You may be able to access this content by login via Shibboleth, Open Athens or with your Emerald account. If you would like to contact us about accessing this content, click the button and fill out the form. Contact us. To rent this content from Deepdyve, please click the button. Rent from Deepdyve. Abstract Purpose — In , Smarandache generalized the Atanassov's intuitionistic fuzzy sets IFSs to neutrosophic sets NS , and other researchers introduced the notion of interval neutrosophic set INSs , which is an instance of NS, and studied various properties. Findings — Relations on INSs and neutrosophic topology.
Practical implications — The main applications are in the mathematical field. My knowledge of this system is based on Sandy P. I reproduce Smarandache's axioms from this paper, with mild stylistic corrections. In the paper mentioned in ref.
Smarandache Notions Journal
However, only the 19 axioms of are denied explidtly. Indeed, negation of Axiom V. For instance, if you change the date of the model proposed below from 31 December to 31 December you obtain at once a second model which is not isomorphic with mine the total number of bank accounts in the United States was surely much less in than in The immediate consequence of this is that models of AntiGeometry can be readily found in all walks of life. I shall now state my interpretation of the undefined terms in Smarandache's and Hilbert's axioms and show, thereupon, that Smarandache's nineteen axioms come out true under this interpretation.
Following Chimienti and Bencze ref. Two lines are said to meet or intersect each other if they have a point in common. In my interpretation the geometrical terms employed in the axioms are made to stand for ordinary, non-geometric objects and relations, with which I assume the reader is familiar. As a matter of fact, Smarandache's system, despite its vaunted vanguardistic libertarianism, still imposes a few existential constraints on admissible models; for example, his Axiom III presupposes the existence of infinitely many of the objects called 'lines'.
This has forced me to introduce three existence postulates which my model is required to comply with, at least one of which is plainly unnatural EP3. I list below the meaning I bestow on Hilbert's property words: i A point is the balance in a particular checking account, expressed in U. Points will be denoted by capital letters.
You can also extend the domain of my model, in direct contradiction with Hilbert's Axiom V. I lighted on the model I shall present below while recovering from a long, delicious and calory-rich lunch with a poet, a psychiatrist and a philosopher, during which not a single word was said about geometry and I drank half a bottle of excellent Chilean merlot. Was this deviant usage adopted because "some of our lines are curves", as Chimienti and Bencze note in their definition of 'angle' follOwing their Axiom IV.
That would bespeak a deep misunderstanding of Hilbertian axiomatics. I hope that my interpretation will make this clear. In it, lines are persons, and we might just as well have called them straights. If A and B are the same point, we say that A and B are identical.
Lines are denoted by lower case italics. Here are the meanings I bestow on the relation words. All relations are supposed to hold at midnight E. Point A lies on line a if and only if person a owns the particular account that shows balance A. For brevity's sake, I shall often say that a owns balance A when he or she owns the said account. Line a lies on plane a if and only if the person a has a checking account with bank a. Point A lies on plane a if and only if the particular checking account that shows balance A is held with bank a.
Items take care of betweenness and the three kinds of incidence we find in Hilbert and Smarandache. Hilbert's relation of congruence does not apply, however, to points, lines or planes, but to two sorts of figures constructed from points and lines, viz. I must therefore define these figures in terms of my points and lines. By our definition of "betweenness", the points belonging to segment AB but not identical with A or B do not lie between A and B.
However, the Smarandache axioms are stated in such a way that none of them contradicts this surprising theorem. If a balance 0 is owned in common by persons hand k, the set formed by h, k and 0 is called the angle h,O,k symbolized LhOk. NOTE 1. NOTE 2. If hand k are distinct persons, such that h besides 0 owns a balance P, not shared with k, and k, besides 0, owns a balance Q, not shared with h, LhOk may be called "the angle PDQ" and be symbolized by LPOQ. In other words, the expression LPOQ" has a referent if and only if there exist persons hand k who respectively own balance P and balance Q separately from one another, and share the balance 0; otherwise, this expression has no referent.
Person a acquired balance A partly from person b if and only if a part of balance A was electronically transferred from funds owned by b to the account owned by a which shows balance A. We shall also need the following definitions:. Two distinct lines a and b are said to be parallel if and only if persons a and b have accounts with the same bank a. Let A be a balance belonging to a person h. Any other balances owned by h can be divided into three classes: i those that are less than A, ii those that are greater than A, and iii those that are equal to A Balances of class i and ii which are held by h in other accounts with the same bank where he has A will be said to lie, respectively, on one and on the other side of A on h.
As I said, the fairly weak but nevertheless inescapable constraints implicit in some of Smarandache's axioms force me to adopt three existence postulates. The first of these is highly plausiblei the second is, as far as I know, false in fact, but not implausiblei while the third is quite unnatural, though not more so than the supposition, involved in Smarandache's Axiom III, that there are infinitely many distinct objects in any model of his system. Existence postulates. John Dee has four checking accounts, with balances of , , and dollars, respectively.
EPI ensures the truth of Smarandache's Axiom There are some checking accounts for whose balance two different banks are held responsible. I shall refer to such accounts as two-bank accounts. We could be more spedfic and stipulate that checks drawn against such accounts will be cashed at the branches of either bank, that the banks share the maintenance costs and monthly service charges, etc.
But all such details are irrelevant for the stated purpose.. There exist infinitely many supernatural persons who may secretly own bank accounts, usually in common. EP3 is needed to take care of the last of the four situations contemplated in Smarandache's Axiom III the Axiom of Parallels , which involves a point that is intersected by infinitely many lines. In our model, this amounts to a balance in current account that is owned in common by infinitely many persons. EP3 is certainly weird, but not more so than say, the postulation of points, lines and a plane at infinity in projective geometry.
As in the latter case, we may regard talk of supernatural persons as a fafon de parler. EP3 will perhaps sound less unlikely if the banks of our model are Swiss instead of American. I shall now show that -with one partial exception I. As we shall see, the said exception is due to an inconsistency in Smarandache's axiom system. There is at least one line h and at least two distinct points A and B of h, such that A and B do not completely determine the line. Three points A, B, C, not on the same line, do not always completely determine a plane a. Three balances belonging to different persons may pertain to accounts they have with different banks.
Axiom 1. There is at least one plane a and at least three points A, B, C, which lie on a but not on the same line, such that A, B, C do not completely determine the plane a. However, according to EP2, the balances A, Band C may pertain to three two-bank accounts held, say, with bank a and bank p. In that case, D could belong to p and not to a. Let two points A, B of a line h lie on a plane a.
This does not entail that every point of h lies on a. Let two planes a and p have a point A in common. This does not entail that a and Phave another point B in common. Balance A could be the balance in the one and only two-bank account for which banks a and Pare jointly responsible see EP2. There exist lines on each one of which there lies only one point, or planes on each one of which there lie only two points, or a space which contains only three points. Nothing in our model precludes the joint fulfilment of the first two disjuncts in this axiom, viz. The third condition, however, cannot be fulfilled, for EP1 demands the existence of at least four points.
Therefore Axiom Thus, the Smarandache axioms of anti-geometry are inconsistent as stated. I propose to delete the last disjunct of I. By the way, 'space' is not a term used in Hilbert's axioms. Indeed, since 'space' stands for the entire domain of application of Smarandache's system it ought not to occur in it either. Axiom This does not entail that B lies also between C andA. Axiom II. Let A and C be two collinear points. Obviously, if a given person owns A and C there is no reason why she or he should own a third checking account, let alone one with a balance that is either equal to A and less than C, or greater than both C and A.
There exist at least three collinear points such that one point lies between the other two, and another point lies also between the other two. In fact, under our definition of betweenness four collinear points can never be ordered in this way. These two conditions are plainly incompatible. Axiom ILS. Let A, B, and C be three non-collinear points, and h a line which lies on the same plane as points A,B, and C but does not pass through any of these points.
Then, the line h may well pass through a point of egment AB, and yet not pass through a point of segment AC, nor through a point of segment BC. Suppose that h does not pass through A, B or C but passes nevertheless through a point of segment AB. This entails that person h owns in common with the owner of both A and B a checking account whose balance X is greater than A and less than B. Obviously, h need not own any balances in common with the owner of both B and C, nor with the owner of both A and C, let alone one that meets the requirements imposed by our definition of segment, viz.
On plane ex there can be drawn through point A either i no line, or li only one line, or iii a finite number of lines, or iv an infinite number of lines which doees not intersect the line h. The line s is are called the parallel s to h through the given pointA. Let A be the balance of a checking account with bank a and h a client of bank a who does not own that account.
The account in question may belong to a person who shares another balance with h case i , or to a person b, or to finitely many persons Cl,. According to EP3, A may also be owned secretly by infinitely many supernatural persons who do not share an account with h case iv. By DEF. N, the lines comprised in cases li , iii and iv all meet the requirements for being parallel with h. InChimienti and Bencze's article ref. Since I do not understand what this condition means, I did not consider it in the preceding discussion. Anyway, the following is clear: No matter how you interpret the terms "point" and "line" and the predicates "coplanar" and "intersect",case i excludes cases li and iv.
However, i implies iii and therefore can occur together with it, if by "finite number" you mean "any natural number" in Peano's sense, i. In contemporary mathematical jargon, this would the usual meaning of the term in this context. By the same token, li implies iii , for "one" is a finite number. Finally, iv certainly implies iii , for any infinite set includes a finite subset. In the light of this, the condition in parenthesis is obvious and trivial and few would think of mentioning it. Therefore, the fact that it is mentioned suggests to me that it is being given some other meaning, which eludes me.
If A, B are two points on a line h, and A' is a point on the same line or on another line h', then, on a given side of A' on line h', we cannot always find a unique B so that the segment AB is congruent to the segmentA'B'. If balances A and B belong to person h, and A' belongs to h' who mayor may not be the same person as h , there is no reason at all why there should exist a unique balance B' such that segments AB and A'B' meet the condition of congruence, viz.
Assume that i the owner of A and B got the monies in the respective accounts partly from a person x and partly from a person y; eli the owner of Ai and B' got these monies partly from x but not from y; iii the owner of A" and B" got these monies partly from y but not from x. If these three conditions are met, Axiom. Again, let B and B' be acquired by hand h', respectively, partly from x and partly from y; A and A: from x but not from y; C and C' from y but not from x.
Axiom NA. Suppose that a definite side of h' on plane 13 is assigned and that a particular point 0' is distinguished on h'. Then there are on 13 either one, or more than one, or even no half-line k' issuing from the point 0' such that i LhOk is congruent with Lh'O'k', and li the interior points of Lh'O'k' lie upon one or both sides of. This axiom is not easy to apply, for it contains the terms 'half-line', 'interior points of an angle ' and 'side of a line on a plane ' which have not been defined and are not used anywhere else in the axioms.
I shall take the half-line k issuing from a point 0 to mean a person k who owns 0 and owns another bank balance less than. As for the other two expressions, since they are otherwise idle, we could simply ignore them. But if the readers do not like this expedient, they may equally well use the following one: Let LaPb be an angle, such that P is the balance held in common by a and b in their checking account with a particular branch of bank a; the interior points of LaPb are the cashiers of that particular branch.
We say that the cashiers who are younger than a, lie on one side of a on a , and that the cashiers who are older than a, lie on the other side of a on a. The condition on interior points in axiom N. If the axiom is understood in this way, its meaning is clear enough. It is so weak that there is no difficulty in satisfying it. Take the arbitrarily assigned side of h' to be younger than. Indeed, there may be several persons kl' k2' Axiom N.
Then it is. The triangle ABC is determined by three distinct balances A, B and C, such that A and B joindy belong to a person c, B and C joindy belong to a person a who is different from c, and C and A joindy belong to a person b who is different from both a and c. It follows that a and b are joint owners of C, band c are joint owners of A, and c and a are joint owners of B. Let A and B be two checking account balances. Consider a series of n checking account balances AI, A An, such that all of them belong to the owner of A, and all except An amount to the same sum as A.
Suppose that An is greater than A. Now, the condition denied in the apodosis, viz. Obviously this is not implied by the initial condition on B, viz. There is a simple moral to be drawn from this exercise. Because Smarandache Anti-Geometry has removed the stringent constraints on points, lines and planes prescribed by the Hilbert axioms, it is child's play to find uninteresting applications for it, like the one proposed above.
When first confronted with this model, Dr. Minh L. Perez wrote me that he had the impression that Smarandache's message was directed against axiomatization. Such an attack would be justified only if we take an equalitarian view of axiom systems. To my mind, equalitarianism in the matter of mathematical axiom systems-though favored by some early twentieth century philosophers-is like placing all games of wit and skill on an equal footing.
The clever Indian who invented chess is said to have demanded corn grains minus 1 for his creation. Who would have the chutzpah to charge even a trillionth of that for tic-tac-toe? But Smarandache's AntiEuclidean geometry does not derogate Hilbert's axiom system for Euclidean geometry. Indeed this system, as well as Hilbert's axiom system for the real number field a , deserve much more -not less- attention and praise in view of the fact that one can also propose consistent yet vapid axiom systems.
Frege, G. Herausgegeben von I. Darmstadt: Wissenschaftliche Buchgesellschaft. Hilbert, D. Leipzig: Teubner. Ther den ZahIbegrifP'. Les principes fondamentaux de la geometrie. Traduction de L. Paris: Gauthier-Villars. Grundlagen tier Geometrie. Zweite Auflage. It is interesting to mention that by using the method outlined in . In the same paper, the authors ask whether it can be shown that.
Although I have read [4J carefully, I found no trace of the aforementioned computation! In this note, we show that -xI is indeed the correct order of magnitude of ogx A x. For any positive real number x let 7I" x be the number of prime numbers less then or equal to x, 5 B x xA x Sen ,. Unfortunately, we have not succeeded in finding a lower bound of the type 8 for. The Proof We begin with the following observation: Lemma. For every prime number P and positive integer k let ep k be the exponent at which P appears in k!. This obviously implies n I m!
Assume that 0'1 ::; Pl. By 1 above, it follows that in fact Sen O'lPl. The asserted inequality follows from. The Proof of the Theorem. In what follows p denotes a prime. We assume proof is to find good bounds on the expression. Certainly, the three subsets above are, in general, not disjoint but their union covers I. D; x Sen. The bound for D 1. Assume that m E C 1. Indeed, this follows from the fact that for Ot 2: 2. Write now m pOk. This shows that there are at most. Applying inequality 20 with x replaced by Proceed by induction on s.
If s 0, there is nothing to prove. Rewriting inequality 23 as. This completes the induction step. Via inequality 23 , inequality 22 implies. Of course, inequalites 27 - 29 may hold even below the smallest values shown above but this needs to be checked computationally. In the same spirit, by using the theorem and the estimation 11" X. A diophantine equation In this section we present an application to a diophantine equation. The application is not of the theorem per se, but rather of the counting method used to prove the theorem. Since S is defined in terms of factorials, it seems natural to ask how often the product S l.
Sen happens to be a factorial. The only solutions of S l. We show that the given equation has no solutions for n? Assume that this is not so. Let P be the largest prime number smaller than n. By Tchebysheff's theorem, we know that P? In particular, P :5 m. Hence, m? We now compute an upper bound for the order of 2 in S I. In this case, 2 3p I S pa. Since 2,,-1? Since k :5 n, the above arguments show that there are at most n. Case 2. By an argument similar to the one employed at Case 1, one gets in this case that a? Since k :5 n, it follows that there are at most n.
From the above anaysis, it follows that the order at which 2 divides S 1. Sen is at most e2.
We now bound each one of the two sums above. We now compute a lower bound for e2. One can now compute 5 1 5 2.
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We conclude suggesting the following problem: Problem. Find all positive integers n such that 5 1 , 5 2 , The published solution was based on the simple observation that the sum of all entries in an n x n latin square has to be a multiple of n. It is unlikely that this argument can be extended to cover the general case.
Finch's result is better than our result which only shows that the limsup of the expression log z A z jz when z goes to infinity is in the interval [0. References  Y. Laurent, "Minoration effective de la distance p-adique entre puissances de nombres algebriques", J. Number Theory 61 , pp. Press, Ibsted, "Surfing on the ocean of numbers", Erhus U. Schoenfeld, Approximate formulas for some functions of prime numbers", minois J. The product between an upper diagonal matrix and a vector is analysed from parallel computation point of view.
An efficient solution for this problem is given by using the inferior Smarandache I-part function. Finally, the efficiency of our solution is proved experimentally by presenting some computational results. Parallel programming has been intensely developed in order to solve difficult problems that contain either a big number of computation or a large volume of data. These often occur both in real word applications e. Weather Prediction or theoretical problems e. Differential Equations. Unfortunately, there is not a standard for writing parallel programs; this depends on the parallel language used or the parallel platform on which the computation is performed.
A common fact of this diversity is represented by easiness to parallelise loops. Loops represent an important source of parallelism occurring in at most all the scientific applications. Many algorithms dealing to the scheduling of loop iterations to processors have been proposed so far.
Consider that there are p processors denoted in the following by PI, P2, These bounds are found distributing equally the work on processors by using It. Suppose that Equation 2. This means that if we have the value 1j' then we find hj as follows: 3. This is an important problem that occurs in many algorithms for solving linear systems. The Smarandache inferior part function is used to distribute equally the work on processors. Smarandache inferior part function represents a natural generalisation of the floor function [Smara1].
Smarandache proposed and studied this generalisation especially in connection to Number Theory functions [Smara1, Smara2]. In the following, we present equation for some Smarandache inferior part functions. The Smarandache J-inferior part function denoted by.
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In the following we study the Smarandache k. Sometime, we will study only the positive inferior part by considering function J: N. Proof The proof is obtained by starting from the double inequality Observe that the equation. A real root of this equation is given by 7. An Efficient Algorithm for the Upper Diagonal Matrix-Vector Product In this section, we present an efficient algorithm for the product y upper diagonal matrix.
The difficult problem for the efficient loop scheduling algorithm is how Equation I is implemented. But, we want to find the upper bounds in at most O p p. For that we use the following theorem. Proof The Smarandache [inferior part function presented in Theorem 1 is used to obtain the proof.
We found that if I k. Thus, the upper bound of processor j is the biggest number k such that all the previous work done by processors I,2, Mathematically, this can be written as follows. According to this theorem, the efficient scheduling is obtained using the upper bound from Equation 9. These bounds certainly give the better approximation of Equation 1. Thus, the part of parallel loop scheduled on processor j is presented in Figure 3. Computation of Processor j. Computational Results and Final Conclusions This section presents some computational results of scheduling the parallel loop from Figure 3.
In order to find that the proposed method is efficient from the practical point of view, two other scheduling algorithms are used. The first scheduling algorithm named uniform scheduling, divides the parallel loop into p chunks with the same size [; ]. Obviously, this represents the simplest scheduling strategy but is inefficient because all the big sums are computed on processor p. The second scheduling algorithm named interleaving, distributes the work on processors from p to p, such that a processor does not compute two consecutive works.
This scheduling distributes the large work equally on processors. All the algorithms have been executed on SGI Power Challenge parallel machine with 16 processors for a upper diagonal matrix of dimension The running time are presented in Table 1. The first important remark that can be outlined is that there is no way to develop efficient methods in Computer Science without Mathematics and this article is a prove for that.
Using a special function named the Smarandache inferior part, it has. The second important remark is that the scheduling proposed in this article is efficient in practice as well. Table 1 shows that the times for the line balanced are smallest It can be seen that the interleaving strategy also offers good times. Table 1 also shows that the uniform strategy gives the largest times.
References [Jaja] J. Smarandache, Only Problems Tabirca and S. Interesting questions have been resolved through the surprising involvement of Fermat numbers. Some properties and values of this function are given in , which also contains an effective computer algorithm for calculation of Zen. The following properties are evident from the definition:. No, there are none, but to my knowledge no proofhas been given.
Before presenting the proof it might be useful to see some elementary results and calculations on Zen.
Explicit calculations of Z 3. This can not happen, therefore it is important to prove the following theorem. Consider the following four cases: 1. We will see that also case 2 be excluded in favor of cases 3 and 4. Case 3 and 4.
Let's illustrate the last statement by a numerical example. Iterating the Pseudo-Smarandache Function The theorem proved in the previous section assures that an iteration of the pseudoSrnarandache function does not result in an invariant, i. Z n :;t:n is true for n:;t:l. It can only go into a loop or cycle ifafter one or more iterations it returns to 2k.
Leaps are represented by t in the diagram. After a number of such iterations the end result will of course be 1. It is what this chain of iterations looks like which is interesting and which will be studied here. To begin with we will look at the iteration ofa few prime powers. Table 1.
Iteration of p6. A horizontal line marks where the rest of the iterated values consist of descending powers of 2 i 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 The characteristic tail of descending powers of 2 applies also to the iterations of composite integers and plays an important role in the ahernating Z-ejI iterations which will be subject of the next section.
The alternating iteration of the Euler cit function followed by the Smarandache Z function. Charles Ashbacher  found that the alternating iteration Z Theorem: The alternating iteration Z In order to have a 2-cycle we must find a solution to the equation. The first five of these are prime numbers. J It should be noted that 2, 8, and can be obtained as iteration results only through iterations of the type. Z cjl n From this we see that. A statistical survey of the frequency of the different 2-cycles, displayed in table 2, indicates that the lower cycles are favored when the initiating numbers grow larger.
Cycle 4 could have appeared in the third interval but as can be seen it is generally scarcely represented. ProhIbitive computer execution times made it impossible to systematically examine an interval were cycle 5 members can be assumed to exist. Table 2. The distribution of cycles for a few intervals of length Interval 3! References 1. American Research Press, Abstract: This study is an extension of work done by Charles Ashbacher.
Iteration results have been re-defined in terms of invariants and loops. Further empirical studies and analysis of results have helped throw light on a few intriguing questions. The Z-sigma sequence originated by n creates a cycle. Ashbacher identified four 2 cycles and one 12 cycle. These are listed in table 1. Iteration cycles C1 - Cs n. Is there another cycle generated by the Zcr sequence? Is there an infinite number ofnumbers n that generate the two cycle 42B20? Are there any other numbers n that generate the two cycle 2B3? Is there a pattern to the first appearance of a new cycle?
Ashbacher concludes his article by stating that these problems have only been touched upon and encourages others to further explore these problems. It is amazing that hundred thousands of integers subject to a fairly simple iteration process all end up with final results that can be descnbed by a few small integers. This merits a closer analysis. In an earlier study of iterations  the author classified iteration results in terms of invariants, loops and divergents.
Applying the iteration to a member of a loop produces another member of the same loop, The cycles descnbed in the previous section are not loops, The members of a cycle are not generated by the same process, half of them are generated by Z a Z. This leads to a situation were the iteration process applied to a member of a cycle may generate a member of another cycle as descnbed in table 2. A Za iteration applied to an element belonging to one cycle may generate an element belonging to another cycle.
This situation makes it impossible to establish a one-to-one correspondence between a number n to which the sequence of iterations is applied and the cycle that it will generate. Henceforth the iteration function will be Z cr n which will be denoted Za n while results included in the above cycles originating from a Z This leads to an unambiguous situation which is shown in table 3.
The Zcr iteration process described in terms of invariants, loops and intermediate elements. No other invariants or loops exist for n;I Each number n;; corresponds to one of the invariants or the loop. The distn"bution of results of the Za iteration has been examined by intervals of size as shown in table 4. The stability of this distribution is amazing. It deserves a closer look and will help bringing us closer to answers to the four questions posed by Ashbacher.
Conjecture: There are infinitely many numbers n which generate the invariant 20 Support: Although the statistics shown in table 4 only skims the surface of the "ocean of numbers" the number of numbers generating this invariant is as stable as for the other invariants and the loop. Let N be the value ofn up to which the search has been completed. No new loops or invariants will be found.
Possibility 2. Possibility 3. This could lead to a new loop or invariant. Let's suppose that a new loop oflength le2 is created. All elements of this loop must be greater than N otherwise the iteration sequence will fall below N and end up on a previously known invariant or loop. Question number 4: No particular patterns were found. Epilog: In empirical studies of numbers the search for patterns and general behaviors is an interesting and important part.
In this iteration study it is amazing that all these. The other amazing thing is the relative stability of distnoution between the three invariants and the loop with increasing n see table 4. When ZO' k n drops below n it catches on to an integer which has already been iterated and which has therefore already been classified to belong to one of the four terminal events. This in my mind explains the relative stability.
In general the end result is obtained after only a few iterations. I may not have brought this subject much further but I hope to have contnouted some light reading in the area of recreational mathematics. Hardy and E. Wright, An Introduction to the Theory of Numbers. Oxford University Press, I "Not even the sky is the limit" expresses the same dilemma as the title of the authors book "Surfing on the ocean of numbers". Even with for ever faster computers and better software for handling large numbers empirical studies remain very limited.
The starting point of this article is represented by a recent work of Finch . Based on two asymptotic results concerning the Erdos function, he proposed some interesting equation concerning the moments of the Smarandache function. The aim of this note is give a bit modified proof and to show some computation results for one of the Finch equation.
We will call the numbers obtained from computation 'Erdos-Smarandache Moments Number'. The ErdosSmarandache moment number of order 1 is obtained to be the Golomb-Dickman constant. These concern the relationship between the Smarandache and the Erdos functions and some asymptotic equations concerning them. These are important functions in Number Theory defined as follows:. Equations are very important because create a similarity between these functions especially for asymptotic properties. Moreover, these equations allow us to translate convergence properties. The main important equations that have been obtained by this translation are presented in the following.
Finch  started from Equation 7 and translated it from the Sarnrandache function. Theorem [Finch, ] If a is a positive number then 8 Proof. Theorem 1. Thus, the series L. J pac; L. The technique that has been applied to the proof of Theorem 2 can be used in the both ways. Theorem 2 started from a property of the Smarandache function and found a property of the.
Obviously, many other properties can be proved using this technique. Moreover, Equations gives a very interesting fact - "the Smarandache and Erdos function may have the same behavior especially on the convergence problems. References Cojocaru, 1. And Cojocaru, S. Erdos, P. Ford, K. Luca, F. Tabirca, S. Based on two asymptotic results concerning the Erdos function, he proposed some interesting equations concerning the moments of the Smarandache function. We will call the numbers obtained from computation 'the Erdos-Smarandache Numbers'. The ErdosSmarandache number of order 1 is obtained to be the Golomb-Dickman constant.
The Smarandache and Erdos functions are important functions in Number Theory defined as follows:. Moreover, these equations allow us to translate convergence properties of the Smarandache function to convergence properties on the Erdos function and vice versa. The main important equations that have been obtained using this translation are presented in the following.
Theorem [Finch, ] If a is a positive integer nwnber then 8 Proof Many terms of the difference The essence of this proof and the proof from [Finch, ] is given by Equation 6. But the above proof is a bit general giving even more. They are presented below. Final Remarks The numbers provided by Equation 7 could have many other names such as the GolombDickman generalized constants or.
We should also say that it is the Finch major contribution in rediscovering a quite old equation and connecting it with the Smarandache function. References Erdos, P. Journal, Vol. Finch, S. Knuth, D. Computer Science, 3, Journal, 10, No. Sabin Tabirca]. Shepp, L. Smarandache Notions Journal, 9, No. Notions Journal, 9, No. Smarandache, F. Timisoara, XVIll. By answering a question by C. Press, AZ. Bencze, OQ. Abstract: An empirical study of Smarandache k-k additive relationships and related data is tabulated and analyzed. It leads to the conclusion that the number of Smarandache additive relations is infinite.
It is also shown that Smarandache k-k relations exist for large values ofk. The sequence of function values starts: n:. A general definition of Smarandache p-q relationships is given by M. Bencze in Vol. He asks for others and questions whether there is a finite or infinite number of them. It is interesting to note that this solution is composed to two pairs of prime twins , and 43,41 , - one ascending and one descending pair.
This is also the case with the third solution found by Bencze. To throw some light on these types of relationships an online program for calculation of Sen  was used to tabulate Smarandache k-k additive relationships. The first surprising observation - at least to the author of these lines - is that the number of solutions does not drop off radically as we increase k. Of course, also the blank squares in the base of the diagram would be filled for n sufficiently large. For the most part, however, the values of Sen are small compared to n.
This corresponds to the large wall running at the back of the diagram. A certain value of Sen may be repeated a great many times in a given interval. It is the occurrence of a great number of values of Sen which are small compared to n that facilitates the occurrence of equal sums of function If this argument is as important as I think it is then chances are good that it might be possible to find, say, a Smarandache additive relationship.
I tried it - there are five of them, see table 9. Of the solutions to the additive relationships 22 are composed of pairs of prime twins. Question 2: What percentage of such prime twin pairs satisfy the Smarandache additive relationship? To elucidate these questions a bit further this empirical study was extended in the following directions. All Smarandache additive relations up to 10 were calculated. There are of which 65 are formed by pairs of prime twins. They are all primes. The results of this extended search are summarized by intervals in table 3 from which we can make the following observations.
The number of composite values of Sen , even as well as odd, are relatively few and decreasing. In the last interval table 3 there are only odd composite values. The number of additive relations is of the order of 0. The additive relations formed by pairs of prime twins is about l3.
Although one has to remember that we are still only "surfing on the ocean of numbers" the following conjecture seems safe to make: Conjecture: The number of Smarandache additive relationships is infinite. Do k-k additive relations exist for all k? If not - which is the largest possible value ofk? When they exist, is the number ofthem infinite or not?
References I. Appendix to article on Smarandache k-k additive relationships Henry Ibstedt The numerical material which was produced in relation to the above study was considered too much to include in the article because the author did not want to distract readers from the essential parts of the study. At the request of ARP the material not included in the article has been edited in the tables below so that the material of this study is complete.
Table 1 Smarandache fimction additive relations 11 n. Part 1 Krassimir T. Box 12 e-mail: krat bgcict. In the author of this remarks wrote reviews for "Zentralblatt fur Mathematik" for books [1 and [2 and this was his first contact with the Smarandache's problems. He solved some of them and he published his solutions in . The present paper contains some of the results from 3. In [1 Florentin Smarandache formulated unsolved problems, while in  C. Dumitrescu and V. Seleacu formulated unsolved problems of his. The second book contains almost all the problems from [1 J, but now each problem has unique number and by this reason the author will use the numeration of the problems from .
Also, in [2J there are some problems, which are not included in [IJ. When the text of  was ready, the author received Charles Ashbacher's book [4 and he corrected a part of the prepared results having in mind [4J. We shall use the usual notations: [xJ and xl for the integer part of the real number x and for the least integer. The 4-th problem from  see also IS-th problem from  is the following: Smarandache's deconstructive sequence:.
Let the n-th term of the above sequence be an. Then we can see that the first digits of the first nine members are, respectively: 1, 2, 4, 7, 2, 7, 4, 2, 1. Let us define the function w as follows: r. Here we shall use the arithmetic function tjJ, discussed shortly in the Appendix and detailed in the author's paper . The problem can be generalized, e. Study this sequence. The form of the general term an of the sequence is: The th and th problems from [2J see also th problem from  are the following: Inferior prime part:.
For any positive real number n one defines pp n as the largest prime number less than or equal to n. Superior prime part:. For any positive real number n one defines Pp n as the smallest prime number greater than or equal to n. Study these sequences. First, we should note that in the first sequence n 2: 2, while in the second one n 2: 0.
It would be better, if the first two members of the second sequence are omitted. Let everywhere below. Then the n-th member of the first sequence is. The checks of these equalities are straightforward, or by induction. Therefore, the values of the n-th partial sums of the two sequences are, respectively,. The th and th problems from [2J see also th problem from  are the following: Inferior square part:. Superior square part:.
The st and nd problems from  see also st problem from  are the following: Inferior cube part:. Superior cube part:. The rd and th problems from [2J see also nd problem from  are the following: Inferior factorial part:. Fp n is the largest factorial less than or equal to n. First, we shall extend the definition of the function "factorial" possibly, it is already defined, but the author does not know this , It is defined only for natural numbers and for a given such number n it has the form: n!
Let the new form of the function "factorial" be the following for the real positive number y:. Obviously, if y is a natural number, y! It can be easily seen that the extended function has the properties similar to these of the standard function. Second, we shall define a new function possibly, it is already defined, too, but the author does not know this. It is an inverse function of the function "factorial" and for the arbitraty positive real numbers x and y it has the form:.
From the above discussion it is clear that we can ignore the new factorial, using the definition. Practically, everywhere below y is a na. The checks of these equalities is direct, or by the method of induction.