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- Mathematics In Europe - Prime numbers

for large m Or to ask the question more precisely how large is the ratio of P m to m Gauss devoted much thought and computation to this question and conjectured the correct relationship However it was not until the end of the nineteenth century that a proof was found Gauss s conjecture now called the prime number theorem was proved independently by Hadamard and de la Vallée Poussin Here is the relationship Let ln m denote the natural logarithm of m Then the ratio of P m to m is well approximated at least for large values of m by 1 ln m For those unfamiliar with the natural logarithm here is another formulation Take any positive integer r for example r 10 and compute m 2 718 r In our example since r 10 we have m 2 718 10 which is very close to 22012 Then the following is true the proportion of prime numbers less than or equal to m in 1 m is approximately 1 r In our example we conclude that about 1 10 of the numbers less than 22012 are prime Primality tests Even with the aid of a computer it can be very tedious to tell whether a very large number m is prime by testing every number less than m asking in each case is n a factor of m However it can easily be seen that one need test only up to the square root of m since if n is a divisor of m then so is m n and both n and m n cannot be greater than the square root of m But even so that can still amount to a large number of tests For example if m has 100 digits then the square root has 50 digits If a computer can test one million values of n per second test is n a factor of m then the computer would have to work longer than the current age of the universe There are however better tests Here is the best known of them The starting point is a fact that was discovered by Fermat It is necessary first to know what is meant by a modulo b for two integers a and b by a modulo b we mean the remainder when a is divided by b For example 15 modulo 2 equals 1 and 139 modulo 4 equals 3 Fermat was able to show that if you take a prime number p and an integer a between 1 and p 1 and then compute a p 1 modulo p the result is 1 Let us test this for p 5 und a 3 First 3 4 equals 81 and if we calculate 81 modulo 5 we indeed get 1 So what makes this a primality test If we wish to test whether m is prime we can choose a number a smaller than m and check whether a m 1 modulo m is equal

Original URL path: http://mathematics-in-europe.eu/tr/81-information/landscape-numbers/155-prime-numbers (2013-11-18)

Open archived version from archive - Mathematics In Europe - Eulers famous equation

among the fundamental constants of mathematics These are zero the identity element for addition one the identity element for multiplication the number e the imaginary unit i the number To understand what comes next one should of course be familiar with these numbers and also know that the exponential function e x is defined also for complex numbers For a complex number z the number e z is to be understood as the sum of the infinite series 1 z z 2 2 z 3 3 This makes sense since sum product and limit are defined for complex numbers Moreover if z happens to be a real multiple of the imaginary unit i that is z i x for some real number x e i x can be expressed in terms of the trigonometric functions sine and cosine as follows e i x cos x i sin x To prove this relationship one needs the series developments of the sine and cosine and after that the rest is easy But after this discussion about complex values of the exponential function the above formula has been proved the sine of is 0 the cosine is 1 and therefore e i 1

Original URL path: http://mathematics-in-europe.eu/tr/81-information/landscape-numbers/156-eulers-famous-equation (2013-11-18)

Open archived version from archive - Mathematics In Europe - Cryptography

been found difficult to factor very large integers compared to the relative ease of finding large prime numbers The RSA procedure uses large numbers n that are products of two large prime numbers p and q That is we have n pq It is assumed in implementing RSA that it is difficult to find the two prime numbers p and q given only their product n Key Creation The first task in implementing the algorithm is to create a public and a private key key pair These can be provided by a key generation program that chooses at random two large prime numbers p and q such that n pq Then φ n read phi of n where φ is Euler s totient function is computed which gives the number of integers from 1 to n that are relatively prime to n For example φ 15 8 since 1 2 4 7 8 11 13 and 14 are the integers between 1 and 15 that are relatively prime to 15 we always include 1 Since a prime number is divisible only by itself and 1 all the positive integers less than that prime are relatively prime to it and so we have φ p p 1 and φ q q 1 It follows that for φ n we have φ n p 1 q 1 Finally we determine two integers e and d such that ed mod φ n 1 Recall that a mod b denotes the remainder when a is divided by b E g one has 2222 mod 100 22 or 2316809 mod 2 1 This means that e and d are inverses with respect to multiplication modulo φ n just as 4 and ¼ are inverses with respect to ordinary multiplication Specifically ed divided by φ n has remainder 1 The numbers e and n form the public key while d and n make up the private key It is unnecessary that the partners in communication have knowledge of the parameters p and q Indeed it is advisable to throw them away after e d and n have been computed since they will no longer be needed and keeping them represents an unnecessary security risk since whoever knows p and q also knows n and such knowledge would compromise the security of the system And how are d and e chosen First one finds an arbitrary integer e that is relatively prime to φ n For example it could be any prime number larger than φ n The determination of d that is the inverse of e modulo φ n is rather difficult it requires some more or less complex procedures Encryption and Decryption To encrypt an arbitrary message we first have to encode it as an integer This is easily accomplished for example we could replace every letter by its Unicode number and string these numbers together into one very long integer We then break a longer message into blocks so that each block is no

Original URL path: http://mathematics-in-europe.eu/tr/41-information/research/286-cryptography (2013-11-18)

Open archived version from archive - Mathematics In Europe - Visualizing Internal Organs with Mathematics

path to travel at the top and bottom than in the middle where it travels through the entire diameter of the circle Thus the middle will appear dark likely a dark green and the edges will be much lighter On the other hand if we view a rectangle from the side we will see a stripe of uniform brightness And now the sixty four dollar question by measuring the differences in brightness from a variety of directions can one determine what shape is present Surprisingly the answer is yes indeed and that is the basis of computer aided tomography a computer tomograph is shown in the picture The problem in this technique of medical diagnosis is quite similar to our glass problem the human body is x rayed from a number of directions measurements are taken of the intensity of absorption and these values are used to compute a three dimensional image of a specific organ of medical interest That s the general idea the details are painfully complex It took a combination of engineering skill computer technology and rather advanced mathematics to create what today is a standard tool of medical practice It took only a few years from the concepts developed in the 1960s to a practical implementation One reason for such a speedy result is that most of the mathematics had already been developed and was just sitting on the intellectual shelf waiting to be exploited Almost one hundred years ago the mathematician Johann Radon 1887 1956 proposed a procedure for reconstructing illuminated objects from measurements of intensity A computer tomograph is thus not merely high tech but also high math so to speak Research in this area continues since there is much room for improvement in both speed and resolution Inverse Problems The problem that arises

Original URL path: http://mathematics-in-europe.eu/tr/41-information/research/289-visualizing-internal-organs-with-mathematics (2013-11-18)

Open archived version from archive - Mathematics In Europe - How Do Quanta Compute?

the quantum world every measurement alters the state of the system and the initial state cannot be reconstituted The result is that there are comparatively few interesting mathematical questions that can be treated by this method Usually in mathematics one requires exact solutions not those that are true only with a certain probability One example in which one can just keep trying for a solution is in the decryption of secret codes And indeed the interest in quantum computers arose when the American Peter Shor shown in the figure devised a quantum algorithm for factorization of large integers which would assist in cracking the RSA code For his work he was awarded the Nevanlinna Prize at the 1998 International Congress of Mathematicians in Berlin What Are Qubits The most important concept in connection with quantum computation is that of the quantum bit or qubit The term is meant to be reminiscent of the bits of everyday computers where a bit is a unit of information storage that can assume one of two values thought of as zero or one Billions of bits are linked together for carrying out complex calculations A qubit is then the quantum computation analogue of the bit One can imagine a qubit as a black box that on receiving a query replies with a zero or a one What is known are the probabilities with which the box will return each of the two values In this sense a classical bit is a specialized qubit namely a qubit about which one is certain whether the output will be a zero or a one This probabilistic definition reflects the fact that the nanoworld is governed by probability not certainty It is only through measurement that it is decided which of the possible values will be concretely realized However the image of the qubit as a black box is inadequate for describing the interaction of multiple qubits For a more refined picture we must imagine that the probability for the output of a one or a zero is determined by a pair of arrows in the plane the square of the length of the arrow labeled 1 gives the probability of a query being answered with a one For example if the length of the arrow labeled 1 is 0 8 then the probability of a one is 0 8 0 8 0 64 Clearly then the probability of a zero is 1 0 64 0 36 and the arrow labeled 0 will have length 0 6 since 0 6 0 6 0 36 In the picture a qubit is shown whose probabilities of zero and one are about equal and so they may be thought of as representing a coin that is tossed to determine whether 0 or 1 is to be output A qubit has direction as well as length and what makes things interesting is that the directions of the zero and one probabilities are independent The interaction of two qubits is represented by their

Original URL path: http://mathematics-in-europe.eu/tr/48-information/research-mathinside/293-how-do-quanta-compute (2013-11-18)

Open archived version from archive - Mathematics In Europe - Quadratic function in elementary economics

article is sold at price P and a and b are positive constant a representing the limit quantity that would be demanded if price were 0 and b expressing the decrease in demand per unit increase in price For example the demanded daily number of say some delicatesse sausages on their price P in say could be Q 80 1 6 P meaning that for each increase of price of 1 b 1 sausage less will be demanded per day and as the price falls the daily demand rises up to a 80 The corresponding graph is note that for prices larger than 50 the demand would be negative which makes no sense so the only sensible part of graph is for prices betweem 0 and 50 euros Say you are selling an article at price P Supposing that your resources suffice for covering the demnad your revenue will be QP for each of the Q articles you will have income P On the other side the productions costs some money The costs will be the sum of fixed costs F independent on the number of produced articles e g the costs of just having your machines run plus QC where C is the production cost of one article In our sausage example the cost of producing one sausage could be say C 0 1 and the daily fixed costs of running the production of sausages could be say F 150 the total daily cost for producing Q sausages would be 150 0 1 Q euros The corresponding graph is taking into account that the demand will never be less than 0 and never more than 80 sausages Taking all this into account you can calculate your profit π i e revenue minus costs π QP QC F Substituting Q

Original URL path: http://mathematics-in-europe.eu/tr/48-information/research-mathinside/994-quadratic-function-in-elementary-economics (2013-11-18)

Open archived version from archive - Mathematics In Europe - Ideal Proportions (by Daniela Della Volpe)

be expressed as a i Phi a i 1 where Phi frac 1 sqrt 5 2 1 61803399 represents the golden section Our reader noticed that the two series of numbers on the banknote resemble the Fibonacci series 1 1 2 3 5 8 13 that is the series of natural numbers f i constructed recursively according to the rule f 1 f 2 1 f i 2 f i f i 1 Bear in mind that if the series is constructed not starting with 1 1 but with any two numbers whatsoever then we are speaking of a generalized Fibonacci series and the two lists of numbers on the banknote could be two generalized series of this kind but if that were the case then in effect there would be an error In fact for the red series we have 43 70 113 43 70 183 70 113 296 113 183 and for the blue series things could work out right if 54 were used instead of 53 54 86 140 54 86 226 86 140 Our reader concluded that the blue series was wrong But The number Phi is effectively related to the Fibonacci series the ratio between two successive terms f i 1 f i of the series does in fact approach Phi as i approaches infinity And this is also true when we consider generalized Fibonacci series However it is not true that given any two successive numbers of the series their ratio is the golden number rather their ratio approaches Phi ever more closely as the series goes forward And this is where our reader has made her mistake On the other hand if the ratios between the numbers of the Fibonacci series were exactly equal to Phi then the golden section would be a rational number since it would be a ratio of two whole numbers and that is false In effect the numbers shown on the banknote are not calculated according to the structure of the Fibonacci series but are rather obtained by multiplications and divisions of Phi beginning with the numbers 226 the blue sequence and 113 the red sequence Here is how this is done 226 Phi 139 67 140 139 67 Phi 86 32 86 86 32 Phi 53 35 53 113 Phi 182 83 183 182 83 Phi 295 83 296 113 Phi 69 83 70 69 83 Phi 43 16 43 But why precisely 226 and 113 The numbers shown on the banknote constitute two famous series Le Corbusier s série rouge and série bleue which are associated to the subdivisions of the drawing in red and blue that appears there This device goes by the name of Modulo r We can think of the Modulor as a modern Vitruvian Man like that of Leonardo da Vinci What it represents is an anthropometric scale of the ideal proportions of the human body The number 226 the last in the blue series in centimeters supposedly corresponds to the height

Original URL path: http://mathematics-in-europe.eu/tr/48-information/research-mathinside/999-ideal-proportions-by-daniela-della-volpe (2013-11-18)

Open archived version from archive - Mathematics In Europe - 1913: Publication of The Concept of a Riemann Surface, by Hermann Weyl

was already defined obtaining values however of the function that differ from those originally obtained at the same points There are two apparent ways to deal with the problem Either one gives up on the idea of uniqueness of function values or else one alters the domain of definition in such a way that it covers the complex plane several perhaps infinitely many times the individual layers being joined in a complicated way somewhat analogously to how the floors of a parking garage are linked Before the appearance of Weyl s book this concept was implemented intuitively in simple cases being exemplified by cardboard models with the multiple layers glued one to the next I well remember that the construction of such models was among the problems in complex analysis that I had to solve during my student years back in the 1950s The essence of Weyl s book was to describe such surfaces called Riemann surfaces after the first mathematician to describe them axiomatically as two dimensional manifolds and to study complex valued functions on them without reference to how they were originally constructed and without reference to their embedding in a three dimensional space which ultimately led to the characterization of surfaces by the functions that exist on them Closed surface of genus 3 Nonorientable closed surface Consequently the book consists of two parts called chapters by Weyl In the first part Riemann surfaces are defined and studied using topological concepts and methods where it is to be kept in mind that before 1913 much was not clearly understood in this area with most ideas resting on intuition Thus this chapter is of considerable independent interest even without reference to functions of a complex variable for the development of the topology of surfaces The second chapter is devoted

Original URL path: http://mathematics-in-europe.eu/tr/anasayfa/47-information/math-calendar/971-1913-publication-of-the-concept-of-a-riemann-surface-by-hermann-weyl (2013-11-18)

Open archived version from archive