Engel Series and Cohen-Egyptian Fraction Expansions
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.
It is well known  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  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 , 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 , 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 .
Theorem 2.1. Each is uniquely representable as an infinite series expansion, called its Engel series (ES) expansion, of the form where Moreover, if and only if for all sufficiently large .
Proof. Define then . If is already defined, put
Observe that is the least integer and
We now prove the folowing.Claim 2. We have .Proof. First, we show that for all by induction. If , we have seen that . Assume now that for . By (2.3), we see that . Since
and we have . If there exists such that , then
and so , contradicting the minimal property of and the Claim is proved.From the Claim and (2.3), we deduce that and . Iterating (2.4), we get
To establish convergence, let
Since and for all , the sequence of real numbers is increasing and bounded above by . Thus, exists and so
By the Claim,
showing that any real number has an ES expansion. To prove uniqueness, assume that we have two infinite such expansions such that
with the restrictions and the same restrictions also for the 's. From the restrictions, we note that
If , then by (2.12) we also have , forcing . If , then (2.12) shows that , forcing again . In either case, cancelling out the terms in (2.12) we get
Since , then
so . But there is exactly one integer satisfying these restrictions. Thus, . Cancelling out the terms and in (2.14) and repeating the arguments we see that for all .
Concerning the rationality characterization, if its ES expansion is infinite periodic of period , it clearly represents a rational number. To prove its converse, let . Since
we see that is a rational number in the interval whose denominator is . In general, from (2.4), we deduce that each is a rational number in the interval whose denominator is . But the number of rational numbers in the interval whose denominator is is finite implying that there are two least suffixes such that . Thus, by (2.3), we have . From (2.2), we know that the sequence is increasing. We must then have and the assertion follows.
Remark 2 s. In passing, we make the following observations. (a)For , we have (b)If , then and so for all . (c)If , then its ES expansion is
To construct a Cohen-Egyptian fraction expansion, we proceed as in  making use of the following lemma.
Lemma 2.2. For any , there exist a unique integer and a unique such that
Proof. Let and . Put and so To prove uniqueness, assume so that Since there is only one integer with this property, we deduce and consequently, proving the lemma.
Theorem 2.3. Each is uniquely representable as a CEF expansion of the form subject to the condition 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 , define
If , then the process stops and we write . If , by Lemma 2.2, there are unique and such that
If then the process stops and we write . If , by Lemma 2.2, there are and such that
the last inequality being followed from and . Observe also that
Continuing this process, we get
If some , then the process stops, otherwise the series convergence follows at once from
To prove uniqueness, let
with the restrictions (2.23) on both digits and . Now 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 Since , then so . But there is exactly one integer satisfying these restrictions. Then and Proceeding in the same manner, we conclude that for all .
Finally, we look at its rationality characterization. If , then , say where . From (2.30), we see that each is a rational number whose denominator is . Using this fact and the second inequality condition in (2.30), we deduce that for some , that is, the expansion terminates. On the other hand, it is clear that each terminating CEF expansion represents a rational number. Now suppose that is irrational and there is a and integer such that for all . Then
Since , it follows that 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 and the notation be as set out in Theorems 2.1 and 2.3.
(i) If , then its ES expansion is infinite periodic of period , while its CEF expansion is finite. More precisely, for , let its ES and CEF expansions be, respectively,
If is the least positive integer such that , then and the digits terminate at .
(ii) If , 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 and let be the least positive integer such that . We treat two seperate cases.
Case 1 (). In this case, we have and . Since , we get and so . We have , and so the CEF expansion terminates. On the other hand, by Remark (a) after Theorem 2.1, we have .Case 2 (). Thus, and . By Lemma 2.2, we have . For , assume that and . Then Since , again by Lemma 2.2, . This shows that Since , we have and thus From the construction of CEF, we know that . Thus, showing that . Furthermore, implying that the CEF terminates at , and by Remark (a) after Theorem 2.1, .
3. The Non-Archimedean Case
We recapitulate some facts about discrete-valued non-archimedean fields taken from [3, Chapter ]. Let be a field complete with respect to a discrete non-archimedean valuation and its ring of integers. The set is an ideal in which is both a maximal ideal and a principal ideal generated by a prime element . The quotient ring is a field, called the residue class field. Let be a set of representatives of . Every is uniquely of the shape
for some , and define the order of by , with . The head part of is defined as the finite series
Denote the set of all head parts by
The Knopfmachers' series expansion algorithm for series expansions in  proceeds as follows. For , let Define If is already defined, put
if , where and which may depend on . Then for
The process ends in a finite expansion if some . If some , then is not defined. To take care of this difficulty, we impose the condition
When , 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 has a finite or an infinite convergent non-archimedean ES expansion of the form where the digits are subject to the restrictions
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 such that , there exist a unique such that and a unique such that
Proof. Let . Then Putting , we show now that . Since , we have where , and so Thus To prove the uniqueness, assume that there exist such that and such that From , we get . If , since we have . Using , we deduce that which is a contradiction. Thus, and so .
For a non-archimedean CEF expansion, we now prove the following.
Theorem 3.3. Each has a non-archimedean CEF expansion of the form where This series representation is unique subject to the digit condition (3.19).
Proof. Define and . Then . If , the process stops and we write . If , by Lemma 3.2, there are and such that where and So If , the process stops and we write . If , by Lemma 3.2, there are and such that where and So Continuing the process, in general, where Thus, We observe that the process terminates if . Next, we show that . By construction, we have . Assume that , then Regarding convergence, consider It remains to prove the uniqueness. Suppose that has two such expansions Since and , we have . Similarly, yielding by uniqueness and . Putting we have and so By Lemma 3.2, since is the unique element in with such property, we deduce . 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 -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 -adic ES expansions is due to Grabner and Knopfmacher .
Theorem 4.1. Let . Then is rational, , if and only if either the -adic ES expansion of is finite, or there exist an and an such that where .
Now for function fields, we need more terminology. Let denote a field and an irreducible polynomial of degree over . There are two types of valuation in the field of rational functions , namely, the -adic valuation , and the degree valuation defined as follows. From the unique representation in ,
set Let and be the completions of , with respect to the -adic and the degree valuations, respectively. The extension of the valuations to and is also denoted by and .
For a characterization of rational elements, we prove the following.
Theorem 4.2. The CEF of or in terminates if and only if .
Proof. Although the assertions in both fields and are the same, their respective proofs are different. In fact, when the field has finite characteristic, both results have already been shown in  and the proof given here is basically the same.
We use the notation of the last section with added subscripts or to distinguish their corresponding meanings.
If the CEF of in either field is finite, then is clearly rational. It remains to prove the converse and we begin with the field . Assume that . By construction, each and so can be uniquely represented in the form
where with gcd. Since and , it is of the form where are polynomials over , not all , of degree and . Thus, yielding By construction, we have Substituting (4.4) and (4.5) into (4.8) and using lead to Since gcd, it follows that , and so successively, we have which together with (4.9) yield Using (3.19) and (4.7), we consequently have This shows that for all large which implies that from some onwards, , and so , that is, the expansion terminates.
Finally for the field , assume that . Without loss of generality, assume . By the Euclidean algorithm, we have
where From the Euclidean algorithm, which is, in the terminology of Lemma 3.2, Again, from the Euclidean algorithm, which is, in the terminology of Lemma 3.2, Proceeding in the same manner, in general we have There must then exist such that , that is, . Thus, the CEF of is where , which is a terminating CEF.
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.
O. Perron, Irrationalzahlen, Chelsea, New York, NY, USA, 1951.