A Division Algorithm Approach to p-adic Sylvester Expansions

Transcription

1 arxiv: v1 [math.nt] 6 Aug 2015 A Division Algorithm Approach to p-adic Sylvester Expansions Eric Errthum Department of Mathematics and Statistics Winona State University Winona, MN eerrthum@winona.edu Astract A method of constructing finite p-adic Sylvester expansions for all rationals is presented. This method parallels the classical Fionacci- Sylvester (greedy) algorithm y iterating a p-adic division algorithm. The method extends to irrational p-adics that have an emedding in the reals. 1 Introduction In [2] and [3] A. Knopfmacher and J. Knopfmacher give algorithms for constructing Egyptian fraction expansions in a p-adic setting that are analogous to those given for the reals y Oppenheim [8]. However, for some positive rational inputs the Knopfmachers algorithm fails to return a finite expansion. In Section 2 we review the asics of the Knopfmachers Sylvester-type algorithm and give such an example. We then introduce a modification of 2010 Mathematics Suject Classification: Primary 11A67; Secondary 11J61. Key words and phrases: p-adic numer, p-adic division algorithm, Sylvester series expansion. 1

2 A Division Algorithm Approach to p-adic Sylvester Expansions 2 their algorithm that will give finite Sylvester expansions for all rationals. The seemingly unnatural correction term in our algorithm is explained y the alternate approach detailed in the final two sections. In Section 3 we riefly recall the Fionacci-Sylvester Greedy Algorithm, especially its relationship to the classical division algorithm. This provides the main motivation for Section 4 wherein we define a new p-adic division algorithm and use the same relationship to construct finite rational p-adic Sylvester expansions. Lastly we show that the division algorithm approach and the given modification to Knopfmachers algorithm yield the same output. 2 p-adic Numers 2.1 Basics of p-adic Numers We egin with some of the necessary asics of p-adic numers. A more thorough exposition can e found in [4]. Let p e a prime and Q p the completion of the rationals with respect to the p-adic asolute value p defined on Q y 0 p = 0 and r p = p ν if r = a pν, where p a. The exponent ν Z is the p-adicvaluation, or order, of r andwill e denoted ν p (r). Proposition 2.1 (c.f. [4]). Let nonzero ζ,ξ Q p. 1. There exists a unique ζ ) Q p such that ν p ( ζ = 0 and ζ = ζp νp(ζ). We will call ζ the unit part of ζ. 2. ν p (ζξ) = ν p (ζ)+ν p (ξ). 3. ν p (ζ +ξ) min{ν p (ζ),ν p (ξ)}. We have equality when ν p (ζ) ν p (ξ).

3 A Division Algorithm Approach to p-adic Sylvester Expansions 3 Every element of Q p has the shape ζ = n=ν p(ζ) c n p n (2.1) for c n {0,1,...,p 1}. This representation is unique, however p-adic numers can also e represented in the form ζ = a 0 + n=1 1 a n (2.2) where a n Z[ 1 ] and the sum is either finite or converges p-adically. There p are a variety of algorithms to decompose a p-adic into this form (c.f. [2]). One reason to study such representations is that they can e considered the p-adic analogue to representing real numers as Egyptian fractions, i.e. as the sum of unit fractions. 2.2 Sylvester-type Series Expansions of p-adic Numers There are various methods of decomposing a rational numer into an Egyptian fraction representation. One of the more naïve methods was given originally y Fionacci and again in modern times y Sylvester [9]. This algorithm is commonly known as the Greedy Algorithm ecause at each inductive step of the decomposition one simply takes the largest unit fraction smaller than the value eing decomposed. Later, Oppenheim [8] generalized this and other Egyptian fraction algorithms to real numers given y their decimal representations. The following inductive algorithm, which results in an expansion like (2.2), is presented y the Knopfmachers in [2] as a p-adic analogue to the Sylvester-Oppenheim algorithm on real numers. Begin y defining the fractional part of a p-adic numer ζ as in (2.1) y ζ = 0 n=ν p(ζ) c n p n. (2.3)

4 A Division Algorithm Approach to p-adic Sylvester Expansions 4 Algorithm 2.2 ([2]). Let ζ Q p. For the initial term, set a 0 = ζ. Then take ζ 1 = ζ a 0 so that ν p (ζ 1 ) 1. Continuing as long as ζ n 0, let 1 a n = and ζ n+1 = ζ n 1. a n ζ n In [3] (see Proposition 5.3) it is stated that the aove Sylvester-type algorithm and a similarly defined Engel-type algorithm terminate if and only if ζ Q. However in [1] Graner and A. Knopfmacher give an example of a rational with nonterminating Engel-type algorithm. Likewise, there are rationals for which this Sylvester-type algorithm does not terminate. Indeed, suppose ζ = a p+a Q with p a. Then a 0 = 1 and ζ 1 < 0. Since all the a n are defined to e positive, a finite sum in (2.2) leads to a contradiction. Although Laohakosol and Kanasri [6] give a complete characterization of the infinite Sylvester-type expansions from the Knopfmachers algorithm that correspond to rational numers, it is less ovious the necessary conditions under which a rational will result in a finite expansion. 2.3 A New Sylvester-Type Algorithm We start y generalizing the definition in (2.3) in the following way. Let ζ k = k 1 n=ν p(ζ) c n p n so that ζ 1 = ζ. In other words, ζ k is the rational image of ζ under the mod p k projection from Q p to Z[ 1 p ]. For real x, define the ceiling function, x, to e the least integer greater than x. Notice that since Q p is not an ordered field, this function is not well-defined for all p-adics. However, if ζ Q p such that there exists an emedding of Q(ζ) into R, then the ceiling function pulls ack to Q(ζ) Q p. We now state our modification to Algorithm 2.2. Algorithm 2.3. Let nonzero ζ Q p such that there exists an emedding ψ : Q(ζ) R and k Z such that k > ν p (ζ). Set ζ 0 = ζ. Inductively for

5 A Division Algorithm Approach to p-adic Sylvester Expansions 5 1 i 0 set t i = ζ i k, 1 ti ψ(ζ i ) q i = t i + p k, p k ψ(ζ i ) and The algorithm terminates if any ζ N = 0. ζ i+1 = ζ i 1 q i. (2.4) Theorem 2.4. Algorithm 2.3 produces a sequence of q i Z[ 1 ] such that p ζ = i=0 q i 1 where the sum (if infinite) converges p-adically. Proof. By (2.4), if the sum converges, it does so to ζ. It suffices to show that ζ i p 0. Suppose ν p (ζ i ) = s > k so that ζ i = ζ i p s. Then and for some m Z. Then t i = 1 p s ζ i ( 1 q i = ζ1 k = k+s 1 ζ1 p s k+s +mp k+s ) p s ζ i q i 1 = ζ 1 i ζ1 + ζ i mp k+s 1 0 mod p k+s k+s so ν p (ζ i q i 1) k +s. Then ν p (ζ i+1 ) k +s ν p (q i ). Since k > s, then ν p (q i ) s. Hence, ν p (ζ i+1 ) k +2s > ν p (ζ i ). Since the order of the ζ i is strictly increasing, ζ i p 0.

6 A Division Algorithm Approach to p-adic Sylvester Expansions 6 Example 2.5. Let k = 1 and consider ξ Q 7 with ξ 2 = 1 and ξ 4 11 mod 7. Then Q(ξ) emeds into R y either ψ(ξ) = 1 11 or ψ(ξ) = For the first choice, Algorithm 2.3 gives The second emedding yields ξ = ξ = Algorithm 2.3 certainly isn t as elegant looking as Knopfmachers Algorithm and has limitations for which p-adics it can e used on. However the importance of Algorithm 2.3 is in the following theorem. Theorem 2.6. Algorithm 2.3 terminates if and only if ζ Q. Instead of proving this theorem directly, in the following sections we will re-frame the approach to finding p-adic Sylvester expansions for rationals to mimic a more classical technique. In Section 4.3 we will show the link etween the two and the proof of Theorem 2.6 will follow easily. 3 The Fionacci-Sylvester Greedy Algorithm The algorithm given y Fionacci and Sylvester for Egyptian fractions of rationals can e interpretted as iterating a modified version of the classical division algorithm (c.f. [7]). We review these classical methods now to provide reference and motivation for the techniques used later. Theorem 3.1 (Modifed Classical Division Algorithm). For all a, Z, a > 0, there exist unique q, r Z such that with = aq r (3.1) 0 r < a. (3.2)

7 A Division Algorithm Approach to p-adic Sylvester Expansions 7 Note that Theorem 3.1 is greedy in the sense that it finds the smallest q such that aq >, i.e. so that a > 1 q. Algorithm 3.2 (F-S Greedy Algorithm). Let 1 < a Q, with a > 0 and gcd(a,) = 1. Iterate Theorem 3.1 in the following way: = aq 0 r 0 q 0 = r 0 q 1 r 1. (3.3) q 0 q 1 q i 1 = r i 1 q i r i The process terminates if any r N = 0. A straightforward computation (c.f. [7]) then gives the following: Theorem 3.3. Algorithm 3.2 terminates in a finite numer of steps and. N a = 1. (3.4) q i 4 A p-adic Division Algorithm Approach i=0 4.1 The p k -Division Algorithm We egin the process of retracing the classical approach y generalizing Theorem 3.1. The p-adic division algorithm defined here is similar to the one given in [5] ut differs consideraly in the restrictions on the quotient and remainder. Theorem 4.1 (p k -Division Algorithm). Let p e prime and k Z. For all a, Z[ 1], a > 0, there exist unique q,r p Z[1 ] such that p = aq r (4.1)

8 A Division Algorithm Approach to p-adic Sylvester Expansions 8 with and 0 r < ap k (4.2) r p ap k p. (4.3) Proof. We first prove existence. Since â and p are relatively prime, positive and negative powers of p are defined mod â. Let α = ν p (a) and β = ν p () and take 0 r < â such that r p β α k mod â. (Case 1: k > β α.) For some m Z we have rp β+α+k + = âm which gives p β = âp α mp β α rp α+k. Thus we can take q = mp β α and r = rp α+k. (Case 2: k β α.) For some m Z we have r+ p β α k = âm which gives p β = âmp α+k rp α+k. (4.4) Thus we can take q = mp k and r = rp α+k. In oth cases, since r < â, oth (4.2) and (4.3) are satisfied. To show uniqueness, suppose that there exist q 1,r 1,q 2,r 2 satisfying (4.1), (4.2), and (4.3). Then a(q 1 q 2 ) = r 1 r 2, thus r 1 r 2 mod â. Also r 1 r 2 mod p α+k since y assumption ν p (r i ) ν p ( ap k ). Therefore, r 1 r 2 mod ap k. Since oth are etween 0 and ap k, r 1 = r 2. Thus q 1 = q 2 and we have uniqueness as desired. Note that the value r may have nontrivial order. When ν p (r) 1 we say a jump occurs. Of course, if p > â then r = r and there is no jump. Also notice that one recovers Theorem 3.1 from Theorem 4.1 y setting k = 0 (or p = 1) and (redundantly) using the standard asolute value in (4.3). Additionally Theorem 4.1 generalizes the classical algorithm in the form of the following corollary.

9 A Division Algorithm Approach to p-adic Sylvester Expansions 9 Corollary 4.2. Suppose a,,p,k Z with p prime and k ν p () ν p (a). Let q p denote the quotient from the p k -division algorithm on a and and let q denote the quotient from the modified classical division algorithm on ap k and. Then q p = q p k. Proof. Since k ν p () ν p (a), Case 2 in the proof of Theorem 4.1 applies and q p = mp k. So it suffices to show that m = q. By (4.4), Thus (3.1) and (3.2) are satisfied. 0 ap k m < ap k. Inaddition, notethatthep k -DivisionAlgorithmcaneextended torational a and in the following way. Suppose = s and a = u for r,s,t,u Z. t v To find the quotient and remainder, clear denominators and compute the desired division algoritm on = sv and a = ut to find q and r. Then q = q and r = r satisfy (4.1), (4.2) and (4.3). However, since we are mostly vt interested in the quotient a, we will assume a, Z[1]. p 4.2 The p k -Greedy Algorithm Since we now have a generalization of Theorem 3.1, we can sustitute it into the iterative process of Algorithm 3.2. Algorithm 4.3 (p k -GreedyAlgorithm). Let a Q, with a > 0, gcd(a,) = 1 ( and k > ν a p ). Iterate Theorem 4.1 as in (3.3). The process terminates if any r N = 0. ( The condition k > ν a p ) plays the synonymous role to 1 < a in the F-S Greedy Algorithm: it prevents the division algorithm from returning a quotient equal to 0. This restriction on k is actually stronger than it needs to e for this alone. However, for reasons related to Corollary 4.2 and explained ( further elow, k ν a p ) is generally undesirale. The Knopfmachers avoided this ostruction y defining their initial a 0 outside of the inductive pattern so that ν p (ζ 1 ) 1. A similar strategy could e used here as well, though we find it more desirale to instead simply choose a different k value.

10 A Division Algorithm Approach to p-adic Sylvester Expansions 10 Theorem 4.4. Algorithm 4.3 terminates after a finite numer of steps. Proof. As opposed to Algorithm 3.2, now we have q i,r i Z[ 1 ] instead of Z. p However r i r i < r i 1 Z, so the sequence of r i s is a decreasing sequence of positive integers much like the classical remainders. Since the algorithm terminates, again we get that (3.4) holds. Example 4.5. Consider a = 473. Performing the3-greedyalgorithmyields, = = = = and thus = However, performing the 3 4 -Greedy Algorithm results in the sum = Applying Corollary 4.2 to Algorithm 4.3 gives another relationship etween the classical and p-adic algorithms. Corollary 4.6. Suppose a > 0 and the situation of k ν a p( ) holds. If no jumps occur than each term of the p k -Greedy Algorithm for a is equal to the corresponding term in the F-S Greedy Algorithm on apk divided y p k. Example 4.7. Consider the fraction a = = If we take p = 11 and k = 1, then k ν 11 ( a ) = 2. Applying the 11-Greedy Algorithm to encounters no jumps and gives =

11 A Division Algorithm Approach to p-adic Sylvester Expansions 11 On 5 11 = 5, the F-S Greedy Algorithm yields = The restriction on jumps is sufficient ut not necessary. For example, the 3-Greedy Algorithm on 22 encounters a jump, yet the relation aove to the 45 classical algorithm on 22 still holds. 15 ( Also notice that, again with k ν a ) p, if ap k > 1 then the F-S Greedy Algorithm returns quotients q i = 1 until the remaining value to e decomposed is less than 1. Hence, y Corollary 4.6 the p k -Greedy Algorithm on a produces a finite string of terms equal to 1 p k. For these reasons we only consider the cases where k > ν p ( a ). This condition is minor, though, in the grand scheme. Each inductive step of Algorithm 4.3 is independent of each other with respect to the value of k. So if the desired k fails the order criteria, it is possile to temporarily use a sufficiently large k in the initial step(s) and then switch ack to the desired k value once the corresponding orders ecome large enough. In this way finite p-adic Sylvester expansions can e found for all rationals. 4.3 Connection Between Approaches WearenowinthepositiontoreturntothemodificationoftheKnopfmachers algorithm given in Section 2. Theorem 4.8. For ζ Q, Algorithm 2.3 and Algorithm 4.3 produce the same output. Proof. Suppose ζ i Q for some i 0. Then ζ i = a for a, Z. Algorithm 2.3 produces /a /a k q i = + p k. a k p k Let r = aq i and σ k =. Then a k a 0 r a = σ σk k + p k < p k. p k

12 A Division Algorithm Approach to p-adic Sylvester Expansions 12 Since ν p (σ k ) k, then r p ap k p. Hence oth (4.2) and (4.3) are satisfied. The proof of Theorem 2.6 is simply a comination of Theorem 4.4 and Theorem 4.8. Further the condition of k > ν p (ζ) in Algorithm 2.3 is explained in light of Corollary 4.6. Indeed, for negative or irrational ζ with k ν p (ζ),astatementanalogoustocorollary4.6holdsandthusalgorithm 2.3 produces a p-adically divergent result. Though, as mentioned aove, this can e worked around y temporarily using alternate k-values. Acknowledgements Thank you to Anthony Martino for eing the first to convince me that Egyptian fractions are interesting and for helping me work out parts of Sections 4.1 and 4.2 in the k = 1 case. This work is dedicated to my children, Gedion and Eomji. References [1] Graner, Peter J.; Knopfmacher, Arnold. Arithmetic and metric properties of p-adic Engel series expansions. Pul. Math. Derecen 63 (2003), no. 3, [2] Knopfmacher, A.; Knopfmacher, J. Series expansions in p-adic and other non-archimedean fields. J. Numer Theory 32 (1989), no. 3, [3] Knopfmacher, A.; Knopfmacher, J. Infinite series expansions for p-adic numers, J. Numer Theory 41 (1992), no. 2, [4] Kolitz, N. p-adic Numers, p-adic Analysis, and Zeta-Functions. Graduate Texts in Mathematics 58, Springer, New York, 1977.

13 A Division Algorithm Approach to p-adic Sylvester Expansions 13 [5] Lager, C. A p-adic Euclidean Algorithm, Rose-Hulman Undergraduate Mathematics Journal 10 (2009), no. 2. [6] Laohakosol, V.; Kanasri, N. R. A characterization of rational numers y p-adic Sylvester series expansions, Acta Arith. 130 (2007), no. 4, [7] Mays, M. A worst case of the Fionacci-Sylvester expansion, J. Comin. Math. Comin. Comput. 1 (1987), [8] Oppenheim, A. The representation of real numers y infinite series of rationals, Acta Arith. 21 (1972) [9] Sylvester, J. J. On a Point in the Theory of Vulgar Fractions. Amer. J. Math. 3 (1880), no. 4,