факторизация, бесконечно много, арифметическая прогрессия, целое число, метод «от противного».
2. Using the above mentioned words and expressions (English translation) make up sentences of your own.
3. Remind yourself of what you were reading in this unit and answer the questions:
a) Who is the founder of the study of prime numbers?
b) Why was the certain period in the development of the study of prime numbers called the Dark Ages?
c) What numbers do we call Mersenne numbers and why?
d) It is possible to solve the following problem: are there infinitely many primes of the form n! + 1?
e) What is the largest known prime? The largest known factorial prime?
4. Name:
a) The scientists considering the study of prime numbers;
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b) the scientists who managed to pose unsolved problems in the theory of prime numbers;
c) the scientists who announced the records of prime numbers.
5. Tell the group everything you know about these mathematicians, i.e. everything found out in this unit.
Smile J
Einstein and telephone
One woman asked Einstein to remember her telephone number: 361-343.
Einstein answered:
- It’s very easy. 19 squared and 7 cubed.
New about limits
At a mathematics exam a professor asks a student to calculate the limit:
The professor is surprised:
- What is it? Why?
The student answers:
- You explained at your lecture that
and I have used this example.
Unit III.
Game Theory
Text 1
Some Basics
Game theory is a branch of applied mathematics fashioned to analyze certain situations in which there is an interplay between parties that may have similar, opposed, or mixed interests. In a typical game, decision-making “players,” who each have their own goals; try to outsmart one another by anticipating each other's decisions. (Encyclopedia Britannica)
Game theory is a distinct and interdisciplinary approach to the study of human behavior. The disciplines most involved in game theory are mathematics, economics and the other social and behavioral sciences. Game theory (like computational theory and so many other contributions) was founded by the great mathematician John von Neumann. The first important book was The Theory of Games and Economic Behavior, which von Neumann wrote in collaboration with the great mathematical economist, Oscar Morgenstern. Certainly Morgenstern brought ideas from neoclassical economics into the partnership, but von Neumann, too, was well aware of them and had made other contributions to neoclassical economics.
Assignments:
1. Answer the questions:
1) What is game theory?
2) What disciplines most involved in game theory and why?
3) Who can be called a founder of game theory?
2. Give a brief summary of the article using the following words and expressions: a branch of applied mathematics, an interplay between parties, the study of human behavior, to be founded by, in collaboration with, to make contribution.
Text 2
What is a game?
Game: A competitive activity involving skill, chance, or endurance on the part of two or more persons who play according to a set of rules, usually for their own amusement or for that of spectators (The Random House Dictionary of the English Language, 1967).
A game is the set of rules that describe it. An instance of the game from beginning to end is known as a play of the game. And a pure strategy is an overall plan specifying moves to be taken in all eventualities that can arise in a play of the game. A game is said to have perfect information if, throughout its play, all the rules, possible choices, and past history of play by any player are known to all participants. Games like tick-tack-toe, backgammon and chess are games with perfect information and such games are solved by pure strategies. But whereas one may be able to describe all such pure strategies for tick-tack-toe, it is not possible to do so for chess, hence the latter's age-old intrigue.
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Games without perfect information, such as matching pennies, stone-paper-scissors or poker offer the players a challenge because there is no pure strategy that ensures a win. Games such as heads-tails and stone-paper-scissors are also called two-person zero-sum games. Zero-sum means that any money Player 1 wins (or loses) is exactly the same amount of money that Player 2 loses (or wins). That is, no money is created or lost by playing the game. Most parlor games are many-person zero-sum games. Not all zero-sum games are fair, although most two-person zero-sum parlor games are fair games. So why do people then play them? They are fun, everyone likes the competition, and, since the games are usually played for a short period of time, the average winnings could be different than 0.
Assignments:
1. Active vocabulary:
A competitive activity, a set of rules, an instance of the game, a play of the game, a pure strategy, eventuality, a game with (without) perfect information, a zero-sum game, tick-tack-toe, backgammon, chess, stone-paper-scissors, matching pennies.
2. Give the definitions of the following notions in English:
A game, a play of the game, a pure strategy, a game with perfect information, a game without perfect information, a zero-sum game, a fair game.
