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...ically and intellectually. There are around 30,000 patients in PVS in the USA. The numbers are far lower in Europe and Japan, partly due to the red... ...s not the only confusing difference between the USA and Europe. The hierarchy of numbers is universal: million, billion, trillion, quadrillion, q... ...Pythagoras, Kepler, and Penrose to current mathematicians dedicated to studying the Fibonacci numbers (permutations of phi). Artists like Goethe, C... ... Kepler, and Penrose to current mathematicians dedicated to studying the Fibonacci numbers (permutations of phi). Artists like Goethe, Cezanne, Bar... .... If you divide phi by one you get 0.6180339887 - that is phi minus 1. Leonardo Fibonacci (1170-1240) developed his famous series in the 12th ce... ...the division are phi-asymptotic). http://www.mcs.surrey.ac.uk/Personal/R.Knott/Fibonacci/phi.html http://jwilson.coe.uga.edu/emt669/Student.F...

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...where SPS(3) = 135642. 0 Question Is there any perfect power among these numbers? I. e. are there integers m, k and n such that m k = SPS(n) Wit... ...vior is typical and provides a powerful proof mechanism when the terminal numbers are known to be fixed. From this, the following lemma is obvious. ... ...is the smallest number k such that nk is a perfect square. The first few numbers in the sequence are 1,2,3,1,5,6,7,2,10,11,3,14,15,1,17, 0 28. Sma... ... is the smallest integer k such that nk is a perfect cube. The first few numbers in this sequence are 1,4,9,2,25,36,49,1,3,100,121,. 0 The computa... ...re easily determined. Theorem 2: The fixed points of SSC(n) are I and all numbers where every prime factor is to the first power Proof: Easily SSC(... ...49639733405162203 and found only the two solutions already mentioned The Fibonacci sequence has a similar definition F(O) = 0, F(I) = L F(n~2) = F(... ... F(O) = 0, F(I) = L F(n~2) = F(n-'-l) -'- F(n) and the first three digit Fibonacci number is F(12) = 144 = 89 -,- 55. It is easy to modify the fol... ...lowing two programs to search for solutions to the given problem that are Fibonacci numbers. Simply modify the initial lines to 38 lucascount= 12 ...

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...lae for n-thtermoftwinprimes 98 A3. Explicit formulae for the n-th term of the twin prime sequence 107 5 6 A4. Some explicit formulae for the composit... ...ic function 124 References 127 Chapter 1 On Some Smarandache's problems In the text below the following notations are used. N - the set of all natura... ...+1)+1) 2 : (4) On the other hand, it is easy to see, that in (2) the number p 8n+1¡1 2 is an integer if and only if n= m(m+1) 2 : Also, for any natura... ...ifk>(s(k)) 2 +4s(k)+2 where a k;i are given in an explicit form by (5). 17 The 10-th Smarandache problem is dual to the above one: Smarandache in¯nit... ... +4s(k)+2 where a k;i are given in an explicit form by (8). 21 The 11-th Smarandache problem is a modi¯cation of the 9-th problem: Smarandache in¯nit... ... ±(k)¡jij ; if ±(k)¸jij ; (14) ifk>(s(k)) 2 +4s(k)+2,then a k;i = 8 < : 0; if "(k)<jij F "(k)¡jij ; if "(k)¸jij ; (15) where F m (m=0;1;2;:::)isthem-t... ...for 1·i·k,thenwesaythat [c l+1 ;c l+2 ;:::;c l+k ] is a base of the sequence c 1 ;c 2 ;:::with a length kandwithrespect to function Ã. For example, th...

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...based series: an even-number (α-particle) progression in nucleo protons and Fibonacci sequence in the outer proton, (octet and face) sites. The proj... .../2 , (33b) where: η is a stable nucleus, {n I , p j } are the numbers of neutrons and protons in the nucleus; T 1/2 is the half-life ... ...s such, no resolution is obtained given the MT nucleus, where near-infinite numbers of strong and weak force carriers, pions, quarks, gluons, etc., ...

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... then ask the same question regarding the number of primes in the sequence. That appears to be a question just as difficult as the one for base ten numbers, and is summarized in the following statement....

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...Contents.4 -- Chapter 1 Sequences Formed by Concatenating All the Positive Integers.5 -- Chapter 2 Sequences Formed by Concatenating Selected Natural Numbers .44 -- Chapter 3 Smarandache Stereograms .70 3.1 -- The Stereogram.70 -- 3.2 Altering the Background.87 -- 3.3 Additional Stereograms .91 -- 3.4 Possibilities for Additional Figures .92 -- Chapter 4 Constants Involvin...

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...rs …………………. 5 Chapter 2 Sequences Formed by Concatenating Selected Natural Numbers …………………. 44 Chapter 3 Smarandache Stereograms ………………….. 70 3.1 The ... ... of Java. It is an extended precision data type that allows for very large numbers to be represented and manipulated. The only limitations to using th... ...randache asked the question: How many primes are there among these numbers? This is an unsolved problem, although clearly they will be very ra... ... 12101112, 1210111220, . . . which are equivalent to the following decimal numbers 6 1, 5, 48, 436, 3929, 35367, 318310, 2864798, . . . The first el... ...which is a perfect cube. Figure 3 66 Figure 4 5) Smarandache Concatenated Fibonacci sequence 1, 11, 112, 1123, 11235, 112358, 11235813, . . . whe... ... . . . where element k is formed by concatenating on the right the first k Fibonacci numbers. Smarandache then asked the question: Are any of the numb... ...ndache then asked the question: Are any of the numbers in the Concatenated Fibonacci sequence (other than the first) a Fibonacci number? The answer t... ... almost certainly no. A program was written to examine the elements of the Fibonacci sequence looking for the first one whose prefix matches the first... ... whose prefix matches the first few digits of the Smarandache Concatenated Fibonacci sequence. For example, the first element of the Fibonacci sequenc...

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... Number theory. The material is divided into ve chapters: Geometric theorems; Diophantine equations; Arithmetic functions; Divisibility properties of numbers and functions; and Some irrationality results. Chapter 1 deals essentially with geometric inequalities for the remarkable elements of triangles or tetrahedrons. Other themes have an arithmetic character (as 9-12) on n...

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...s; Diophantine equations; Arithmetic functions; Divisibility properties of numbers and functions; and Some irrationality results. Chapter 1 deals esse... ...sibility problems or questions related especially to the sequence of prime numbers. The author has solved various conjectures by Smarandache, Bencze, ... ...metic functions . . . . . 175 18 On multiplicatively de cient and abundant numbers . . . . . . . . . . . . . 179 19 On values of arithmetical function... ...ctions . . . . . . . . . . . . . 225 Chapter 4. Divisibility properties of numbers and functions 227 1 On a divisibility property . . . . . . . . . . ... ... . . . . . . . . . . . . . . 232 4 On a conjecture of Smarandache on prime numbers . . . . . . . . . . . . . 234 5 On consecutive primes . . . . . . .... ...ays even for n 3. Relation (2) is true for f(a) = 2 a 1 andf(a) =F(a) = Fibonacci number of ordera (see Applications of Theorem 2). Clearly (2) ) (... ... 1 ; B = (u b ) y 1 +::: + (v b ) y 1 : Application 2. Let f(a) =F(a) = Fibonacci number of order a. These numbers are de ned by F(1) =F(2) = 1,F(n... ....2, 4.3, 5.1, 5.6 F P. Fermat 1.8, 1.10, 2.2, 2.8, 2.12, 3.15, 4.2, 4.6 L. Fibonacci 3.22 S. Florit a 2.7 E. Fouvry 4.10 A. Froda 5.4 G K.F. Gauss ...

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... of Civil Engineering and Architecture, Beijing, 100044, P.R.China E-mail: maolinfan@163.com Abstract: An interesting symmetry on multiplication of numbers found by Prof.Smarandache recently. By considering integers or elements in groups on graphs, we extend this symmetry on graphs and find geometrical symmetries. For extending further, Smarandache’s or combinatorial sys...

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... . . . . . . . . . 33 Elementary Abelian Regular Coverings of Cube BY FURONG WANG AND LIN ZHANG. . . . . . . . . . . . . . . . . . . . . .49 Super Fibonacci Graceful Labeling of Some Special Class of Graphs BY R.SRIDEVI, S.NAVANEETHAKRISHNAN AND K.NAGARAJAN. . . . . . . . 59 Surface Embeddability of Graphs via Tree-travels BY YANPEI LIU. . . . . . . . . . . . . . . ....

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...ne (V) to the study of some conjectures and open questions proposed always by F. Smarandache. In particular we will analyse some conjectures on prime numbers and the generalizations of Goldbach conjecture. This book would be a telescope to explore and enlarge our knowledge on the Smarandache's universe. So let's start our observation....

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...by F. Smarandache. In particular we will analyse some conjectures on prime numbers and the generalizations of Goldbach conjecture. This book would be ... ... ) ( n n n n n n a n n Zt p and as known the sum of reciprocals of natural numbers diverges. Conjecture 1.1.2 K 8737 . 0 ) ( 1 » » = a where N ... ...e range of Zt(n) function is N-{0} where N is the set of positive integers numbers. Proof. The theorem is a direct consequence of Walfistz result [3]... ...d radical [6] for the Pseudo- Smarandache and Pseudo-Smarandache-Totient numbers. Problem 37. Is the number 0.1243549107585.... where the sequence... ... b(n) the SACRP sequence. 6) Smarandache alternate consecutive and reverse Fibonacci sequence 1, 11, 112, 3211, 11235, 853211, 11235813, 2113853211... ...32134, ...... Again as for the sequences #4 and #5 but this time using the Fibonacci sequence. • Evaluate ) 1 ( ) ( lim + ¥ fi n a n a n . Is this... ...n a n a n . Is this limit convergent? • Is any terms of that sequence a Fibonacci number itself? • Is this sequence a Smarandache A and C sequence ... ...y by the alternate concatenation of terms of a(n). 8) Smarandache additive Fibonacci sequence 1, 1, 2, 3, 5, 8, 21, 233, 317811, 3524578, ............ ...ce 1, 1, 2, 3, 5, 8, 21, 233, 317811, 3524578, ......... Sequence of Fibonacci numbers whose digits sum is a Fibonacci number too. • Let <N+> be...

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... Someday our earth may be burdened with people (but I will not be there). Numbers cheapen us. Collective folly weakens us. Man and art drift apart.... ...i- ble...horses...rain...all night the two of us rode through the rain. Fibonacci’s dog-eared book, Liber Abaci, still interests me: what tattered ... ... preceding num- bers, continuing ad infinitum. And it is true I can divide Fibonacci’s number (after the fourteenth in his sequence) by the next high... ...ur country. Lincoln: 1,866,452 votes Douglas: 1,376,957 votes Those numbers are printed in my mind’s eye. I am proud that I beat Stephen Dougl... ...ied their black cattle by the million; and are every day increasing their numbers, and extending their cattle field into the wilderness...” I respo...