Talk:Egyptian fraction

message: This:
: Assume 7. 1 and 1/7 of 7 is 8.
was recently changed to this:
: Assume 7. 7 and 1/7 of 7 is 8.
with description "A hard-to-spot mistake, I haven't read the papyrus but this is the only thing which makes sense)".
This is not necessarily so, for another interpretation of "1 and 1/7 of 7" could be "11/7 of 7". So I think we should stick with the old version. --Romanm 07:47, 23 Jul 2004 (UTC)
thread_title: Untitled

message: The article says:
:Ahmes converted 19/20 by simply testing the first partition  1/2 within a well known ancient method, known as Hultsch-Bruins, or
::19/20 = 1/2 + 18/40, 
:and solved 18/40 by looking for the divisors of 40, or 20, 10, 8, 5, 4, 2, 1 that add up to 18. Note that by replacing 18 by (10 + 8), or
:: 19/20 = 1/2 + (10 + 8)/40
:::: = 1/2 + 1/4 + 1/5
The choice of 18/40 here puzzles me, and is given without explanation.  Why wouldn't Ahmes have started with 9/20 and tried to find a subset of {10, 5, 4, 2, 1} that added to 9?  This gives:
: 19/20 =  1/2 + (5 + 4)/20
::: = 1/2 + 1/4 + 1/5
more simply.  What am I missing here?  -- Dominus 00:29, 8 May 2006 (UTC)
replies: text: I have no idea. Maybe Ahmes had a penchant for never writing fractions in lowest terms? --King Bee 15:15, 3 December 2006 (UTC)
thread_title:  Puzzling example 

message: The section at the end of "Fractions in Egypt" dealing with 5/7 seems to be out of place or incomplete.  It was in "Notation" and seemed out of place there so I moved it to "Fractions in Egypt." There still seems to be some loss on continuity.  --Jbergquist 23:08, 17 June 2006 (UTC)
thread_title: Fragment out of place?

message: I feel too close to this material (note my page on the subject is one of only three external references) and too uninterested in its non-mathematical aspects to be a good editor for it, but I am quite unhappy with the present treatment. I feel that it fails both WP:NPOV and WP:OR, that it contains outright falsehoods, and that by paying attention only to analysis of ancient methods it slights some very notable mathematical aspects of the problem.
::The ancient scribes said what they said. To not report the scribal view of Egyptian fractions as the ancient texts report its methods seems unfair, and inappropriate. (Milo)
Some specific complaints with what's included here:
*This quote:
::But first, it should be noted that modern Harmonic series can be used to prove every positive rational number can be written in the form of an Egyption fraction. However, modern methodologies produce awkward and relatively longer unit fraction series compared to the shorter and more concise ancient methods. (Milo)
:is false. Modern computer algorithms are easily capable of finding optimally short series for numbers of the magnitude used in the examples here, and good mathematical bounds on the size of the expansion are proven by G. Tenenbaum and H. Yokota. Length and denominators of Egyptian fractions. J. Number Th. 35, 1990, pp. 150–­156, and by M. Vose. Egyptian fractions. Bull. Lond. Math. Soc. 17, 1985, p. 21. Also the attitude that the ancients knew better than we know now fails WP:NPOV.
*Too much off-topic discussion of the origins of algebra. It's notable, just not on-topic for this page.
*No citations for Hultsch-Bruins. Is this original research?
::No this is not original research, F. Hultsch first published it 
in 1895. Its 2/p form is documented in the RMP 2/nth table was independently confirmed by E.M. Bruins in 1944, hence the name modern Hultsch-Bruins, as generalized by
n/p - 1/A = (nA -p)/aP
with A being chosen in the range p/n < A< p and the divisors of A
being used to optimally find the vale (nA -p).(Milo, 11/10/07)
*Long sequences of calculations that add verbiage but not a lot of understanding.
...and what's not included:
Egyptian fraction notation changed during Egyptian eras. To be
brief Middle Kingdom Egyptians were first to use the fully
developed method of writing n/p and n/pq by exact and concise
Egyptian fraction series, methods that included a proof that
each statement was correct. By the time of the Greeks and Archimedes
the exact way of writing Egyptian fraction series were stressed in proofs for infinite series situtations (for example Archimedes'
Lemma 4A/3 = A + A/4 + A/16, ... became A + A/4 + A/12). Later, 
Byzantine and Middle Ages mathematicians tended to drop the last 
step of writing its algebra and every day math, the conversion of
vulgar fractions to Egyptian fraction series. This time period tended 
to write its final algebra and higher math answers as vulgar fractions, or as we call today improper fractions, steps that had long been used in earlier algebra. As a user of several notations, Fibonacci 1202 AD was the one of the last Middle Ages writer to fully use the Egyptian fraction notation, while at the same time experimenting with modern greedy algorithms and other ideas form a wider region ( such as math that came from Islamic and Vedic mathematicians, as Diophantus
had earlier used Chinese math and the Chinese Remainder Theorem to solve
indeterminate equations). By 1600 AD modern base 10 decimals came onto the scene and attempted to remove Egyptian fraction notation from our math history memories. Sadly, the 3,600 year history of finding optimal partitioning values A, and its aliquot parts, needed to solve n/p and n/pq conversions to exact unit fraction series remain only as faint ways in most math history books. The death of the Egyptian fraction notation, and its many versions and uses, has sadly not been properly noted. Egyptian fractions had survived for over 3,600 years, longer than any notation in the Western Tradition. Yet, many modern obituaries on the topic have often been negative, such as Otto Neugebaur's "Exact Science in Antiquity" - where Neugebaur falsely suggests, without proof, that Egyptian fractions marked intellectual decline in the RMP, rather than its actual historical role, intellecutal advancement. (Milo 11/11/06).
*Sufficient historical context. Who, exactly, used this notation for what time period, and why did they stop?
The improvement of Huygens' telescope was one of the major reasons 
while Egyptian fractions are not used today. Napier developed logs scales, needed to find planetary bodies within base 10 decimals, 
thereby creating more accurate and smaller units of measure. The new
notation freshly used zero as a place-holder. The new base 10
decimal system notation had not been used or anticipated in any
way by Egyptian, Greek, Roman and Middle Age mathematicians in their practical and theoretical mathematics. Yet, all the mathematics of 
the ancients are worth studying, since they developed both practical
and theoretical arithmetic, alegbra, geometry and calculus within its Egyptian fraction notation that intellectually empowered it.(Milo 11/11/06).
*More references. The subject has a big literature which is barely touched on here. See http://www.ics.uci.edu/~eppstein/numth/egypt/refs.html for starters. As another instance of a reference that might make for an interesting cross-link with another WP entry, Ron Graham's Ph.D. thesis was on Egyptian fractions.
*The existence of modern methods other than greedy. The greedy algorithm is interesting mostly for its pathological behavior. Modern number theorists have developed much more effective methods. In particular the connection with continued fractions seems important enough to mention: the sequence of continued fraction convergents to a number gives something almost exactly like an Egyptian fraction series, but with both positive and negative signs, and it can be converted to a true Egyptian fraction using the secondary series of the continued fraction. (M. N. Bleicher. A new algorithm for the expansion of continued fractions. J. Number Th. 4, 1972, pp. 342–­382.)
::Modern number methods do not commmonly factor the rational number being converted to a unit fraction series. Hence the greedy algorithm, as Stan Wagon "Old and New Unsolved Problems in Plane Geometry and Number Theory mangling easy problems like 5/121, writing out an improved 1/25 + 1/1225 + 1/3477 + 1/7081 + 1/11737, from the very awkward greedy algorithm with a 25 digit last term denominator. Egyptian scribes simply factored 5/121 into 1/11 x 5/11 and wrote 5/11 = 1/3 + 1/11 + 1/33, and 5/121 = 1/33 + 1/121 + 1/363. (Milo. 11/10/06)
:::The greedy method is not modern, and some modern methods do in fact depend on factorization.
. On the other hand, it would not surprise me at all if the scribes did exactly as you say. But do you have a source for that specific expansion? —David Eppstein 20:57, 8 November 2006 (UTC)
replies: text: Too right. Article started out as more an essay on egyptain mathematics, see for example this version. It still needs a lot of work. Edit at will. --Salix alba (talk) 07:35, 8 September 2006 (UTC)

text: I can help with some of this, but much is beyond my level of expertise. I understand your reticence, but I hope you will edit as much as you feel comfortable doing. Other than avoiding linking to your own material, I know of no other restrictions. Tom Harrison Talk 14:21, 8 September 2006 (UTC)
replies: text: Ok, I'll think about it for a few more days but probably do some editing. For future reference I note that the 4/n problem has its own article: Erdős-Strauss conjecture (which also needs work; I'll probably deal with that earlier), and therefore doesn't need a lot of detail here. —David Eppstein 23:29, 8 September 2006 (UTC)
replies: text: Good stuff. It finally looks like a decient article. --Salix alba (talk) 01:10, 1 October 2006 (UTC)
replies: text: Thanks! There's a little more left to do (the greedy algorithm redlinks), but I'm pretty pleased with how this came out. —David Eppstein 05:02, 1 October 2006 (UTC)
thread_title: Unhappy

message: it is very confusing!!!!!!!!!!!!!!!!! —The preceding unsigned comment was added by 195.147.207.170 (talk • contribs) 12:20, 14 November 2006 (UTC)
replies: text: Could you be more specific? Which sections do you find confusing, and what parts of them do you have trouble understanding? —David Eppstein 15:55, 14 November 2006 (UTC)
thread_title: Confusing

message: Dear David: Your use of Struik, 1967, as a guide to unfairly degrades ancient Egyptian fractions. Struik's views looks and acts a great deal like Otto Neugebauer's improper reporting of the 2/th table in Exact Sciences in Antiquity. There is no question that base 10 decimals replaced the Egyptian fracion numeration system, the later having domimated Western math for 3,600 years. Yet, Struik and Neugebauer like to declare that only the Greeks developed abstract forms of arithmetic, and that ancient Egyptian fractions showed intellectual decline in the RMP, rather than reporting the fact that every noted Greek mathematician studied in Egypt, learning its mathematics beginning with its numeration system. That is, a broader view of the history of math and the history of science needs to be offered before your  'time warp' jump to modern greedy algorithm studies like your own, as I  first read in Intelligencer in 1991. Best Regards, Milo
*** Dear David, I understand that you are using Struik in a way that is not consistent with the above. However, Struik's point of view only cites a late Hellene text, and not the 2,200 year older Egyptian fraction system, say as Brown discussed. The Babylonian base 60 system rounded off every n/p and n/pq conversion in its awkward and incomplete unit fraction system. Whenever p and pq were not multiples of 2,3 and 5 Babylonian methods, that Struik oddly prized highly, offered  poor approximations. Egyptians beginning in 2,000 BCE always wrote exact unit fraction series for its n/p and n/pq conversion. Hence, Struik was incorrect in his evaluation, and his name and conclusion should be removed from Wikpedia. Best Regards, Milo 11/24/06.
****
::This differs how from the use of rounding in modern decimal notation? —David Eppstein 17:19, 23 November 2006 (UTC)
replies: text: I am using Struik only as a reference for the assertion that these fractions continued to be used practically into medieval times, and for Ptolemy's complaints. I don't think there is anything in the article about Greek supremacy or intellectual decline, though perhaps there is content like that elsewhere in Struik. In what sense is the current treatment unfair, or missing any important history of this specific notation? This is not the place for a broader history of mathematics or of the Egyptians' contributions to abstract thought. —David Eppstein 15:50, 16 November 2006 (UTC)

text: Like it or not, modern number theory is an important aspect of this subject and much of the number theory involved is far from recreational. The use of "Egyptian fraction" to refer to all aspects of mathematics involving sums of distinct unit fractions, not just ancient methods, is not and should not be in question. It is standard mathematical usage, and anything else would be a violation of WP:NPOV, as I already discussed much earlier under the "unhappy" heading. Similarly, I personally am not especially interested in the ancient history, but I have included it prominently in the article (in the longest and first section) because it is also an important part of the subject, and omitting it would be a violation of WP:NPOV. I am not prepared to seriously consider any suggestion of scaling back this article to discuss only one side of the subject. If you want an article that reflects only the historical aspects, unadulterated with modernism, go to Egyptian mathematics or go to some other site than Wikipedia. —David Eppstein 00:55, 17 November 2006 (UTC)

text: Why do you continue to claim that the greedy method is modern? Your insistence on using the greedy algorithm as the paragon of modernism, when so much better algorithms are known now, comes off as intellectually dishonest. If the article were claiming that the Egyptians used the greedy method, or that the greedy method was somehow superior to what the Egyptians did, it would be mistaken and need fixing, but I don't believe that the article as written supports either of those statements. —David Eppstein 23:31, 17 November 2006 (UTC)

text: That last sentence epitomizes what I find objectionable about the writing of the article before I made such major revisions to it, writing that I believe you were largely responsible for: (1) It is run-on, badly written, and difficult to comprehend. (2) It tries to pass off the entire history of Egyptian fractions as a conflict between Hultsch and Fibonacci, ignoring any recent contributions to the subject. (3) Its claim that a method described in 1200 is the epitomy of modernism while a method described in 1895 is ancient comes off as strange and ahistorical. (4) Its claim that modern methods are incapable of finding optimally short expansions is false.
*** David, show me any modern number theory person that have duplicated the RMP 2/n table's optimal series, as David Brown discussed. Please also note that the Coptic Akhmim Papyrus wrote n/p conversion tables listing n upto and including. 32. Brown published the Coptic n= 17 and n= 19 tables, available somewhere on his web page. Clearly, by inspection, the
ancient scribes found relatively more optimal series than published by any modern mathematician used the lastest 'brute force' algorithms,
as implied by Brown and stated by myself. Milo 11/18/06
**** 
 (5) Its claim that modern methods are incapable of finding expansions with the minimum possible denominator is false.
***
Claiming that some idea is false is easy, in words, but very hard to 
achieve by modern algorithms/proofs. Ancients used algebraic identities,
as F. Hultsch showed in 1895, per the rule, n/p = 1/A + (nA-p)/Ap
as we previously discussed 10 years ago (even cititng your following example, as I recall.) Yes, we have already discussed the 31/311 case.
It can be found by H-B by taking a second differential, 1/A and 1/B.
Our current questions deal with the published 2/n table from 1650 BCE, where n ranged upto and including 101, and n = 32 with often more optimal series in the 400 AD Akhmim Papyrus. Where are the modern algorithms
that duplicate and or improve on the ancients? Since there are none,
to my knowledge, the ball is in your court. Best Wishes, Milo 11/18/06. **** 
 (6) Its implication that there always exists a single expansion that is both optimally short and having the minimum possible denominator is false. Example: 31/311 = 1/12 + 1/90 + 1/311 + 1/1244 + 1/1866 + 1/2799 + 1/3110 = 1/12 + 1/63 + 1/2799 + 1/8708; there is no four-term expansion having maximum denominator 3110. —David Eppstein 17:03, 18 November 2006 (UTC)

text: By the way, regarding Hultsch's method — it certainly has some explanatory power, as not all expansions have the form he describes. For instance the two expansions I list above for 31/311 do not include 1/A as terms in the expansion for A=180 and A=252 respectively. So I suppose that 1/12 + 1/63 + 1/2799 + 1/8708 would not be generated by this method, 
***
This looks like the solution that I sent you 10 years ago, and a second
subtaction step, using the second step, 1/B = 1/63 and its combined divisors with 1/A = 1/12, solves the problem in the H-B style. Best
Regards, Milo 11/19/06.
***
and in fact Hultsch's method seems unable to find a four-term expansion for 31/331 no matter which A is chosen. But more generally the method does not specify how to decide which A to use, nor how to find the best subset of divisors, so I would not call it an algorithm.
***Agreed, on the point concerning the rule for selecting A or B. There are no known rules other than by inspection, thereby showing the degree of work put in by the student/scribe(a form of recreational math?). Here I agree with you, on one point. The ancient H-B method and its clones tended not to follow algorithmic rules. Your methods certainly are algorithms. Hence we have been discussing apples and oranges QED. Milo 11/18/06. 
****
 The latter part of the problem, selecting an appropriate subset of divisors, is itself nontrivial: see subset sum problem. —David Eppstein 18:34, 18 November 2006 (UTC)

text: I am not citing Weisstein for any historical information, but as a relevant web page on related material. I am aware that his site contains mistakes. Brown is not an adequate citation for historical information for reasons I expand on here, but I welcome the addition of a link to his site. —David Eppstein 16:45, 20 November 2006 (UTC)
thread_title:  Struik 

message: The distinction between a method that involves two subtractions and a method that involves n subtractions is subtle to nonexistent, as far as I am concerned. Fibonacci, in the translation I have, takes six lines to perform the first step for the expansion of 17/29, into 1/2 + 5/58 or as it is written in my translation \scriptstyle\frac{5}{58}\,\frac{1}{2}. He then states "\scriptstyle\frac{5}{58} must be made into unit fraction parts, namely by the same distinction" (distinction is the word used for the seven methods he lists, of which the greedy algorithm is the seventh).  That is, it is very obvious that he is going through his list of methods and applying the first one that fits. Writing down a list of methods, applying the first one that fits, and (if the method happens to be the seventh) going through the same procedure for the remainder is certainly an algorithm, though it is not quite the same as the greedy algorithm because of the possibility that one of the other methods might be applied in one iteration. He says nothing about applying this method only if the number of subtractions can be limited to two, he merely applies the same sequence of choices to the remainder 5/58. That to me implies possibly unbounded iteration. I have seen modern algorithms expressed less clearly, that are still clearly intended as algorithms.
His other example of 4/13 is possibly more interesting for the history of the greedy algorithm specifically, though, because he expands it two ways. First, he expands it as I have described above: using the first applicable distinction (the seventh greedy one) and then using the first applicable distinction on the remainder. This greedy step produces the unit fraction 1/4 and a remaining fraction which he writes in his mixed radix notation as \scriptstyle\frac{3\ 0}{4\ 13} (note the zero as placeholder contradicting your claim that this usage did not occur until much later), meaning the vulgar fraction 3/52.
Keeping it in mixed notation he expands it as \scriptstyle\frac{1\ 0}{2\ 13} + \scriptstyle\frac{1\ 0}{4\ 13} using his second distinction, from which he finds the non-greedy expansion \scriptstyle\frac{1}{52}\,\frac{1}{26}\,\frac{1}{4}. So far, applying the same method as the 17/29 example, which is not quite the same as the greedy method because of the possibility of switching to a different distinction at later iterations. But the part that convinces me that Fibonacci had in mind the pure greedy algorithm, unbounded iteration and all, occurs when he, after this, mentions that the expansion of 4/13 can be found in another way, in which one applies the seventh distinction again to the remainder 3/52, and after several lines of calculation he reaches the pure greedy expansion \scriptstyle\frac{1}{468}\,\frac{1}{18}\,\frac{1}{4}.
You can cavil all you want that he only ever applies two subtractions. But he is presenting finite examples: of course they do not continue indefinitely. If he had presented expansions with at most five terms, would you then call them "the fourth subtraction pseudo-greedy proto-algorithm algebraic identity method"? Well, given what I've seen of your writing so far, you probably would, but it would be wrong. To me it is clear: he describes a method (his seventh distinction, in the version described for the second 4/13 example and in the 17/29 example) which implicitly requires applying the same method to whatever remainder is left, if it not be a unit. That is all that is needed to described the greedy algorithm as we would call it modernly.
Re your idiosyncratic attempt at using the word "algorithm" in some imagined historic sense: we are writing in modern English, for readers of modern English. Therefore "algorithm" must be taken in its modern meaning.
As for whether "first subtraction" and "second subtraction" are acceptable terms to use in an article for a general audience without some definition of what those terms mean: no. And WP:NEO to me implies that you should avoid such definitions unless absolutely necessary. I see no such necessity here.
Until I can be persuaded that you are willing to be reasoned with, and I find your recent additions to the talk page very unpersuasive in this respect, I am going to eschew such long responses as a waste of my time, and return to a policy of reverting your mistakes without the courtesy of an explanation. And I am going to continue to be frustrated at having to waste my time policing what would remain a perfectly satisfactory article if you could only be persuaded to leave it alone.
—David Eppstein 01:01, 17 December 2006 (UTC)
thread_title:  Zero as placeholder, and the distinction between two subtractions and n subtractions 

message: LeadSongDog added the following references as part of a new "further reading" section:
#
#  
#  
#  
#  
#  taken from "Die Altagyptschen Hohlmass, Buske-Verlag, 2005. 
#  
# 
My impression is that this is heavily weighted towards the history of Egyptian mathematics (not the primary subject of this article) and away from the mathematics of Egyptian fractions specifically. Additionally, several of these references duplicate ones already present in the references section. Could I see some explanation, please, of what specifically these references have to add to the subject of Egyptian fractions (as distinct from Egyptian mathematics more generally) and why they should be listed as further reading rather than cited as references within the article? In the meantime, I have undone the changes. —David Eppstein (talk) 19:36, 7 October 2008 (UTC)
replies: text: It's an attempt to put an end to the blog add/delete cycle by providing the underlying information to support direct referencing.  If you prefer them on the talk page, thats cool with me. I stumbled in to what appeared to be a revert war and tried to find a way out.  Do as you like.LeadSongDog (talk) 19:49, 7 October 2008 (UTC)
replies: text: You mean the edit war from last July? —David Eppstein (talk) 03:17, 8 October 2008 (UTC)
replies: text: Gads!  Don't I look the dolt now...  Sorry for the fuss.  LeadSongDog (talk) 03:38, 8 October 2008 (UTC)
thread_title:  Further reading 

message: Question: Is it possible to decide, whether a fraction n/p can be represented as a sum of exact two fractions? If so, how could these fractions be computed? There a known special cases, e.g. 2/(2k-1) = 1/k + 1/(k(2k-1)), but what about the general case?  —Preceding unsigned comment added by 141.20.50.147 (talk) 11:50, 5 December 2008 (UTC)
replies: text: From http://www.ics.uci.edu/~eppstein/numth/egypt/force.html : to solve the equation x/y=1/a + 1/b; rewrite it as (ax-y)(bx-y)=y^2, and letting the two factors of y^2 be r and s we can solve a=(r+y)/x, b=(s+y)/x. Simply try all factors r of y^2 for which r<y and see which ones work. It's also possible to use a slower but simpler brute force search in which you try all values of a between y/x and 2y/x. —David Eppstein (talk) 16:08, 5 December 2008 (UTC)
thread_title:  Sum of exact two fractions 

message: The article does not mention any applications of Egyptian fractions, but applications do presumably exist. I am not aware of what they are and would be glad to know. If anyone can include mention of applications, that would be appreciated. Thanks. 74.195.16.39 (talk) 16:32, 20 April 2009 (UTC)
replies: text: What are the applications of arithmetic? Egyptian fractions are primarily a system of arithmetic, used long ago and now replaced by different notation. The modern number-theoretic study of them is largely motivated by concerns of pure mathematics rather than applications. —David Eppstein (talk) 16:42, 20 April 2009 (UTC)
thread_title: Applications

message: Are dyadic rationals the set of numbers of the form 1/2^n as this page says or m/2^n as the dyadic rational page says?
Dakane2 (talk) 04:36, 10 May 2011 (UTC)
replies: text: The m/2^n definition is the correct one. But the numbers 1/2^n are examples of dyadic rationals (just not all possible dyadic rationals) and using sums of them any dyadic rational can be formed. —David Eppstein (talk) 04:46, 10 May 2011 (UTC)
thread_title:  Dyadic rational definition 

message: (To David Eppstein). Look now. I have been doing and publishing research in analytic number theory for forty years, and one of the main topics I studied is Omega results. I know exactly what the symbol means. It is never used to replace a number. You can write that the number of terms needed = Omega(loglogx), meaning that there is a relation between the number of terms and the function loglogx, or alternately that the function counting the number of terms belongs to a certain set of functions.  But you cannot use Omega(loglogx) in order to denote a number of objects. It is just never done in the literature. Also, there is no need to be insulting in your comments. Sapphorain (talk) 07:02, 19 April 2016 (UTC)
replies: text: How could "the number of terms needed = Omega(loglogx)" be any different than "the number of terms needed is Omega(loglogx)"? Especially given that "=Omega" is in my experience usually pronounced out loud as "is Omega" as It's not actually Intended as an equalIty? But I admIt that my experIence wIth Omega notatIon Is In theoretIcal computer scIence and combInatorIcs, where the conventIons may be dIfferent (certaInly the defInItIon of Omega Is somewhat dIfferent) and Its usual use Is Indeed to lower bound (not denote) a number of objects, In exactly the same way that O-notatIon Is used In the same paragraph to upper bound the same number of objects. ThIs artIcle Is not partIcularly analytIc, but It Is number theory, so I suppose that takes precedence. —David Eppstein (talk) 07:21, 19 April 2016 (UTC)
thread_title:  Incorrect use of the symbol Omega 

message: Is the Erdős–Graham Conjecture stated correctly?  If the sets contain unit fractions, then their reciprocals are natural numbers greater than 1.  It would be pretty hard to find a set of those that sum to 1.  I think we must either define the sets as containing natural numbers greater than 1, or take out the "reciprocals". Sicherman (talk) 03:53, 30 October 2017 (UTC)
replies: text: Yes. Changed to integers instead of unit fractions. Thanks for catching this. —David Eppstein (talk) 05:02, 30 October 2017 (UTC)
replies: text: Glad to help!  But you omitted the critical phrase "greater than 1." I have supplied it. Sicherman (talk) 02:07, 31 October 2017 (UTC)
thread_title: Erdős–Graham Conjecture

message: From what I can tell "Egyptian fractions" were the main representation used throughout the Mediterranean and Greek- and Arabic-writing world throughout the medieval period (coexisting with Mesopotamian sexagesimal arithmetic), only displaced late by "common fraction" techniques using Hindu–Arabic numerals. This article hand-waves thousands of years of history with not even a sentence as "continued to be used in Greek times and into the Middle Ages", with organization that forestalls future expansion about it. The substantial focus on Fibonacci per se, without any mention of e.g. medieval Arabic sources, seems like a "neutral point of view" problem. More generally, the focus on algorithms expressed in modern notation pulls the subject from context and gives an anachronistic impression of its substance. The approach seems appropriate for a modern math book, but not really sufficient for a general encyclopedia. –jacobolus (t) 13:51, 26 October 2023 (UTC)
replies: text: Example source: . –jacobolus (t) 15:55, 26 October 2023 (UTC)
thread_title:  History sections seem substantially inadequate