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Journal of Integer Sequences, Vol. 14 (2011), Article 11.9.1
23 11
Inequalities and Identities Involving Sums of Integer Functions Mircea Merca Department of Informatics Constantin Istrati Technical College Grivit¸ei 91, 105600 Campina Romania
[email protected] Abstract The aim of this paper is to present a method of generating inequalities and, under certain conditions, some identities with sums that involve floor, ceiling and round functions. We apply this method to sequences of nonnegative integers that could be turned into periodical sequences.
1
Introduction
The floor and ceiling functions map a real number to the largest previous or the smallest following integer, respectively. More precisely, the floor function ⌊x⌋ is the largest integer not greater than x and the ceiling function ⌈x⌉ is the smallest integer not less than x. Iverson [7, p. 127] introduced this notation and the terms floor and ceiling in the early 1960’s and now this notation is standard in most areas of mathematics. Many properties of floor and ceiling functions are presented in [1, 5]. If x, y are real numbers and n is an integer so that y − x < 1 and x ≤ n ≤ y then ⌈x⌉ = ⌊y⌋ = n .
(1)
For all integers m and all positive integers n the following identities can be used to convert floors to ceilings and vice-versa lmm m + n − 1 . (2) = n n 1
When m is integer and n is a positive integer the quotient of m divided by n is m and n the value jmk m mod n = m − n n is the remainder (or residue) of the division. If the integers a and b have the same remainder when divided by n (i.e., if n divides a − b) we say that a is congruent to b modulo n and we write a≡b
(mod n) .
The mod in a ≡ b (mod n) defines a binary relation, whereas the mod in a mod b is a binary operation. The nearest integer function [x], also called nint or the round function, is the integer closest to x. Since this definition is ambiguous for half-integers, the additional rule is necessary to adopt. In this work, the nearest integer function is defined by 1 . (3) [x] = x + 2 If x is a real number and n is an integer so that |x − n|
1, i m m i=1 i=1
(28)
is true if Mf (m, m, k) = m ,
Rf (m, k) = 0
and
Lf (m, k) > −1 .
(29)
Using Maple, we find values for m and k so that condition (29) is fulfilled. This time we do not find anything. Is identity (28) false for any positive integer k and any integer m, m > 1? Example 3. Solution to Problem 1. When f is the floor function, k = 2 and m = 12, using Maple, we get the following: M=
19 , 6
L=−
13 72
We have (f,12)
b2,1
= (−1)2 B2 − M = 16
and
R=
4 . 9
1 19 − = −3 . 6 6
The values of b2,2 and b2,3 are those calculated in Example 1. n 2 X i 1 3 1 S1 (n) = = n + n2 − 12 36 24 i=1
By (21) we obtain 1 n . 4
When f is the round function, k = 2 and m = 12, we get the following: M=
43 , 3
L=−
We have (f,12)
b2,1
= (−1)2 B2 +
1 8
and
R=
2 . 9
12 − M 1 7 = − = −1 , 2 6 6
By (21) we obtain S2 (n) =
n 2 X i i=1
and the problem is solved.
3.2
1 1 1 3 n + n2 − n = 12 36 24 12
,
Positive integers powers
For every positive integer m and every integer a relatively prime to m, we denote by ordm (a) the multiplicative order of a modulo m, i.e., the smallest positive integer n such that an ≡ 1 (mod m), namely ordm (a) = min {n ∈ N∗ | an ≡ 1 (mod m)} Note that ordm (a) divides ϕ(m), ϕ being the Euler’s totient function. If ordm (a) = ϕ(m) then a is called a primitive root modulo m. If m is a prime, then ϕ(m) = m − 1. For every positive integer m, m > 1, and every integer k relatively prime to m, k > 1, the sequence (xn )n>0 , xn = k n , has the property xn+ordm (k) ≡ xn (mod m). The relation n X i=1
ki =
k n+1 − k k−1
(30)
and Corollary 12 allow us to obtain the following linear homogeneous recurrence: an − (k + 1)an−1 + kan−2 − an−ordm (k) + (k + 1)an−ordm (k)−1 − kan−ordm (k)−2 = 0 , i P where an = ni=1 f km and f is any of these floor, ceiling or round functions. If m is a prime and k is a primitive root modulo m, then we have an − (k + 1)an−1 + kan−2 − an−m+1 + (k + 1)an−m − kan−m−1 = 0 . For every positive integer m, m > 1, and every integer k relatively prime to m, k > 1, (f,k,m) (f,k,m) the sequence x = xn , n>0
x(f,k,m) n
( kn, if f is the floor function, = 2k n + m, if f is the round function, 17
has the property (f,k,m)
xn+ordm (k)
( (f,k,m) xn ≡ (f,k,m) xn
(mod m), if f is the floor function, (mod 2m), if f is the round function.
As m and k are relatively primes, it follows that n n k k = +1 , m m which enables us to write
n i X k i=1
m
=n+
n i X k i=1
m
and then to note that it is enough to investigate only those sums which involve the floor or the round function. We denote ( M m, ordm (k), x(f,k,m) , if f is the floor function, (31) Mf (m, k) = M 2m, ordm (k), x(f,k,m) , if f is the round function, (
F Ff (m, n, k) = F ( L Lf (m, k) = L ( R Rf (m, k) = R
m, ordm (k), n, x(f,k,m) , if f is the floor function, 2m, ordm (k), n, x(f,k,m) , if f is the round function, m, ordm (k), x(f,k,m) , if f is the floor function, 2m, ordm (k), x(f,k,m) , if f is the round function, m, ordm (k), x(f,k,m) , if f is the floor function, 2m, ordm (k), x(f,k,m) , if f is the round function.
(32)
According to (30), (31) and (32) we obtain: n+1 if f is the floor function, n −Mf (m, k), k −k + · 1 Ff (m, n, k) = m(k − 1) m (m − Mf (m, k)) , if f is the round function. 2
If a is a positive integer so that a ≡ k (mod m), then ordm (a) = ordm (k). Thus, for a ≡ k (mod m), we deduce that Mf (m, a) = Mf (m, k) ,
Lf (m, a) = Lf (m, k) and Rf (m, a) = Rf (m, k) ,
where f is the floor or the round function. Example 4. If k ≡ 1 (mod m), i.e., ordm (k) = 1, then it is clear that: Mf (m, k) = 1
and
Lf (m, k) = Rf (m, k) = 0 , 18
(33)
where f is the floor function. According to Theorem 1 we get the following identity n i X k k n+1 − (n + 1)k + n = , k ≡ 1 (mod m) , k > 1 , m m(k − 1) i=1
(34)
that can be rewritten in this way: n X (km + 1)n+1 − (n + 1)km − 1 (km + 1)i = , k>0. m m2 k i=1 For instance, by (34), for m = 5 and k = 6 we get: n i X 6n+1 − 5n − 6 6 . = 5 25 i=1 Example 5. If m is a prime and k is a primitive root modulo m, then we have m−1 1 X m Mf (m, k) = i= , m − 1 i=1 2
where f is the floor function. According to Corollary 9, we get the following identity n+1 n i X n k −k k − = . m m(k − 1) 2 i=1 Example 6. If k is an odd number, then we get ord2 (k) = 1 ,
(2k + 2) mod 4 = 0 ,
Mf (2, k) = 0 and
Lf (2, k) = Rf (2, k) = 0 ,
where f is the round function. By Theorem 1, we get the identity n i X k kn − 1 n k + , k ≡ 1 (mod 2) , k > 1 . = · 2 2 k−1 2 i=1 Example 7. If m > 2 and k ≡ 1 (mod m), then we get Mf (m, k) = (2k + m) mod 2m = m + 2 and
Lf (m, k) = Rf (m, k) = 0 ,
where f is the round function. By Theorem 1, we get the identity n i X k n+1 − (n + 1)k + n k = , k ≡ 1 (mod m) , k > 1 , m > 2. m m(k − 1) i=1 From (34) and (35), we deduce the identity n i n i X X k k = , k≡1 m m i=1 i=1 19
(mod m) , k > 1 , m > 2.
(35)
Example 8. If k = m − 1 then k 2 ≡ 1 (mod m), i.e., ordm (k) = 2. Moreover, we have (2(m − 1) + m) mod 2m = m − 2 and 2(m − 1)2 + m mod 2m = m + 2 . Thus we deduce that
Mf (m, m − 1) = m ,
Lf (m, m − 1) = −
1 , m
Rf (m, m − 1) = 0 ,
where f is the round function. Now, according to (33), the following inequalities are direct consequences of the Theorem 3 n 1 k n+1 − k X k i − ≤ ≤ 0 , k ≡ −1 (mod m) , − m m(k − 1) i=1 m n i k n+1 − k X 1 1 k + − , ≤ m(k − 1) 2m m 2m i=1
k ≡ −1
(mod m) ,
and by Corollary 8 and Corollary 11, we get the identities !# n+1 " n n i X 1 1 1 X i k −k k + + = = k m m(k − 1) 2m m 2 i=1 i=1 !% n+1 $ n X 1 k −k 1 = = + 1+ ki m(k − 1) m m i=1 ' & X n+1 n 1 k −k = ki = m(k − 1) m i=1 # n+1 " X n k −k 1 = = k i , k ≡ −1 (mod m) . m(k − 1) m i=1 Using Maple to determine the values ordm (k), Lf (m, k) and Rf (m, k) we can generate many inequalities, but also few identities. Example 9. The integers m = 7 and k = 2 are relatively primes, but 2 is not a primitive root modulo 7, i.e., ϕ(7) = 6 and ord7 (2) = 3. When f is the floor function we get M=
7 , 3
L=−
1 21
and
R=
4 . 21
By Theorem 1 and Corollary 4, we get the inequalities n 4 1 2n+1 n 2 X 2i − ≤ ≤ − − − , 21 7 3 7 i=1 7 21 n i 2n+1 n X 5 2 5 − − − , ≤ 7 3 14 i=1 7 42 20
and by Corrollary 6 and Corollary 9 we get the identities n i X 2 i=1
3.3
7
=
2n+1 n 5 − − 7 3 14
2n+1 n 5 = − − 7 3 21 n+1 n 10 n 2 2 − − − − = . 3 21 7 3 7
=
2n+1 7
Fibonacci numbers
By definition, the first two Fibonacci numbers are 0 and 1, and each subsequent number is the sum of the previous two (sequence A000045 in [10]). In mathematical terms, the sequence (Fn )n≥0 of Fibonacci numbers is defined by the recurrence relation Fn = Fn−1 + Fn−2 with seed values F0 = 0 and
F1 = 1 .
One way to discover some fascinating properties of the Fibonacci sequence is to consider the sequence of least nonnegative residues of the Fibonacci numbers under some modulus. One of the first modern inquiries into this area of research was made by D. D. Wall [11] in 1960, though J. L. Lagrange [4] made some observations on these types of sequences in the eighteenth century. So, we know that F ( mod m) is periodic, and the period is known as the Pisano period π(m) (see [4] and sequence A001175 in [10]). On the other hand, there are known a lot of identities that involve sums with Fibonacci numbers. Together with Theorem 1, these allow the establishing of numerous inequalities, but also few identities with sums that imply integer functions and Fibonacci numbers. The following applications base on the identity n X
Fi = Fn+2 − 1 ,
(36)
i=1
which, corroborated with Corollary 12, allows us to get the relation n−π(m) n X Fi π(m) X Fi X 1 Fi f f Fn+2 − Fn+2−π(m) − Fπ(m)+2 + 1 , − − = f m m m m i=1 i=1 i=1 where f is any of these floor, ceiling or round functions. (f,m) For every positive integer m the sequence x(f,m) = xn
n>0
x(f,m) n
,
if f is the floor function, Fn , = Fn + m − 1, if f is the ceiling function, 2Fn + m, if f is the round function 21
has the property (f,m) xn+π(m)
( (f,m) xn ≡ (f,m) xn
(mod m), (mod 2m),
if f is the floor or the ceiling function, if f is the round function.
We denote ( M m, π(m), x(f,m) , if f is the floor or the ceiling function, Mf (m) = M 2m, π(m), x(f,m) , if f is the round function,
( F m, π(m), n, x(f,m) , Ff (m, n) = F 2m, π(m), n, x(f,m) , ( L m, π(m), x(f,m) , Lf (m) = L 2m, π(m), x(f,m) , ( R m, π(m), x(f,m) , Rf (m) = R 2m, π(m), x(f,m) ,
if f is the floor or the ceiling function, if f is the round function,
(37)
(38)
if f is the floor or the ceiling function, if f is the round function, if f is the floor or the ceiling function, if f is the round function.
According to (36), (37) and (38) we obtain: if f is the floor function, −Mf (m), n m − 1 − M (m), if f is the ceiling function, Fn+2 − 1 f + · Ff (m, n) = m m 1 (m − Mf (m)) , if f is the round function. 2 In order to generate identities, we use Maple to find values for m so that Rf (m)−Lf (m) < 1. We find these values: if f is the floor function, {2, 3, 4} , m ∈ {2, 3, 4, 11} , if f is the ceiling function, {2, 4, 11} , if f is the round function.
Example 10. Taking into account that π(4) = 6, when f is the round function, we get 1 1 M =4, L=− and R = . 4 2 By Theorem 3, we get the inequality n Fn+2 − 1 X Fi 1 1 − ≤ , − ≤ 4 4 4 2 i=1 that can be rewritten in the following way n F X 3 F 3 n+2 i − − ≤ . 4 8 i=1 4 8 By Corollary 8, we get n X Fi i=1
4
Fn+2 3 Fn+2 Fn+2 − 3 = = = . − 4 8 4 4 22
4
Observations and conjectures
Let a and m be relatively prime positive integers. To determine the arithmetic mean defined by (31) for floor function we have to determine the sum ordm (a)
X
ai mod m .
i=1
Taking into account the relation ordm (a)
X
ai =
i=1
and that
a(aordm (a) − 1) , a−1 ordm (a)
ordm (a)
X
(39)
i
a mod m ≡
i=1
X
ai
(mod m) ,
i=1
we deduce that, when a − 1 and m are relatively primes, the sum (39) is divisible to m. Using Maple to determine this sum, we notice the following relation. Conjecture 1. Let a and m be relatively prime positive integers. If a−1 and m are relatively prime and ordm (a) is even then ordm (a)
X i=1
m · ordm (a) ai mod m = . 2
Using Maple to determine the value of some sums as ordm (a)
X i=1
2ai + m mod 2m ,
necessary to determine the arithmetic mean (31) for round function, we notice another interesting identity. Conjecture 2. Let a and m be relatively prime positive integers. If m is prime and ordm (a) is even then ordm (a) X 2ai + m mod 2m = m · ordm (a) . i=1
By (36), we deduce that the sum of π(m) consecutive Fibonacci numbers is a multiple of m. This allows us to state the sum π(m)
X
(Fi mod m)
i=1
is multiple of m. Using Maple to determine the arithmetic mean defined by (37), we notice the following identity: 23
Conjecture 3. Let m be a positive integer, m > 1. Then π(m)
X
(Fi mod m) = m · w(m) ,
i=1
where w(m) is Fibonacci winding number (sequence A088551 in [10]).
5
Acknowledgements
The author would like to thank the anonymous referees for their valuable comments and suggestions for improving the original version of this manuscript. The author would like to express his gratitude for the careful reading and helpful remarks to Oana Merca, which have resulted in what is hopefully a clearer paper.
References [1] T. Andreescu, D. Andrica, and Z. Feng, 104 Number Theory Problems, Birkh¨auser Boston, 2007. [2] D. H. Bailey, J. M. Borwein, and R. Girgensohn, Experimental evaluation of Euler sums, Experiment. Math. 3 (1994), 17–30. [3] D. H. Bailey, P. B. Borwein, and S. Plouffe, The rapid computation of various polylogarithmic constants, Math. Comp. 66 (1997), 903–913. [4] J. D. Fulton and W. L. Morris, On arithmetical functions related to the Fibonacci numbers, Acta Arith. 16 (1969), 105–110. [5] R. L. Graham, D. E. Knuth, and O. Patashnik, Concrete Mathematics, Addison-Wesley, 1989. [6] D. E. Knuth, Johann Faulhaber and sums of powers, Math. Comp. 61 (1993), 277–294. [7] T. Koshy, Discrete Mathematics with Applications, Elsevier Academic Press, 2004. [8] M. Merca, Problem 78, Eur. Math. Soc. Newsl. 79 (2011), 52. [9] M. Petkovˇsek, H. S. Wilf, D. Zeilberger, A=B, A. K. Peters Ltd., 1996. [10] N. J. A. Sloane, The On-Line Encyclopedia of Integer Sequences. Published electronically at http://oeis.org, 2010. [11] D. D. Wall, Fibonacci series modulo m, Amer. Math. Monthly 67 (1960), 525-532.
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2010 Mathematics Subject Classification: Primary 11B50; Secondary 11B34, 11B37, 11B39, 11B68, 26D07, 26D15. Keywords: Sequences (mod m), Fibonacci numbers, Bernoulli numbers, Pisano period, power sums, integer function. (Concerned with sequences A000045, A000975, A001175, A026039, A033113, A033114, A033115, A033116, A033117, A070333, A077854, A088551, A097137, A097138, A097139, A156002, A171965, A172046, A172131, A173196, A173645, A173722, A175287, A175724, A175766, A175780, A175812, A175829, A175831, A175842, A175846, A175848, A175864, A177041, A177100, A177166, A177176, A177189, A177205, A177332, A177337, A177339, A177881, A178222, A178397, A178543, A178577, A178703, A178704, A178706, A178710, A178742, A178744, A178750, A178826, A178827, A178828, A178874, A178875, A178982, A179001, A179006, A179018, A179111, A181120, A181286 and A181640.)
A004697, A033118, A122046, A173690, A175822, A175868, A177237, A178420, A178711, A178829, A179041,
A014817, A033138, A131941, A173691, A175826, A175869, A177239, A178452, A178719, A178872, A179042,
A014785, A033119, A153234, A173721, A175827, A175870, A177277, A178455, A178730, A178873, A179053,
Received February 16 2011; revised versions received July 7 2011; September 5 2011. Published in Journal of Integer Sequences, October 16 2011. Return to Journal of Integer Sequences home page.
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