SMARANDACHE NUMBERS REVISITED A.A.K. MAJUMDAR Ritsumeikan Asia-Pacific University 1–1 Jumonjibaru, Bepp u-shi 874–8577, Japan E-mail : aakmajumdar@gmai l.com/[email protected] & Guest Professor (April – September, 2018) Department of Mathematics , Jahangirnagar University Savar, Dhaka 13 42, Bangladesh 2018 2 Smarandache Numbers Revisited Copyright 2018 by the Author and the Publisher) P ons Publishing House / Pons asbl Q uai du Batelage, 5 1 000 - Bruxelles B elgium P eer Reviewers : 1 . Professor Dr. M.A. Matin, Department of Mathematics, Dhaka University, Dhaka 1000, Bangladesh. Bangladesh. Savar, Dhaka 1342, Bangladesh. 2. Professor Dr. Syed Sabbir Ahmed, Department of Mathematics, Jahangirnagar University, 1. S avar, Dhaka 1342, Bangladesh. 3 . Professor Dr. M. Kaykobad, Department of CSE (Computer Science & Engineering), BUET (Bangladesh University of Engineering & Technology), Dhaka 1000, Bangladesh. Front cover design © K.M. Anwarus Salam Cover design © Dr. K.M. Anwarus Salam ISBN : 978–1–59973–573–3 3 Chapter 4 : The Pseudo Smarandache Function PREFACE The primary source of all mathematics is the integers More than seven years ago, my first book on some of the Smarandache notions was published. The book consisted of five chapters, and the topics covered were as follows : (1) some recursive type Smarandache sequences, (2) Smarandache determinant sequences, (3) the Smarandache function, (4) the pseudo Smarandache function, and (5) the Smarandache function related and the pseudo Smarandache function related triangles. Since then, new and diversified results have been published by different researchers. The aim of this book to update some of the contents of my previous book, and add some new results. In Chapter 1, some recurrence type of Smarandache sequences are considered. These are : The Smarandache odd sequence, the Smarandache even sequence, the Smarandache circular seqence, the Smarandache square product sequences, the Smarandache permutation sequence, the Smarandache reverse sequence, the Smarandache symmetric sequence, and the Smarandache prime product sequence. It has been conjectured that none of these sequences contain infinitely many Fibonacci or Lucas numbers. In my earlier book, it has been shown that none of these sequences satisfies the recurrence relations of the Fibonacci and Lucas numbers. Here, we show that this result, in fact, follow from the common characteristic of these recursive sequences. Chapter 2 deals with two types of geometric type Smarandache determinant sequences; it is also shown that the results of some particular type of Smarandache determinant sequences are simplified with the introduction of the circulant matrices. In Chapter 3, some new expressions of the Smarandache function S(n) are given. Chapter 4 gives some new results on the pseudo Smarandache function, Z(n), including solutions of the Diophantine equations Z(n) + SL(n) = n and Z(n) = SL(n), where SL(n) is the Smarandache LCM function. In Chapter 4, we also consider the equation Z(mn) = mk Z(n), where m, n and k are positive integers. In connection with the Smarandache number related triangles, it is known that, if a, b and c are the sides of the 60-degree and 120-degree triangles, then the Diophantine equations c2 = a2 + b2  ab are satisfied by a, b and c. Chapter 5 gives partial solutions to these Diophantine equations. Finally, in Chapter 6, some miscellaneous topics are treated. The five topics covered in Chapter 6 are (1) the triangular numbers and the Smarandache T-sequence, (2) the Smarandache friendly numbers, (3) the Smarandache reciprocal partition sets of unity, (4) the Smarandache LCM ratio, and (5) the Sandor-Smarandache function. Most of the results appeared before, but some results, particularly some in Chapter 2, Chapter 5 and Chapter 6, are new. Particular mention must be made of Section 6.5 dealing with the Sandor-Smarandache function. In writing the book, I took the freedom of including the more recent results, found by other researchers, to keep the expositions up-to-date. In the previous book, several open problems / conjectures / questions were listed, most of which still remain unsolved. In this book, we add some new open problems and conjectures at the end of Chapter 1, Chapter 3, Chapter 4, Chapter 5 and Chapter 6. I would like to take this opportunity to thank the Department of Mathematics, Jahangirnagar University, Bangladesh, for hosting me as a guest Professor during the Academic Development Leave from the Ritsumeikan Asia-Pacific University, Japan, from April to September, 2018. A.A.K. Majumdar Ritsumeikan Asia-Pacific University, Japan 4 Smarandache Numbers Revisited Dedicated to the Memory of My Departed Parents who were like umbrellas and canopies enveloping us 5 Chapter 4 : The Pseudo Smarandache Function TABLE OF CONTENTS Chapter 0 Introduction 7–10 Chapter 1 Some Recursive Smarandache Sequences 11–14 Smarandache Odd Sequence, Even Sequence, Circular Sequence 11 Smarandache Square Product Sequences, Permutation Sequence 12 Smarandache Reverse Sequence, Symmetric Sequence 13 Smarandache Prime Product Sequence 13 References 14 Chapter 2 Smarandache Determinant Sequences 15–27 2.1 Smarandache Bisymmetric Geometric Determinant Sequence 16–24 2.2 Smarandache Circulant Geometric Determinant Sequence 24–25 2.3 Smarandache Circulant Arithmetic Determinant Sequence 25–26 References 27 Chapter 3 The Smarandache Function 28–36 3.1 Some Explicit Expressions for S(n) 29–34 3.2 Some Remarks 35 References 36 Chapter 4 The Pseudo Smarandache Function 37–56 4.1 Some Explicit Expressions for Z(n) 37–49 4.2 Miscellaneous Topics 50–56 References 56 Chapter 5 Smarandache Number Related Triangles 57–76 5.1 60-Degree and 120-Degree Triangles 58–55 5.2 Partial Solution of a2 = b2 + c2  bc 59–68 5.3 Some Remarks 68–76 References 76 Chapter 6 Miscellaneous Topics 77–131 6.1 Triangular Numbers and Smarandache T-Sequence 77–81 6.2 Smarandache Friendly Numbers 82–84 6.3 Smarandache Reciprocal Partition Sets of Unity 85–90 6.4 Smarandache LCM Ratio 91–105 6.5 Sandor-Smarandache Function 106–131 References 132 SUBJECT INDEX 133 5 6 Smarandache Numbers Revisited Notations and Symbols F =F(n) : The n-th term of the sequence of Fibonacci numbers n L =L(n) : The n-th term of the sequence of Lucas numbers n A = (a ) : The matrix A (of order IJ) whose entries are a , 1≤i≤I, 1≤j≤J ij ij   C C :The columns C and C (of the matrix A = a ) are interchanged i j i j ij   R R :The rows R and R (of the matrix A = a ) are interchanged i j i j ij C C+ kC :The column C is multiplied by the constant k and is added to the column C i i j j i R R + kR :The row R is multiplied by the constant k and is added to the row R i i j j i D d : The determinant D with entries d , 1≤i, j≤n ij ij x : The integer part of the real number x > 0 (the floor of x)  : The set of positive integers m n : The integer m divides the integer n S(.) : The Smarandache function Z(.) : The pseudo Smarandache function SL(n) : The Smarandache LCM function (N , N , …, N ) : GCD (Greatest Common Divisor) of the n (positive) integers N , N , …, N 1 2 n 1 2 n [N , N , …, N ] : LCM (Least Common Multiple) of the n (positive) integers N , N , …, N 1 2 n 1 2 n (m, n) = 1 : The integers m and n are relatively prime T(n, r) : The Smarandache LCM ratio function of degree r SL(n, r) : The Smarandache LCM ratio function of the second type SRRPS(n) : The Smarandache repeatable reciprocal function of unity with n arguments F (n) : The order of the set SRRPS(n) RP SDRPS(n) : The Smarandache distinct reciprocal partition of unity with n integers f (n) : The order of the set SDRPS(n) DP  : The empty set n n n!   : The binomial coefficient   ; 0kn k k k! (n  k)! 6 7 Chapter 4 : The Pseudo Smarandache Function Chapter 0 Introduction Eight Smarandache sequences were considered in Majumdar(*). They are : (1) the Smarandache odd sequence, (2) the Smarandache even sequence, (3) the Smarandache circular sequence, (4) the Smarandache square product sequences, (5) the Smarandache permutation sequence, (6) the Smarandache reverse sequence, (7) the Smarandache symmetric sequence, and (8) the Smarandache prime product sequence. These sequences share the common characteristic that they are all recurrence type, that is, in each case, the n–th term can be expressed in terms of one or more of the preceding terms. In case of the Smarandache odd, even, circular and symmetric sequences, we showed that none of these sequences satisfies the recurrence relationship for Fibonacci or Lucas numbers. We proved further that none of the Smarandache prime product and reverse sequences contains Fibonacci or Lucas numbers (in a consecutive row of three or more). In Chapter 1, we show the general result that the recurrence type sequences do not satisfy the recurrence relations of the Fibonacci or Lucas numbers. We recall that the sequence of Fibonacci numbers, F(n) , and the sequence of Lucas n1 numbers L(n) , are defined through the following recurrence relations : n1 F(1)=1, F(2) = 1; F(n+2) = F(n+1) +F(n), n1, (0.1) L(1)= 1, L(2) = 3; L(n+2) = L(n+1)+L(n), n1. (0.2) From the recurrence relation (0.1), we see that F(n) is increasing in n  1; in fact, F(n) is strictly increasing in n  2, since F(n + 1) – F(n) = F(n – 1) > 0 for all n  2. Moreover, we have the following result, which shows that F(n) is strictly convex in the sense of the inequality. Lemma 0.1 : For n1, F (n + 2) – F(n + 1) > F(n + 1) – F(n). Proof : Since for n  2, F(n + 2) – F(n + 1) = F(n) > F(n – 1) = F(n + 1) – F(n), the result follows. ■ In a similar manner, from (0.2), we see that L(n) is strictly increasing in n  1, with L(n + 2) – L(n + 1) > L(n + 1) – L(n) for all n  1. Lemmas 0.2–0.4 give some properties satisfied by the terms of F(n) and L(n) . n1 n1 Lemma 0.2 : In the sequence of Fibonacci numbers, F(n) , the terms F(3n–2) and n1 F(3n–1) are odd, and the terms F(3n) are even, for all n≥1. Proof : is by induction on n. From (0.1), we see that the result is true for n=1. So, we assume that the result is true for some integer n. Now, since F(3n+1)=F(3n)+F(3n–1), F(3n+2)=F(3n+1)+F(3n), F(3(n+1))=F(3n+2)+F(3n+1). it follows that the result is true for n+1 as well, completing induction. ■ 7 8 Smarandache Numbers Revisited Lemma 0.3 : In the sequence of Lucas numbers, L(n) , the terms L(3n – 2) and n1 L(3n–1) are odd, and the terms L(3n) are even, for all n≥1. Proof : is by induction on n, similar to that of Lemma 0.2, and is omitted here. ■ Lemma 0.4 : For all n≥1, (1) 3 divides F(4n), (2) 5 divides F(5n), (3) 4 divides F(6n). Proof : Since F(4)=3, F(5)=5, F(6)=8, we see that the result is true for n = 1. To proceed by induction on n, we assume that the result is true for some integer n. Now, since F(4(n+1))=F(4n+3)+F(4n+2) =[F(4n +2)+F(4n+1)]+F(4n+2) =2F(4n +2)+F(4n+1) =2[F(4n +1)+F(4n)]+F(4n+1) =3F(4n +1)+2F(4n), F(5(n+1))=F(5n+4)+F(5n+3) =[F(5n +3)+F(5n+2)]+F(5n+3) =2F(5n +3)+F(5n+2) =2[F(5n +2)+F(5n+1)]+F(5n+2) =3[F(5n +1)+F(5n)]+2F(5n+1) =5F(5n+1)+3F(5n), F(6(n+1))=F(6n+5)+F(6n+4) =[F(6n +4)+F(6n+3)]+F(6n+4) =2F(6n +4)+F(6n+3) =2[F(6n +3)+F(6n+2)]+F(6n+3) =3F(6n +3)+2F(6n+2) =3[F(6n+2)+F(6n+1)]+2F(6n + 2) =5F(6n+2)+3F(6n+1)] =5[F(6n+1)+F(6n)]+3F(6n + 1) =8F(6n+1)+5F(6n), we see that the result is true for n+1 as well, thereby completing induction. ■ Let a  be the sequence such that n n1 a =A, a >10a for all n1 (A>0). (0.3) 1 n+1 n Lemma 0.5 : For the sequence a  (defined in (0.3)), n n1 (1) a >a , n  1, n+1 n (2) a – a >a – a , n1, n+2 n+1 n+1 n (3) a >a +a , n1. n+2 n+1 n 9 Chapter 4 : The Pseudo CShmaaprtaenr d0a c:h eI nFturnocdtuiocnti on Proof : Clearly, all the terms of the sequence a  are positive. n n1 Part (1) of the lemma follows from the fact that a – a > 9a > 0 for all n  1. n+1 n n Since a – a >9a >a – a , n+2 n+1 n+1 n+1 n part (2) of the lemma follows. Part (3) follows by virtue of the following chain of inequalities : a – a >(a – a ) + (a – a )>9(a + a ) > a . n+2 n n+2 n+1 n+1 n n+1 n n+1 All these complete the proof of the lemma. ■ From the proof of Lemma 0.5, we see that the inequality in part (3) holds for any sequence satisfying the condition (0.3). However, it should be kept in mind that the inequality in part (3) of Lemma 0.5 may hold true even for a sequence which does not satisfy the condition (0.3). Lemma 0.6 : Consider the sequence a  (defined in (0.3)) with A  1. Then, n n1 (1) a >F(n+1) for all n1, n+1 (2) a >L(n+1) for all n1. n+1 Proof : First, note that a F(1)=L(1), a >9A>F(2), a >L(2). 1 2 2 The proof is now by induction on n. So, we assume that the result is true for some n (including all numbers less than n). Since (by virtue of part (3) of Lemma 0.5, together with the recurrence relation (0.1)) a – F(n + 2) > [a – F(n + 1)] + [a – F(n)], n+2 n+1 n by the induction hypothesis, it follows that a – F(n + 2) > 0, n+2 so that the result is true for n + 1 as well. The proof of part (2) is similar and is left to the reader. ■ The proof of Lemma 0.6 shows that the inequalities therein depend on the inequality given in part (3) of Lemma 0.5. Thus, the result in Lemma 0.6 may be true for a sequence not satisfying the condition in (0.3). Lemma 0.7 : Consider the sequence a  (defined in (0.3)). n n1 (1) I f a ≥ F(n + m) for some integers n≥1 and m≥1, then n a >F(n+m+1), n+1 (2) If a ≥ L(n + m) for some integers n≥1 and m≥1, then n a >L(n+m+1). n+1 Proof : The proofs of part (1) and part (2) are similar, and we prove part (1) only. To prove part (1), we observe that 2F(n+m)≥F(n+m)+F(n+m–1)=F(n+m+1) for all n≥1, m≥1. Now, a >10a ≥10F(n+m)≥5F(n+m+1)>F(n+m+1), n+1 n and we get the result desired. ■