In mathematics, a prime number (or a prime) is a natural number which has exactly two distinct natural number divisors: 1 and itself. An infinitude of prime numbers exists, as demonstrated by Euclid around 300 BC. The first twenty-five prime numbers are: 22, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97 …3571113171923 29313741434753 59616771737983 8997
giuseppina trifiletti N objects are set in a circle one object every M objects is removed and the circle is closed Which object will be the last? Which is the elimination order?
giuseppina trifiletti There isn't an all-cases formule There isn't an all-cases formule The problem can be solved using a precise algorythm The problem can be solved using a precise algorythm the right one for each counts, which, when translated to a programming language, allows the computer to give us an answer in a short time. the right one for each counts, which, when translated to a programming language, allows the computer to give us an answer in a short time.
The Lager Counts In a work camp for war prisoners there were 10 people. The prisoners had to be killed. The guards' chief, sadistic but a math-lover, decided to save one prisoner's life. The prisoners must be in a circle, To pick the lucky one he decided to make a counts with 17 beats. He said he begins from the first at his left. He, of course, would not have counted himself.
giuseppina trifiletti Then, before starting the counts, he asked: “Does someone want to change their place?” Only one of them lifted his hand and moved to the chief's right. Why ? Some other prisoners understood why, but didn't want to challenge for the safety-place, they chose instead to die with their friends.
giuseppina trifiletti This is for you an hard black counts Now in a ring all you by hand Will not die out in this bad band The one who now will not go out The Lager Counts
17 beats for your end \\\\ Thisisforyouanhardblackcounts 1234 Nowinaringallyoubyhand 5678 Willnotdieoutinthisbadband 9101112 Theonewhonowwillnotgoout 1314151617 Scan go-out separating
giuseppina trifiletti We must now make a practical demonstration Some volunteers come here please
What if the prisoners were, for example, 24? Could they react with such steadyness? No. They had to calculate differently because 24 > 17
For the previous work strategy it is necessaire that The number of beats - must be prime - must be greater than the number of prisoners The first of the counts must be always the first at the left of the chief If it is not so, the rest of the counts will no more be always different from 0 and so, sooner or later, the one before the first in the counts will be picked out and so he won't be saved.
C B A D chief removed B C B A D chief removed D C B A D chief removed C “A” would not have been picked out in any case. If the prisoners were 4, or any other number smaller than 17 (and of course >1) C B A D chief 4 prisoners
giuseppina trifiletti They leave in the following order B, D, C B first (17 : 4 = 4 and rest is 1) D second (17 : 3 = 5 and rest is 2) C third (17 : 2 = 8 and rest is 1) The last is A
giuseppina trifiletti Fortunately the guards' chief loved to make bad jokes to try his prisoners spirit and sharpness. In the end he punished the one who chose to save himself and gave a free day to those who resigned to death.
giuseppina trifiletti A good chance to talk about the algorythms and about the prime numbers
Hot new – sexy prime numbers Mathematicians sometimes also like to have fun. In spite of their name though, sexy primes are not connected to sex but to number 6 which, in latin, is called “sex”. A couple of prime numbers is called “sexy” if the difference between the two is equal to 6. So sexy couples are of the kind: (p, p+6). Do you want a list of the first few sexy couples? Here it is:... –(5, 11), (7, 13), (11, 17), (13, 19), (17, 23), (23, 29), (31, 37), (37, 43), …
The importance of Prime Numbers The importance of prime numbers comes from the fact that they are the bricks that form all natural numbers. It means that each natural number (2,3,4,5,6,7,8,...) can be built using those bricks. Fundamental Arithmetic Theorem –Each natural number can be represented only in one way as a product of prime numbers. –Examples: 38 = 19x272 = 3 2 x 2 3 38 = 19 times 2 72 = 3 at the second power times 2 at the third power
giuseppina trifiletti The importance of prime numbers in cryptography and security During the last 20 years the Prime numbers theory attracted also interest of many not- mathematicians, because of its application in computer science and cryptography, linked for example to security inside the internet, digital signatures, transmission of encrypted data...
giuseppina trifiletti Great prime numbers Factorization and Security In reality you have to choose great prime numbers (p and q), so that it becomes impossible to factorize, separate into factors a number, so as 15 =3 x 5, n = p x q, in a reasonable amount of time. The method is based on the assumption that it is extremely difficult to get to know p and q knowing their product. Try to discover which factors form the number 2047, for example. You'll need some time. Just imagine what will happen if the number is made of millions of digits!
giuseppina trifiletti 28 settembre 2008 46mo Mersenne prime number Marin Mersenne 1588-1684 Some Californian mathematicians discovered a prime number with 13 million of digits. When they will publisch their results they’ll receive a 100-thousand dollars prize
giuseppina trifiletti Not all prime numbers can be expressed as 2 p - 1 and not all 2 p -1 numbers are prime numbers \Not all Mersenne's numbers are prime numbers and not all prime numbers can be written as Mersenne's numbers. – Number 5 (which is a prime number), for example, can't be expressed as 2 p -1. –2 11 -1=..... is not a prime number and it is the smallest of this kind of numbers not to be a prime number. Mersenne's formule though can find great prime numbers (thanks to the exponential formule) which are very useful for cryptography and not only for that.
giuseppina trifiletti THE SAME NUMBERS OF FRIENDS How much would you bet that in our group at least 2 people have the same number of friends? How much would you bet that in any group of people at least 2 people have the same number of friends?
Any group with 5 people and the drawers principle A group made of 5 people A,B,C,D,E Nobody in the group is friend with himself If one person in the group, for example A, has no friends then B,C,D,E can't have 4 friends, and if one person in the group has 4 friends there can't be anyone with no friends. So each person can have – 0,1,2, 3 friends or 1,2,3, 4 friends Only 4 drawers for 5 people One drawer has to be used for at least 2 people
The number for friends can be 0123 or 1234 Only 4 drawers for 5 people only m-1 drawers if the people are m
giuseppina trifiletti Aldo, Bruno, Carla, Dario, Elisa, meet by chance. I can bet 1 million euros, or even much more, because I am perfectly certain, that at least 2 of them have the same number of friends in the group itself, even if the group is made of random people.
giuseppina trifiletti We investigate and discover (with a fast heartbeat, because of the high bet) that Aldo has no friends in the group, B has one friend, C has 2 friends and D has 3 friends. So unlucky! Until now everyone has a different number of friends!
giuseppina trifiletti In this way we filled all the drawers. 0 friends1 friend2 friends3 friends AldoBrunoCarlaDario But what about Elisa? How many friends does she have?
0 friends1 friend2 friends3 friends AldoBruno Carla Elisa Dario I can conclude, with some considerable relief, that I can only put her in one of the 4 drawers. As a matter of fact Elisa states that she has two friends, so she'll be put in Carla's drawer. 4 drawers
I bet and I won a million of dollars. Who, among you, would like to give them to me?
SOLUTION: for every individual I in the group, made of m people, let n(I) be the number of I's friends inside the group. So, not allowing I to be friends with himself, n(I) can only be 1,2,...m-1. We can observe that, for a fixed group with a determined number of elements/ndividuals, n(I) can't be 0 and also m-1. If there was a person with m-1 friends, every other person would have at least 1 friend. In the same way if one person had no friends (0 friends), there couldn't be people with m-1 friends. If we use the drawers principle (where the m number of people is the number of objects to put in the drawers and the drawers are the possible m-1 values assigned to n(I)), you can see that at least 2 people have to stay in the same drawer, which means that they have the same number of friends.
THE BIRTHDAY THE SAME MONTH How much would you bet that in our group –At least two people’s birthdays happen in the same month? How many people should be there to have the certainty of this? –At least three people’s birthdays happened in the same month? How many people should be there to have the certainty of this? THE SAME DAY How much would you bet that in our group –At least two people’s birthdays happened on the same day? How many people should be there to have the certainty of this?
giuseppina trifiletti It is a paradox of probability theory defined in the 1939 by Richard von Mises. 1939Richard von Mises The paradox states –the probability at least 2 people in a group have the birthday in the same day is much higher than the intuition suggests: in fact already in a group of 23 people the probability is approximately 51%; with 30 people excedes the 70%, with 50 people it touches the 97% –Even if to arrive to the certainty it needs a group of 366 people (thanks to the Principle of the drawers) Il paradosso del compleanno