All or almost on the prime numbers, these numbers divisible only by one and themselves (theorems, demonstrations, problems, anecdotes). What one already knows about them, which one is unaware of or which one does not manage to show. Their applications (for example the coding of information) and their central place into arithmetic. With them, they is 2000 years of history of mathematics which one visits.

Prime numbers, these numbers without other factors that one and seems not to conceal any mystery, one fails to find a regularity unspecified in their succession. Known as of the beginnings of arithmetic, the prime numbers have excited the curiosity of thousands of mathematicians. They are in the heart of the science of the numbers, because entire breaks up in a single way into a product of factors first. They are also at the origin of some of the most difficult problems of mathematics and acquired, with progress of cryptography, a considerable economic importance.

In this work, the author mixes theoretical explanations and prickly anecdotes, in order to restore all the colors of the chatoyant universe of the prime numbers.

Jean-Paul Delahaye is a professor of data processing at the University of sciences and technologies of Lille and researcher at the Laboratory of fundamental data processing of CNRS, in Lille.

Code P017 - 155 F - 336 pages

Foreword

"the mathematicians tried, in vain so far, to discover a regularity in the continuation of the prime numbers, and we have good reasons to believe that there is a mystery which the human spirit will not never penetrate. It is enough besides, to be convinced of it, to throw a glance on a table of prime numbers (that some took the trouble to calculate until several hundreds of thousands); one is then convinced instantaneously that it reigns there neither command nor rule. "

Leonard Euler (1707-1783)

"the problem of the distinction between prime numbers and numbers composed, and that of the decomposition of a number in product of factors first are most significant and most useful of all the arithmetic one. [... ] the honor of science seems to require that one cultivate with zeal any progress in the solution of these elegant and famous questions. "

Carl Friedrich Gauss (1777-1855)

Very early, as of the first sharings of toys or delicacies, one learns that certain integers, such 6 = 2 X 3, " break " easily in two factors . On the other hand, one will never manage to break up thus numbers 2, 3, 5, 7... These numbers are named first. Euler and Gauss, two of the largest mathematicians of all times, had included/understood the importance of the prime numbers well, like their mystery. The prime numbers have a central importance into arithmetic, because any number breaks up in a single way into product of one or several factors first (150 = 2 X 3 X 5 X 5; 7 = 7). As for their mystery, one perceives it by considering the beginning of their continuation, for example the 25 prime numbers lower than 100:

2 3 5 7 11 13 17 19 23 29 31 37 41 43 47 53 59 61 67 71 73 79 83 89 97.

Thus try to predict the differences between these numbers! Null rule does not seem to control the succession of the prime numbers.

As of the access, one has a presentiment of that the world of the prime numbers is infinitely rich. I propose to you to guide yourselves in the explored parts of this world, to the edge of the unknown, where the mathematicians are reduced to the assumptions, for once stripped of the weapon of the proof. Only quality necessary for this voyage is curiosity. I will make sothat it amuses you (the entertainment was always one of the engines of arithmetic) while informing you of a little mathematics and their history.

Moreover, you will discover, behind the variegated fauna of the prime numbers, the applications become crucial for the development of data processing and the modern communications.

The voyage holds many surprises, because humanity, intrigued since millenia by these basic numbers (it is the direction of the adjective first), acquired thousand knowledge on them. The prime numbers are of an unsuspected daily utility, and are in the heart of a broad range of techniques and products, systems design of corrections of errors in the computers with the encoding of information or the development of effective algorithms in fields such as the processing of the signal or numerical calculation.

To the school, one enters surreptitiously the universe of the prime numbers by learning simplification from the fractions: for example, how to write most simply possible 150/275? The receipt is known: one starts by writing that 150 = 2 X 3 X 5 X 5 and that 275 = 5 X 5 X 11. Fraction 150/275 becomes: (2 X 3 X 5 X 5) / (5 X 5 X 11). The simplification of the 5 gives 6/11 then, fraction more pleasant to use than the first (especially if you must share a ground or cut out a cake). These decompositions thanks to which one simplifies the fractions let foresee that the prime numbers, these "elementary particles " of the numerical universe, play a significant role in all kinds of situations.

The prime numbers are in an infinite number, like had already shown it Euclide by proving that there is not " greater prime number ". One will thus not be right of them by a simple enumeration. Within sight of the irregularity of the beginning of their list, underlined in introduction, one suspects that it is not convenient to know them in detail, nor to obtain precise information on their distribution. This fear is justified: many conjectures about the prime numbers remain irresolute, and the results established on their subject were it generally only at the end of considerable efforts, sometimes continued during several centuries, even several millenia. Thus one is still unaware of, in spite of significant efforts of search, if there is an infinity of prime numbers " twins ", i.e. for couples of prime numbers separated by two units, such as 3 and 5, or 17 and 19.

The prime numbers know a renewed interest since one found properties to them useful for which wishes to transmit, code or hide information. Following the invention of the algorithm of cryptography rsa in 1978, search on the prime numbers, hitherto relegated to the field of mathematical curiosities, aroused a considerable interest. In 20 years, it was not contradicted, quite to the contrary: as the data-processing telecommunications networks wove their fabrics everywhere, on the Earth and in space, on started to use the prime numbers larger than all those which were known 20 years ago (it is not rare - and it is even advised today in certain applications - to use prime numbers of more than 100 digits). Between the methods to find great prime numbers which, multiplied, will create the key of encoding (the " shield " to protect information), and the algorithms of factorization of the great numbers (" the sword " to decipher it), the war is ceaseless. It is capital from the economic and military point of view: if an agency of espionage succeeded in doing much better than the others as regards decomposition in factors first - it is perhaps the case without it being known it -, it would take an advantage determining on its enemies. Hundred other examples presented here will prove it to you: the world which becomes animated around the prime numbers is rich of history, topicality, future and, do not forget, of recreations.