Engel Series and Cohen-Egyptian Fraction Expansions

Abstract

Two kinds of series representations, referred to as the Engel series and the Cohen-Egyptian fraction expansions, of elements in two different fields, namely, the real number and the discrete-valued non-archimedean fields are constructed. Both representations are shown to be identical in all cases except the case of real rational numbers.

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Vichian Laohakosol 1 and Tuangrat Chaichana 2, 3 and Jittinart Rattanamoong 2, 3 and Narakorn Rompurk Kanasri 4

Recommended by Stéphane Louboutin

1, Department of Mathematics, Kasetsart University, Bangkok 10900, Thailand
2, Department of Mathematics, Faculty of Science, Chulalongkorn University, Bangkok 10330, Thailand
3, The Centre of Excellence in Mathematics, CHE, Si Ayutthaya Road, Bangkok 10400, Thailand
4, Department of Mathematics, Khon Kaen University, Khon Kaen 40002, Thailand

Received 11 September 2009; Accepted 2 December 2009

1. Introduction

It is well known [1] that each nonzero real number can be uniquely written as an Engel series expansion, or ES expansion for short, and an ES expansion represents a rational number if and only if each digit in such expansion is identical from certain point onward. In 1973, Cohen [2] devised an algorithm to uniquely represent each nonzero real number as a sum of Egyptian fractions, which we refer to as its Cohen-Egyptian fraction (or CEF) expansion. Cohen also characterized the real rational numbers as those with finite CEF expansions. At a glance, the shapes of both expansions seem quite similar. This naturally leads to the question whether the two expansions are related. We answer this question affirmatively for elements in two different fields. In Section 2, we treat the case of real numbers and show that for irrational numbers both kinds of expansion are identical, while for rational numbers, their ES expansions are infinite, periodic of period 1 , but their CEF expansions always terminate. In Section 3, we treat the case of a discrete-valued non-archimedean field. After devising ES and CEF expansions for nonzero elements in this field, we see immediately that both expansions are identical. In Section 4, we characterize rational elements in three different non-archimedean fields.

2. The Case of Real Numbers

Recall the following result, see, for example, Kapitel IV of [1], which asserts that each nonzero real number can be uniquely represented as an infinite ES expansion and rational numbers have periodic ES expansions of period 1 .

Theorem 2.1.

Each A∈...\{0} is uniquely representable as an infinite series expansion, called its Engel series (ES) expansion, of the form [figure omitted; refer to PDF] where [figure omitted; refer to PDF] Moreover, A∈... if and only if _an+1 =_an (≥2) for all sufficiently large n .

Proof.

Define _A1 =A-_a0 , then 0<_A1 ≤1 . If _An ≠0 (n≥1) is already defined, put [figure omitted; refer to PDF] [figure omitted; refer to PDF] Observe that _an is the least integer >1/_An and [figure omitted; refer to PDF] We now prove the folowing.

Claim 2.

We have 0<...≤_An+1 ≤_An ≤...≤_A2 ≤_A1 ≤1 .

Proof.

First, we show that _An >0 for all n≥1 by induction. If n=1 , we have seen that _A1 >0 . Assume now that _An >0 for n≥1 . By (2.3), we see that _an ∈... . Since [figure omitted; refer to PDF] and 1/_an <_An , we have _An+1 >0 . If there exists k∈... such that _Ak+1 >_Ak , then [figure omitted; refer to PDF] and so _ak -1>1/_Ak , contradicting the minimal property of _an and the Claim is proved.

From the Claim and (2.3), we deduce that _a1 ≥2 and _an+1 ≥_an (n≥1) . Iterating (2.4), we get [figure omitted; refer to PDF] To establish convergence, let [figure omitted; refer to PDF] Since _An >0 and _an ∈... for all n≥1 , the sequence of real numbers (_Bn ) is increasing and bounded above by _A1 . Thus, _{lim n[arrow right]∞Bn} exists and so [figure omitted; refer to PDF] By the Claim, [figure omitted; refer to PDF] showing that any real number has an ES expansion. To prove uniqueness, assume that we have two infinite such expansions such that [figure omitted; refer to PDF] with the restrictions _a0 ∈..., _a1 ≥2, _an+1 ≥_an (n≥1) and the same restrictions also for the _bn 's. From the restrictions, we note that [figure omitted; refer to PDF] If _A1 =1 , then by (2.12) we also have _∑n≥1 1/_b1b2 ..._bn =1 , forcing _a0 =_b0 . If 0<_A1 <1 , then (2.12) shows that 0<_∑n≥1 1/_b1b2 ..._bn <1 , forcing again _a0 =_b0 . In either case, cancelling out the terms _a0 ,_b0 in (2.12) we get [figure omitted; refer to PDF] Since _an+1 ≥_an , then [figure omitted; refer to PDF] so 0<_a1 -1/_A1 ≤1 . But there is exactly one integer _a1 satisfying these restrictions. Thus, _a1 =_b1 . Cancelling out the terms _a1 and _b1 in (2.14) and repeating the arguments we see that _ai =_bi for all i .

Concerning the rationality characterization, if its ES expansion is infinite periodic of period 1 , it clearly represents a rational number. To prove its converse, let A=a/b∈...\{0} . Since

[figure omitted; refer to PDF] we see that _A1 is a rational number in the interval (0,1] whose denominator is b . In general, from (2.4), we deduce that each _An (n≥1) is a rational number in the interval (0,1] whose denominator is b . But the number of rational numbers in the interval (0,1] whose denominator is b is finite implying that there are two least suffixes h,k∈... such that _Ah+k =_Ah . Thus, by (2.3), we have _ah+k =_ah . From (2.2), we know that the sequence {_an } is increasing. We must then have k=1 and the assertion follows.

Remark 2 s.

In passing, we make the following observations.

(a) For n≥1 , we have [figure omitted; refer to PDF]

(b) If A∈^...c , then _An ∈^...c and so 1/_An ∉... for all n≥1 .

(c) If A∈... , then its ES expansion is [figure omitted; refer to PDF]

To construct a Cohen-Egyptian fraction expansion, we proceed as in [2] making use of the following lemma.

Lemma 2.2.

For any y∈(0,1) , there exist a unique integer n≥2 and a unique r∈... such that [figure omitted; refer to PDF]

Proof.

Let n=[left ceiling]1/y[right ceiling]∈... and r=ny-1 . Put ...1/y...:=n-1/y∈[0,1) and so [figure omitted; refer to PDF] To prove uniqueness, assume ny-r=1=my-s so that [figure omitted; refer to PDF] Since there is only one integer with this property, we deduce n=m and consequently, r=s proving the lemma.

Theorem 2.3.

Each A∈...\{0} is uniquely representable as a CEF expansion of the form [figure omitted; refer to PDF] subject to the condition [figure omitted; refer to PDF] and no term of the sequence appears infinitely often. Moreover, each CEF expansion terminates if and only if it represents a rational number.

Proof.

To construct a CEF expansion for A∈...\{0} , define [figure omitted; refer to PDF] If _r0 =0 , then the process stops and we write A=_n0 . If _r0 ≠0 , by Lemma 2.2, there are unique _n1 ∈... and _r1 ∈... such that [figure omitted; refer to PDF] Thus, [figure omitted; refer to PDF] If _r1 =0, then the process stops and we write A=_n0 +1/_n1 . If _r1 ≠0 , by Lemma 2.2, there are _n2 ∈... and _r2 ∈... such that [figure omitted; refer to PDF] the last inequality being followed from _n1 =[left ceiling]1/_r0 [right ceiling], _n2 =[left ceiling]1/_r1 [right ceiling], and _r1 <_r0 . Observe also that [figure omitted; refer to PDF] Continuing this process, we get [figure omitted; refer to PDF] with [figure omitted; refer to PDF] If some _rk =0 , then the process stops, otherwise the series convergence follows at once from [figure omitted; refer to PDF]

To prove uniqueness, let

[figure omitted; refer to PDF] with the restrictions (2.23) on both digits _ni and _mj . Now [figure omitted; refer to PDF] It is clear that the restrictions (2.23) imply the strict inequality in (2.33). This also applies to the right-hand sum in (2.32). Equating integer and fractional parts in (2.32), we get [figure omitted; refer to PDF] Since _nk+1 ≥_nk , then [figure omitted; refer to PDF] so 0<_n1 -1/w≤1 . But there is exactly one integer _n1 satisfying these restrictions. Then _n1 =_m1 and [figure omitted; refer to PDF] Proceeding in the same manner, we conclude that _ni =_mi for all i .

Finally, we look at its rationality characterization. If A∈... , then _r0 ∈... , say _r0 :=p/q, where p,q∈... . From (2.30), we see that each _ri is a rational number whose denominator is q . Using this fact and the second inequality condition in (2.30), we deduce that _rj =0 for some j≤p , that is, the expansion terminates. On the other hand, it is clear that each terminating CEF expansion represents a rational number. Now suppose that A is irrational and there is a j and integer n such that _ni =n for all i≥j . Then

[figure omitted; refer to PDF] Since _∑k≥1 1/^nk =1/(n-1) , it follows that A is rational, which is impossible.

The connection and distinction between ES and CEF expansions of a real number are described in the next theorem.

Theorem 2.4.

Let A∈...\{0} and the notation be as set out in Theorems 2.1 and 2.3.

(i) If A∈... , then its ES expansion is infinite periodic of period 1 , while its CEF expansion is finite. More precisely, for A∈...\... , let its ES and CEF expansions be, respectively,

[figure omitted; refer to PDF] If m is the least positive integer such that 1/_Am ∈... , then [figure omitted; refer to PDF] and the digits _ni terminate at _nm .

(ii) If A∈...\... , then its ES and its CEF expansions are identical.

Proof.

Both assertions follow mostly from Theorems 2.1, 2.3, and Remark (b) except for the result related to the expansions in (2.38) which we show now.

Let A∈...\... and let m be the least positive integer such that 1/_Am ∈... . We treat two seperate cases.

Case 1 (m=1 ).

In this case, we have 1/_A1 ∈... and _a1 =1+[1/_A1 ]=1+1/_A1 . Since _r0 =A-_n0 =A-[A]=A-_a0 =_A1 , we get _n1 =1/_A1 and so _a1 =_n1 +1 . We have _r1 =_n1r0 -1=0 , and so the CEF expansion terminates. On the other hand, by Remark (a) after Theorem 2.1, we have _a1 =_ai (i≥2) .

Case 2 (m>1 ).

Thus, 1/_A1 ∉... and _A1 =_r0 . By Lemma 2.2, we have _a1 =_n1 . For 1≤i≤m-2 , assume that _Ai =_ri-1 and _ai =_ni . Then [figure omitted; refer to PDF] Since 1/_Ai+1 ∉... , again by Lemma 2.2, _ai+1 =_ni+1 . This shows that _a1 =_n1 ,...,_am-1 =_nm-1 . Since 1/_Am ∈... , we have _am =1+[1/_Am ]=1+1/_Am and thus [figure omitted; refer to PDF] From the construction of CEF, we know that _nm =[left ceiling]1/_rm-1 [right ceiling] . Thus, _nm =1/_Am showing that _am =_nm +1 . Furthermore, _rm =_nmrm-1 -1=0, implying that the CEF terminates at _nm , and by Remark (a) after Theorem 2.1, _am =_am+i (i≥1) .

3. The Non-Archimedean Case

We recapitulate some facts about discrete-valued non-archimedean fields taken from [3, Chapter 4 ]. Let K be a field complete with respect to a discrete non-archimedean valuation |·| and ...AA;:={A∈K;|A|≤1} its ring of integers. The set P:={A∈K;|A|<1} is an ideal in ...AA; which is both a maximal ideal and a principal ideal generated by a prime element π∈K . The quotient ring ...AA;/P is a field, called the residue class field. Let ...9C;⊂...AA; be a set of representatives of ...AA;/P . Every A∈K\{0} is uniquely of the shape

[figure omitted; refer to PDF] for some N∈... , and define the order v(A) of A by |A|=^2-v(A) =^2-N , with v(0):=+∞ . The head part ...A... of A is defined as the finite series

[figure omitted; refer to PDF] Denote the set of all head parts by

[figure omitted; refer to PDF] The Knopfmachers' series expansion algorithm for series expansions in K [4] proceeds as follows. For A∈K , let [figure omitted; refer to PDF] Define [figure omitted; refer to PDF] If _An ≠0 (n≥1) is already defined, put

[figure omitted; refer to PDF] if _an ≠0 , where _rn and _sn ∈K\{0} which may depend on _a1 ,...,_an . Then for n≥1

[figure omitted; refer to PDF] The process ends in a finite expansion if some _An+1 =0 . If some _an =0 , then _An+1 is not defined. To take care of this difficulty, we impose the condition

[figure omitted; refer to PDF] Thus

[figure omitted; refer to PDF] When _rn =1, _sn =_an , the algorithm yields a well-defined (with respect to the valuation) and unique series expansion, termed non-archimedean Engel series expansion . Summing up, we have the following.

Theorem 3.1.

Every A∈K\{0} has a finite or an infinite convergent non-archimedean ES expansion of the form [figure omitted; refer to PDF] where the digits _an are subject to the restrictions [figure omitted; refer to PDF]

Now we turn to the construction of a non-archimedean Cohen-Egyptian fraction expansion, in the same spirit as that of the real numbers, that is, by way of Lemma 2.2. To this end, we start with the following lemma.

Lemma 3.2.

For any y∈K\{0} such that v(y)≥1 , there exist a unique n∈S such that v(n)≤-1 and a unique r∈K such that [figure omitted; refer to PDF]

Proof.

Let n=...1/y... . Then v(n)=v(1/y)=-v(y)≤-1. Putting r=ny-1 , we show now that v(r)≥v(y)+1 . Since n=...1/y... , we have [figure omitted; refer to PDF] where _ck ∈...9C; , and so [figure omitted; refer to PDF] Thus [figure omitted; refer to PDF] To prove the uniqueness, assume that there exist _n1 ∈S such that v(_n1 )≤-1 and _r1 ∈K such that [figure omitted; refer to PDF] From ny-r=1=_n1 y-_r1 , we get (n-_n1 )y=r-_r1 . If n≠_n1 , since n,_n1 ∈S we have |n-_n1 |≥1 . Using |y|>|r-_r1 | , we deduce that [figure omitted; refer to PDF] which is a contradiction. Thus, n=_n1 and so r=_r1 .

For a non-archimedean CEF expansion, we now prove the following.

Theorem 3.3.

Each y∈K\{0} has a non-archimedean CEF expansion of the form [figure omitted; refer to PDF] where [figure omitted; refer to PDF] This series representation is unique subject to the digit condition (3.19).

Proof.

Define _n0 =...y... and _r0 =y-_n0 . Then v(_r0 )≥1 . If _r0 =0 , the process stops and we write y=_n0 . If _r0 ≠0 , by Lemma 3.2, there are _n1 ∈S and _r1 ∈K such that [figure omitted; refer to PDF] where v(_n1 )≤-1 and v(_r1 )≥v(_r0 )+1. So [figure omitted; refer to PDF] If _r1 =0 , the process stops and we write y=_n0 +1/_n1 . If _r1 ≠0 , by Lemma 3.2, there are _n2 ∈S and _r2 ∈K such that [figure omitted; refer to PDF] where v(_n2 )≤-1 and v(_r2 )≥v(_r1 )+1. So [figure omitted; refer to PDF] Continuing the process, in general, [figure omitted; refer to PDF] where [figure omitted; refer to PDF] Thus, [figure omitted; refer to PDF] We observe that the process terminates if _rk =0 . Next, we show that v(_nk )≤-k (k≥1) . By construction, we have v(_n1 )≤-1 . Assume that v(_nk )≤-k , then [figure omitted; refer to PDF] Regarding convergence, consider [figure omitted; refer to PDF] It remains to prove the uniqueness. Suppose that x∈K\{0} has two such expansions [figure omitted; refer to PDF] Since v(_∑j 1/_n1n2 ..._nj )=v(1/_n1 )≥1 and _n0 ∈S , we have _n0 =...y... . Similarly, _m0 =...y... yielding by uniqueness _n0 =_m0 and _∑j≥1 1/_n1n2 ..._nj =_∑i≥1 1/_m1m2 ..._mi . Putting [figure omitted; refer to PDF] we have _n1 ω=1+_∑j≥2 1/_n2 ..._nj and so [figure omitted; refer to PDF] By Lemma 3.2, since _n1 is the unique element in S with such property, we deduce _n1 =_m1 . Continuing in the same manner, we conclude that the two expansions are identical.

It is clear that the construction of non-archimedean ES and CEF expansions is identical which implies at once that the two representations are exactly the same in the non-archimedean case.

4. Rationality Characterization in the Non-Archimedean Case

In the case of real numbers, we have seen that both ES and CEF expansions can be used to characterize rational numbers with quite different outcomes. In the non-archimedean situation, though ES and CEF expansions are identical, their use to characterize rational elements depend significantly on the underlying nature of each specific field. We end this paper by providing information on the rationality characterization in three different non-archimedean fields, namely, the field of p -adic numbers and the two function fields, one completed with respect to the degree valuation and the other with respect to a prime-adic valuation.

The following characterization of rational numbers by p -adic ES expansions is due to Grabner and Knopfmacher [5].

Theorem 4.1.

Let x∈p_...p \{0} . Then x is rational, x=α/β , if and only if either the p -adic ES expansion of x is finite, or there exist an m and an s≥m, such that [figure omitted; refer to PDF] where γ|"β .

Now for function fields, we need more terminology. Let ... denote a field and π(x) an irreducible polynomial of degree d over ... . There are two types of valuation in the field of rational functions ...(x) , namely, the π(x) -adic valuation _|·|π , and the degree valuation _|·|1/x defined as follows. From the unique representation in ...(x) ,

[figure omitted; refer to PDF] set [figure omitted; refer to PDF] Let ...((π)) and ...((1/x)) be the completions of ...(x) , with respect to the π(x) -adic and the degree valuations, respectively. The extension of the valuations to ...((π)) and ...((1/x)) is also denoted by _|·|π and _|·|1/x .

For a characterization of rational elements, we prove the following.

Theorem 4.2.

The CEF of y∈...((π)) or in ...((1/x)) terminates if and only if y∈...(x) .

Proof.

Although the assertions in both fields ...((π)) and ...((1/x)) are the same, their respective proofs are different. In fact, when the field ... has finite characteristic, both results have already been shown in [6] and the proof given here is basically the same.

We use the notation of the last section with added subscripts π or 1/x to distinguish their corresponding meanings.

If the CEF of y in either field is finite, then y is clearly rational. It remains to prove the converse and we begin with the field ...((π)) . Assume that y∈...(x)\{0} . By construction, each _rk ∈...(x) and so can be uniquely represented in the form

[figure omitted; refer to PDF] where _pk (x), _qk (x) (≠0)∈...[x] with gcd (_pk (x),_qk (x))=1, π(x)[does not divide]_pk (x)_qk (x) . Since _nk =...1/_rk-1 ...∈_Sπ and v(_nk )≤-k , it is of the form [figure omitted; refer to PDF] where _{sv(nk )} (x),...,_s0 (x) are polynomials over ... , not all 0 , of degree <d and _mk (x)∈...[x] . Thus, [figure omitted; refer to PDF] yielding [figure omitted; refer to PDF] By construction, we have [figure omitted; refer to PDF] Substituting (4.4) and (4.5) into (4.8) and using v(_rk-1 )=-v(_nk ) lead to [figure omitted; refer to PDF] Since gcd (π(x^{)-v(_nk+1 )}_pk (x),_qk (x))=1 , it follows that _qk (x)|"_qk-1 (x) , and so successively, we have [figure omitted; refer to PDF] which together with (4.9) yield [figure omitted; refer to PDF] Using (3.19) and (4.7), we consequently have [figure omitted; refer to PDF] This shows that _{|pk (x)|1/x} ≤(1/2)_{|pk-1 (x)|1/x} for all large k which implies that from some k onwards, _pk (x)=0 , and so _rk =0 , that is, the expansion terminates.

Finally for the field ...((1/x)) , assume that y=p(x)/q(x)∈...(x)\{0} . Without loss of generality, assume deg p(x)≥deg q(x) . By the Euclidean algorithm, we have

[figure omitted; refer to PDF] where [figure omitted; refer to PDF] From the Euclidean algorithm, [figure omitted; refer to PDF] which is, in the terminology of Lemma 3.2, [figure omitted; refer to PDF] Again, from the Euclidean algorithm, [figure omitted; refer to PDF] which is, in the terminology of Lemma 3.2, [figure omitted; refer to PDF] Proceeding in the same manner, in general we have [figure omitted; refer to PDF] There must then exist k∈... such that deg _Rk =0 , that is, _Rk ∈...\{0} . Thus, the CEF of y is [figure omitted; refer to PDF] where _nk+1 =(-1^)kRk-1 q∈...[x] , which is a terminating CEF.

Acknowledgments

This work was supported by the Commission on Higher Education and the Thailand Research Fund RTA5180005 and by the Centre of Excellence In Mathematics, the Commission on Higher Education.

References

[1] O. Perron Irrationalzahlen , Chelsea, New York, NY, USA, 1951.

[2] R. Cohen, "Egyptian fraction expansions," Mathematics Magazine , vol. 46, pp. 76-80, 1973.

[3] J. W. S. Cassels Local Fields , vol. 3, of London Mathematical Society Student Texts, pp. xiv+360, Cambridge University Press, Cambridge, UK, 1986.

[4] A. Knopfmacher, J. Knopfmacher, "Series expansions in p -adic and other non-archimedean fields," Journal of Number Theory , vol. 32, no. 3, pp. 297-306, 1989.

[5] P. J. Grabner, A. Knopfmacher, "Arithmetic and metric properties of p -adic Engel series expansions," Publicationes Mathematicae Debrecen , vol. 63, no. 3, pp. 363-377, 2003.

[6] V. Laohakosol, N. Rompurk, A. Harnchoowong, "Characterizing rational elements using Knopfmachers' expansions in function fields," Thai Journal of Mathematics , vol. 4, no. 1, pp. 223-244, 2006.