Numerele naturale au fascinat dintotdeauna omenirea, ce le-a considerat, pe bună dreptate, ca fii... more Numerele naturale au fascinat dintotdeauna omenirea, ce le-a considerat, pe bună dreptate, ca fiind mai mult decât mijloace de a studia cantităţile, le-a considerat entităţi având o personalitate proprie. Mistica tuturor popoarelor abundă de proprietăţi supranaturale atribuite numerelor. Într-adevăr, pare că, pe măsură ce le cercetezi mai adânc, descoperi că au „consistenţă”, că, departe de a fi o creaţie conceptuală a omului, un simplu instrument la îndemâna sa, se conduc de fapt după legităţi proprii, pe care nu-ţi permit să le influenţezi, ci doar să le descoperi. Printre primii oameni care au simţit acest lucru se numără Pitagora, ce a început prin a cerceta numerele şi a sfârşit prin a întemeia o mişcare religioasă puternic fondată pe simbolistica numerelor. Pasiunea sa pentru numerele naturale era atât de mare (accepta, totuşi, şi existenţa numerelor raţionale, ce sunt, în fond, tot un raport de numere naturale), încât circulă o legendă conform căreia şi-ar fi înecat un discipol pentru „vina” de a-i fi relevat existenţa numerelor iraţionale. Mult mai aproape pe scara istoriei, în secolul XIX, matematicianul german Leopold Kronecker este creditat a fi spus: „Dumnezeu a creat numerele naturale; toate celelalte sunt opera omului”. Departe de considerente de filozofie a matematicii, ne-am limitat enciclopedia la numerele întregi pentru că am considerat că este un domeniu îndeajuns de vast în sine pentru a face obiectul unei astfel de lucrări. Am împărţit enciclopedia în două părţi, „Clase de numere” (se subînţelege, întregi), respectiv „Clase de prime şi pseudoprime”, prima cuprinzând principalele clase de numere cu care se operează actualmente în teoria numerelor (ramura matematicii ce studiază în principal numerele întregi), cea de-a doua câteva tipuri consacrate de prime (numere care au „sfidat” dintotdeauna matematicienii prin rezistenţa lor în a se lăsa înţelese şi ordonate) şi tipurile cunoscute de pseudoprime (o categorie aparte de numere, ce împart însă multe atribute cu numerele prime). Am încercat să folosim noţiuni cât mai simple şi operaţii elementare pentru a nu îndepărta cititorii prin simboluri şi denumiri de funcţii; din acelaşi motiv am definit toate clasele de numere doar în sistemul comun, zecimal, şi nu am considerat clasele de numere care, deşi întregi, se definesc apelând la numere iraţionale sau complexe.
In two of my previous published books, “Two hundred conjectures and one hundred and fifty open pr... more In two of my previous published books, “Two hundred conjectures and one hundred and fifty open problems on Fermat pseudoprimes”, respectively “Conjectures on primes and Fermat pseudoprimes, many based on Smarandache function”, I already expressed my passion for integer numbers, especially for primes and Fermat pseudoprimes, fascinating numbers that seem to be a little bit more willing to let themselves ordered and understood than the prime numbers. This book brings together sixty-two papers on prime numbers, many of them supporting the author’s belief, expressed above, namely that new ordered patterns can be discovered in the “undisciplined” set of prime numbers, observing the ordered patterns in the set of Fermat pseudoprimes, especially in the set of Carmichael numbers, the absolute Fermat pseudoprimes, and in the set of Poulet (sometimes also called Sarrus) numbers, the relative Fermat pseudoprimes to base two. Other papers, which are not based on the observation of Fermat pseudoprimes, are based on the observation of Mersenne numbers, Fermat numbers, Smarandache generalized Fermat numbers, and other well known or less known classes of integers which are very much related with the study of primes. Part One of this book of collected papers contains two hundred and thirteen conjectures on primes and Part Two of this book brings together the articles regarding primes, submitted by the author to the preprint scientific database Vixra, representing the context of the conjectures listed in Part One, papers regarding squares of primes, semiprimes, twin primes, sequences of primes, types of duplets or triplets of primes, special classes of composites, ways to write primes, formulas for generating large primes, generalizations of the twin primes and de Polignac’s conjecture, generalizations of Cunningham chains and Fermat numbers and many other classic issues regarding prime numbers. Finally, in the last eight from these collected papers, I defined a new function, the MC function, and I showed some of its possible applications (for instance, I conjectured that for any pair of twin primes p and p + 2, where p ≥ 5, there exist a positive integer n of the form 15 + 18*k such that the value of Smarandache function for n is equal to p and the value of MC function for n is equal to p + 2, I also made a Diophantine analysis of few Smarandache type sequences using the MC function).
About the works of Florentin Smarandache have been written a lot of books (he himself wrote dozen... more About the works of Florentin Smarandache have been written a lot of books (he himself wrote dozens of books and articles regarding math, physics, literature, philosophy). Being a globally recognized personality in both mathematics (there are countless functions and concepts that bear his name) and literature, it is natural that the volume of writings about his research is huge. What we try to do with this encyclopedia is to gather together as much as we can both from Smarandache’s mathematical work and the works of many mathematicians around the world inspired by the Smarandache notions. Because this is too vast to be covered in one book, we divide encyclopedia in more volumes. We are going to structure this volume of encyclopedia in six parts: the first will cover the Smarandache type sequences and series (obviously, among them there are the well-known sequences of numbers obtained through concatenation but also numerous other sequences), the second part will cover the Smarandache type functions and constants, the third part will cover the conjectures on Smarandache notions and the conjectures on number theory due to Florentin Smarandache, the fourth part will cover the theorems on Smarandache notions and the theorems on number theory due to Florentin Smarandache, the fifth part will cover the criteria, formulas and algorithms for computing due to Florentin Smarandache and the sixth part will cover the unsolved problems regarding Smarandache notions and the open problems on number theory due to Florentin Smarandache. Obviously, the division into these chapters has mostly the role to organise the matters treated, not to delineate them one from another, because all are related; for instance, a function treated in chapter about functions may create a sequence treated in chapter about sequences or a conjecture about primes treated in the chapter about primes may involve a diophantine equation, though these ones have their own chapter. Similarly, we presented some conjectures, theorems and problems on sequences or functions in the chapters dedicated to definition of the latter, while we presented other conjectures, theorems and problems on the same sequences or functions in separate chapters; we could say we had a certain vision doing so (for instance that we wanted to keep a proportion between the sizes of the sections treating different sequences or functions and not to interrupt the definitions between two related sequences or functions by a too large suite of problems) but it would not be entirely true: the truth is that a work, once started, gets its own life and one could say that almost it dictates you to obey its internal order.
Part One of this book of collected papers aims to show new applications of Smarandache function i... more Part One of this book of collected papers aims to show new applications of Smarandache function in the study of some well known classes of numbers, like prime numbers, Poulet numbers, Carmichael numbers, Sophie Germain primes etc. Beside the well known notions of number theory, we defined in these papers the following new concepts: “Smarandache-Coman divisors of order k of a composite integer n with m prime factors”, “Smarandache-Coman congruence on primes”, “Smarandache-Germain primes”, “Coman-Smarandache criterion for primality”, “Smarandache-Korselt criterion”, “Smarandache-Coman constants”. Part Two of this book brings together several articles regarding primes, submitted by the author to the preprint scientific database Vixra. Apparently heterogeneous, these articles have, objectively speaking, a thing in common: they are all directed toward the same goal – discovery of new ordered patterns in the “undisciplined” set of the prime numbers, using the same means – the old and reliable integer numbers. Subjectively speaking, these papers have, of course, another thing in common: the patterns coming from the very mathematical thoughts and obsessions of the author himself: such “mathematical thoughts and obsessions” are: trying to find correspondends of the patterns found in sequences of Fermat pseudoprimes (a little more “disciplined” class of numbers than the class of primes) in sequences of prime numbers; trying to find chains of consecutive or exclusive primes defined by a recurrent formula; trying to show the importance of classification of primes in eight essential subsets (the sequences of the form 30*k + d, where d has the values 1, 7, 11, 13, 17, 19, 23, 29) etc. This collection of articles seeks to expand the knowledge on some well known classes of primes, like for instance Sophie Germain primes, but also to define new classes of primes, like for instance “ACPOW chains of primes”, or classes of integers directly related to primes, like for instance “chameleonic numbers”.
Prime numbers have always fascinated mankind. For mathematicians, they are a kind of “black sheep... more Prime numbers have always fascinated mankind. For mathematicians, they are a kind of “black sheep” of the family of integers by their constant refusal to let themselves to be disciplined, ordered and understood. However, we have at hand a powerful tool, insufficiently investigated yet, which can help us in understanding them: Fermat pseudoprimes. It was a night of Easter, many years ago, when I rediscovered Fermat’s "little" theorem. Excited, I found the first few Fermat absolute pseudoprimes (561, 1105, 1729, 2465, 2821, 6601, 8911…) before I found out that these numbers are already known. Since then, the passion for study these numbers constantly accompanied me. Exceptions to the above mentioned theorem, Fermat pseudoprimes seem to be more malleable than prime numbers, more willing to let themselves to be ordered than them, and their depth study will shed light on many properties of the primes, because it seems natural to look for the rule studying it’s exceptions, as a virologist search for a cure for a virus studying the organisms that have immunity to the virus. In this book I gathered together 30 of my articles posted on VIXRA about Poulet numbers (Fermat pseudoprimes to base 2), Carmichael numbers (absolute Fermat pseudoprimes), Fermat pseudoprimes to base 3 and other relative Fermat pseudoprimes, also 30 sequences of such numbers posted by me on OEIS and 150 open problems regarding, of course, these numbers, problems listed in the last part, Part four, of this book. I titled the book in this way to show how many new and exciting things one can say more about this class of numbers, but, though indeed these collected papers contain 200 conjectures about Fermat pseudoprimes, listed in the Part one of this book, these collected papers contain also many observations about the properties of Fermat pseudoprimes and generic formulas for many subclasses of such numbers. Also, among these 200 conjectures there are some conjectures concerning strictly prime numbers, more precisely few types of primes which have arisen in the study of Fermat pseudoprimes.
In three of my previous published books, namely “Two hundred conjectures and one hundred and fift... more In three of my previous published books, namely “Two hundred conjectures and one hundred and fifty open problems on Fermat pseudoprimes”, “Conjectures on primes and Fermat pseudoprimes, many based on Smarandache function” and “Two hundred and thirteen conjectures on primes”, I showed my passion for conjectures on sequences of integers. In spite the fact that some mathematicians stubbornly understand mathematics as being just the science of solving and proving, my books of conjectures have been well received by many enthusiasts of elementary number theory, which gave me confidence to continue in this direction. Part One of this book brings together papers regarding conjectures on primes, twin primes, squares of primes, semiprimes, different types of pairs or triplets of primes, recurrent sequences, sequences of integers created through concatenation and other sequences of integers related to primes. Part Two of this book brings together several articles which present the notions of c-primes, m-primes, c-composites and m-composites (c/m-integers), also the notions of g-primes, s-primes, g-composites and s-composites (g/s-integers) and show some of the applications of these notions (because this is not a book structured unitary from the beginning but a book of collected papers, I defined the notions mentioned in various papers, but the best definition of them can be found in Addenda to the paper numbered tweny-nine), in the study of the squares of primes, Fermat pseudoprimes and generally in Diophantine analysis. Part Three of this book presents the notions of “Coman constants” and “Smarandache-Coman constants”, useful to highlight the periodicity of some infinite sequences of positive integers (sequences of squares, cubes, triangular numbers, polygonal numbers), respectively in the analysis of Smarandache concatenated sequences. Part Four of this book presents the notion of Smarandache-Coman sequences, id est sequences of primes formed through different arithmetical operations on the terms of Smarandache concatenated sequences. Part Five of this book presents the notion of Smarandache-Coman function, a function based on the well known Smarandache function which seems to be particularly interesting: beside other characteristics, it seems to have as values all the prime numbers and, more than that, they seem to appear, leaving aside the non-prime values, in natural order.
The definition of “concatenation” in mathematics is, according to Wikipedia, “the joining of two ... more The definition of “concatenation” in mathematics is, according to Wikipedia, “the joining of two numbers by their numerals. That is, the concatenation of 69 and 420 is 69420”. Though the method of concatenation is widely considered as a part of so called “recreational mathematics”, in fact this method can often lead to very “serious” results, and even more than that, to really amazing results. This is the purpose of this book: to show that this method, unfairly neglected, can be a powerful tool in number theory. In particular, as revealed by the title, I used the method of concatenation in this book to obtain possible infinite sequences of primes. Part One of this book, “Primes in Smarandache concatenated sequences and Smarandache-Coman sequences”, contains 12 papers on various sequences of primes that are distinguished among the terms of the well known Smarandache concatenated sequences (S sequences) but also on “Smarandache-Coman sequences of primes” (SC sequences), defined by the author as “all sequences of primes obtained from the terms of Smarandache sequences using any arithmetical operation”: the SC sequences presented in this book are related, of course, to concatenation, but in three different ways: the S sequence is obtained by the method of concatenation but the operation applied on its terms is some other arithmetical operation; the S sequence is not obtained by the method of concatenation but the operation applied on its terms is concatenation, or both S sequence and SC sequence are using the method of concatenation. Part Two of this book, “Sequences of primes obtained by the method of concatenation”, brings together 51 articles which aim, using the mentioned method, to highlight sequences of numbers that are rich in primes or are liable to lead to large primes. The method of concatenation is applied to different classes of numbers, e.g. squares of primes, Poulet numbers, triangular numbers, reversible primes, twin primes, repdigits, factorials, primorials, in order to obtain sequences, possible infinite, of primes. Part Two of this book also contains a paper which lists a number of 33 sequences of primes obtained by the method of concatenation, sequences presented and analyzed in more detail in my previous papers, gathered together in five books of collected papers: “Two hundred conjectures and one hundred and fifty open problems on Fermat pseudoprimes”, “Two hundred and thirteen conjectures on primes”, “Conjectures on primes and Fermat pseudoprimes, many based on Smarandache function”, “Sequences of integers, conjectures and new arithmetical tools”, “Formulas and polynomials which generate primes and Fermat pseudoprimes”.
To make an introduction to a book about arithmetic it is always difficult, because even most appa... more To make an introduction to a book about arithmetic it is always difficult, because even most apparently simple assertions in this area of study may hide unsuspected inaccuracies, so one must always approach arithmetic with attention and care; and seriousness, because, in spite of the many games based on numbers, arithmetic is not a game. For this reason, I will avoid to do a naive and enthusiastic apology of arithmetic and also to get into a scholarly dissertation on the nature or the purpose of arithmetic. Instead of this, I will summarize this book, which brings together several articles regarding primes and Fermat pseudoprimes, submitted by the author to the preprint scientific database Research Gate. Part One of this book, “Sequences of primes and conjectures on them”, brings together thirty-two papers regarding sequences of primes, sequences of squares of primes, sequences of certain types of semiprimes, also few types of pairs, triplets and quadruplets of primes and conjectures on all of these sequences. There are also few papers regarding possible methods to obtain large primes or very large numbers with very few prime factors, some of them based on concatenation, some of them on other arithmetic operations. It is also introduced a new notion: “Smarandache-Coman sequences of primes”, defined as “all sequences of primes obtained from the terms of Smarandache sequences using any arithmetical operation” (for instance, the sequence of primes obtained concatenating to the right with the digit one the terms of Smarandache consecutive numbers sequence). Part Two of this book, “Sequences of Fermat pseudoprimes and conjectures on them”, brings together seventeen papers on sequences of Poulet numbers and Carmichael numbers, i.e. the Fermat pseudoprimes to base 2 and the absolute Fermat pseudoprimes, two classes of numbers that fascinated the author for long time. Among these papers there is a list of thirty-six polynomials and formulas that generate sequences of Fermat pseudoprimes. Part Three of this book, “Prime producing quadratic polynomials”, contains three papers which list some already known such polynomials, that generate more than 20, 30 or even 40 primes in a row, and few such polynomials discovered by the author himself (in a review of records in the field of prime generating polynomials, written by Dress and Landreau, two French mathematicians well known for records in this field, review that can be found on the web address <http://villemin.gerard.free.fr/Wwwgvmm/Premier/formule.htm>, the author – he says this proudly, of course – is mentioned with 18 prime producing quadratic polynomials). One of the papers proposes seventeen generic formulas that may generate prime-producing quadratic polynomials.
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Books by Marius Coman