Egyptian Fraction

The Rhind Mathematical Papyrus

A finite sum of distinct unit fractions is, rather unimaginatively, termed an Egyptian fraction . Consider, if you must, an expression such as:

$${\frac {1}{2}}+{\frac {1}{3}}+{\frac {1}{16}}.$$

Each component in this rather particular sum adheres to a rigid set of rules: its numerator must be precisely 1, and its denominator must be a positive integer . Furthermore, and this is where the distinct part becomes inconveniently relevant, all these denominators must differ from one another. The result of such an arcane summation is inevitably a positive rational number , typically represented as $a/b$. For instance, the example provided, that collection of unit fractions, inconveniently adds up to $43/48$. It’s a testament to the stubbornness of mathematics that every positive rational number can be expressed in this manner. These sums, along with similar constructions that occasionally permitted the inclusion of $2/3$ and $3/4$ as summands , weren’t just ancient mathematical curiosities; they formed a serious, albeit cumbersome, system for representing rational numbers for the ancient Egyptians. This peculiar notation persisted, finding use in various other civilizations well into medieval times. In the grand march of modern mathematical notation, Egyptian fractions have, predictably, been superseded by the more straightforward vulgar fractions and the ubiquitous decimal notation. Yet, like a persistent echo from the past, Egyptian fractions continue to be a subject of study in contemporary number theory and recreational mathematics , and, of course, they remain a fascinating, if slightly baffling, artifact in modern historical analyses of ancient mathematics .

Applications

Beyond their purely historical significance, Egyptian fractions occasionally offer some rather unexpected practical advantages over other methods of representing fractional numbers. One might even call them useful, if one were inclined to such optimism.

For example, when faced with the arduous task of dividing sustenance or other tangible objects into equitable portions, Egyptian fractions can, surprisingly, lend a hand. Imagine, if you dare, the scenario: one is tasked with distributing 5 pizzas equally among 8 eager diners. The Egyptian fraction representation of this predicament, $5/8 = 1/2 + 1/8$, offers a rather elegant, if slightly indirect, solution. It dictates that each diner receives precisely half a pizza, plus an additional eighth of a pizza. This can be pragmatically achieved by, say, bisecting 4 of the pizzas to yield 8 halves, and then meticulously dividing the single remaining pizza into 8 eighths. Exercises in this specific type of fair division of food are not merely theoretical; they are a standard, if somewhat simplistic, classroom example employed to introduce students to the fascinating, or perhaps frustrating, world of unit fractions.

Furthermore, Egyptian fractions can provide a rather ingenious solution to certain rope-burning puzzles . These puzzles typically involve measuring a specific duration of time by igniting non-uniform ropes, each of which burns completely after a unit time, but at an uneven rate. Any rational fraction of a unit of time can be measured by first expanding the fraction into a sum of unit fractions. Then, for each unit fraction $1/x$, a rope is burned in such a way that it maintains $x$ simultaneously lit points along its length. It’s worth noting that for this particular application, the unit fractions are not strictly required to be distinct from one another. However, one must brace oneself for the inherent inefficiency: this solution may necessitate an infinite number of re-lighting steps, a minor detail often overlooked in the initial enthusiasm.

Early history

The notation for Egyptian fractions , a system whose persistence still baffles some, saw its initial development during the illustrious Middle Kingdom of Egypt . Among the earliest surviving mathematical texts that showcase the use of these fractions are the venerable Egyptian Mathematical Leather Roll , the Moscow Mathematical Papyrus , the Reisner Papyrus , the Kahun Papyrus , and the Akhmim Wooden Tablet . A somewhat later, and arguably more refined, text, the Rhind Mathematical Papyrus , introduced what were considered improved methods for writing Egyptian fractions . This particular papyrus, meticulously penned by the scribe Ahmes , dates from the Second Intermediate Period of ancient Egypt. Its contents are rather comprehensive, including an extensive table of Egyptian fraction expansions specifically for rational numbers of the form $2/n$. Beyond this table, the papyrus also contains 84 word problems , each accompanied by its solution. These solutions were recorded in a concise scribal shorthand, with the final answers to all 84 problems consistently expressed using the Egyptian fraction notation. Similar tables of expansions for $2/n$ can also be found in some of the other aforementioned ancient texts. However, one must not be misled into believing Egyptian fractions were the only game in town; as the Kahun Papyrus clearly demonstrates, vulgar fractions were still very much in use by scribes within their computational processes. Apparently, even ancient Egyptians understood the concept of having multiple tools for a task, even if some were more cumbersome than others.

Notation

To denote the unit fractions that formed the backbone of their Egyptian fraction notation, the ancient Egyptians employed a rather distinctive approach in their hieroglyphic script. They would place a specific hieroglyph , which resembled an open mouth and was pronounced “er” (meaning “[one] among”) or possibly “re”, directly above a numerical symbol. This visual convention served to represent the reciprocal of the number beneath it. For example, to express $1/3$, one would see:

$$\text{=}{\frac {1}{3}}$$

And similarly, for $1/10$:

$$\text{=}{\frac {1}{10}}$$

In the more cursive hieratic script, this concept of reciprocity was conveyed by simply drawing a line over the character that represented the number. A more efficient, if less visually dramatic, solution.

The Egyptians, ever practical within their own complex system, also developed special, dedicated symbols for $1/2$, $2/3$, and $3/4$. These particular symbols were not merely decorative; they served a crucial function in reducing the overall complexity and visual length of Egyptian fraction series, especially when dealing with numbers greater than $1/2$. By subtracting one of these special fractions first, the remaining, typically smaller, number could then be expressed as a sum of distinct unit fractions using the standard Egyptian fraction notation.

$$\text{=}{\frac {1}{2}}$$

$$\text{=}{\frac {2}{3}}$$

$$\text{=}{\frac {3}{4}}$$

Beyond these standard notations, the Egyptians also employed an alternative system, a modification dating from the Old Kingdom , to denote a specific set of fractions. These were of the form $1/2^k$ (for $k=1, 2, \ldots, 6$) and sums derived from these numbers, which are, by their very nature, dyadic rational numbers. These have, rather romantically, been dubbed “Horus-Eye fractions.” This nomenclature arose from a theory, now widely discredited (as so many human theories eventually are), suggesting that these symbols were visually derived from the segmented parts of the Eye of Horus . Regardless of their disputed origin, these fractions were utilized during the Middle Kingdom in conjunction with the later notation for Egyptian fractions to subdivide a hekat . The hekat was the primary ancient Egyptian unit of volume, used for measuring grains, bread, and other smaller quantities of goods, as meticulously documented in the Akhmim Wooden Tablet . Should any remainder stubbornly persist after a quantity had been expressed using these Eye of Horus fractions of a hekat, that remainder was then written using the conventional Egyptian fraction notation, expressed as multiples of a ro, a smaller unit precisely equal to $1/320$ of a hekat. Because, of course, simplicity was never the Egyptian way.

Calculation methods

Modern historians of mathematics, with a dedication that borders on the obsessive, have meticulously scrutinized the Rhind papyrus and other ancient sources. Their noble, if somewhat Sisyphean, goal: to uncover the precise methods the Egyptians employed in their calculations involving Egyptian fractions . A significant portion of this scholarly endeavor has focused on deciphering the tables of expansions for numbers of the form $2/n$ found within the Rhind papyrus . While these expansions can, with the benefit of hindsight and modern algebraic tools, generally be articulated as algebraic identities, it’s crucial to acknowledge that the actual methods utilized by the ancient Egyptians might not directly align with these contemporary formulations. Furthermore, the expansions presented in the table are not uniformly governed by a single, overarching identity. Instead, different identities appear to apply to expansions for prime denominators versus composite denominators, and, to add another layer of complexity, more than one identity often fits the numbers within each category. It seems even ancient mathematicians enjoyed a bit of variety in their algorithms.

Let’s dissect some of these methods, as far as modern analysis can discern:

• For small odd prime denominators $p$, a straightforward expansion was frequently employed: $${\frac {2}{p}}={\frac {1}{(p+1)/2}}+{\frac {1}{p(p+1)/2}}$$ This identity provides a direct, two-term expansion for these specific primes, a rare moment of relative simplicity in the Egyptian system.

• For larger prime denominators, the Egyptians often resorted to a more involved method, utilizing an expansion of the general form: $${\frac {2}{p}}={\frac {1}{A}}+{\frac {2A-p}{Ap}}$$ Here, $A$ represents a number deliberately chosen for its abundance of divisors (such as a practical number ), positioned strategically between $p/2$ and $p$. The remaining, somewhat awkward, term $(2A-p)/Ap$ was then further expanded. This was achieved by expressing the numerator, $2A-p$, as a sum of the divisors of $A$. Subsequently, a fraction of the form $d/Ap$ was constructed for each such divisor $d$ in this sum. As a concrete illustration, consider Ahmes ’ expansion for $2/37$, which he presented as $1/24 + 1/111 + 1/296$. This perfectly aligns with the described pattern if we choose $A=24$. In this case, $2A-p = 2(24) - 37 = 48 - 37 = 11$. This 11 can be expressed as the sum of divisors of 24, specifically $8+3$. Thus, we see how $1/111$ arises from $8/(24 \cdot 37)$ and $1/296$ from $3/(24 \cdot 37)$. It’s worth noting that multiple expansions of this type could exist for a given $p$. However, as K. S. Brown astutely observed, the expansion favored by the Egyptians was often the one that resulted in the smallest possible largest denominator among all valid expansions fitting this pattern. A preference for “least worst,” perhaps.

• For certain composite denominators, specifically those that could be factored as $p \cdot q$, the expansion for $2/(pq)$ often mirrored an existing expansion for $2/p$, with the rather simple modification that each denominator in the $2/p$ expansion was merely multiplied by $q$. This method appears to have been a common strategy for many of the composite numbers found in the Rhind papyrus . However, one must always account for the exceptions that prove the rule, notably $2/35$, $2/91$, and $2/95$, which stubbornly refused to conform to this pattern.

• Another method for composite denominators involved the identity: $${\frac {2}{pq}}={\frac {1}{p(p+q)/2}}+{\frac {1}{q(p+q)/2}}$$ For instance, Ahmes applied this to expand $2/35 = 2/(5 \cdot 7)$ as $1/30 + 1/42$. Later scribes, demonstrating a subtle evolution in their mathematical thinking, utilized a more generalized form of this expansion: $${\frac {n}{pq}}={\frac {1}{p(p+q)/n}}+{\frac {1}{q(p+q)/n}}$$ This more versatile identity proved effective whenever $p+q$ happened to be a multiple of $n$.

• Finally, the very last (prime) expansion recorded in the Rhind papyrus , for $2/101$, deviates from all the preceding forms. Instead, it employs a distinct expansion: $${\frac {2}{p}}={\frac {1}{p}}+{\frac {1}{2p}}+{\frac {1}{3p}}+{\frac {1}{6p}}$$ This particular identity possesses the rather convenient property of being applicable regardless of the specific value of $p$. Thus, $2/101$ becomes $1/101 + 1/202 + 1/303 + 1/606$. A related, though not identical, expansion was also observed in the Egyptian Mathematical Leather Roll for several cases, suggesting a recurring, if not universally applied, principle.

Later usage

The notation for Egyptian fractions , with its inherent awkwardness, managed to persist through Greek times and well into the Middle Ages . This enduring presence is somewhat remarkable, especially considering the complaints voiced as early as Ptolemy ’s monumental work, the Almagest . Ptolemy, apparently a proponent of efficiency, bemoaned the clumsiness of the Egyptian fraction system when contrasted with alternatives, such as the more elegant Babylonian base-60 notation . It seems the struggle between tradition and practicality is as old as mathematics itself.

The decomposition of numbers into unit fractions also captured the attention of mathematicians in 9th-century India. The Jain mathematician Mahāvīra , for instance, delved into related problems during this period.

A pivotal text in medieval European mathematics, the Liber Abaci (published in 1202) by Leonardo of Pisa , more widely recognized as Fibonacci, offers invaluable insights into the continued application of Egyptian fractions during the Middle Ages. More significantly, it introduced concepts that remain relevant in the modern mathematical study of these series.

While the primary focus of the Liber Abaci was on calculations involving decimal and vulgar fraction notation—systems that would eventually, and mercifully, replace Egyptian fractions —Fibonacci himself employed a rather intricate notation. This involved a hybrid system combining a mixed radix notation with sums of fractions. Many of the computations documented throughout Fibonacci’s extensive work feature numbers expressed as Egyptian fractions . One dedicated section of this book provides a comprehensive list of methods designed for converting vulgar fractions into Egyptian fraction series.

Let’s examine some of Fibonacci’s conversion strategies:

• Splitting the numerator: If the number in question was not already a unit fraction, Fibonacci’s initial approach was to attempt to decompose the numerator into a sum of divisors of the denominator. This method was feasible whenever the denominator happened to be a practical number . The Liber Abaci even included helpful tables of such expansions for specific practical numbers like 6, 8, 12, 20, 24, 60, and 100. A quaint, but effective, shortcut.

• Algebraic identities: Fibonacci also leveraged several algebraic identities. One notable example is: $${\frac {a}{ab-1}}={\frac {1}{b}}+{\frac {1}{b(ab-1)}}$$ To illustrate, Fibonacci represented the fraction $8/11$ by splitting its numerator, 8, into a sum of two numbers, each of which conveniently divides one plus the denominator (11+1=12). So, $8/11$ was seen as $6/11 + 2/11$. Applying the algebraic identity to each of these two parts then yielded the expansion: $8/11 = 1/2 + 1/22 + 1/6 + 1/66$. He described similar methods for denominators that were two or three less than a number possessing numerous factors. Because, why make it simple when you can make it clever?

• The “greedy” algorithm: In those rare, unfortunate instances where all other methods proved futile, Fibonacci reluctantly suggested a “greedy” algorithm for generating Egyptian fraction expansions. This algorithm operates by iteratively selecting the unit fraction with the smallest denominator that is not larger than the remaining fraction requiring expansion. In more contemporary notation, a fraction $x/y$ is replaced by the expansion: $${\frac {x}{y}}={\frac {1}{,\left\lceil {\frac {y}{x}}\right\rceil ,}}+{\frac {(-y),{\bmod {,}}x}{y\left\lceil {\frac {y}{x}}\right\rceil }}$$ Here, $\lceil \cdot \rceil$ denotes the ceiling function . Since $(-y) \bmod x < x$, this method is guaranteed to produce a finite expansion. Fibonacci, ever the pragmatist, recommended switching to an alternative method after the initial expansion. However, he also provided examples where this greedy expansion was iterated until a complete Egyptian fraction was meticulously constructed. Consider these examples from his work: $4/13 = 1/4 + 1/18 + 1/468$ and $17/29 = 1/2 + 1/12 + 1/348$.

  Compared to the more refined ancient Egyptian expansions or even some modern techniques, this greedy method can generate expansions that are remarkably long, often featuring rather large denominators. Fibonacci himself, with a hint of exasperation, acknowledged the inherent awkwardness of the expansions produced by this method. For instance, the greedy algorithm expands $5/121$ into a truly sprawling series: $${\frac {5}{121}}={\frac {1}{25}}+{\frac {1}{757}}+{\frac {1}{763,309}}+{\frac {1}{873,960,180,913}}+{\frac {1}{1,527,612,795,642,093,418,846,225}}$$ Meanwhile, other, more discerning methods yield a significantly shorter and more manageable expansion for the same fraction: $5/121 = 1/33 + 1/121 + 1/363$. The Sylvester’s sequence (2, 3, 7, 43, 1807, …) can be conceptualized as being generated by an infinite greedy expansion of the number 1, where, at each step, the denominator is chosen as $\lfloor y/x \rfloor + 1$ instead of $\lceil y/x \rceil$. Perhaps due to this connection, Fibonacci’s greedy algorithm is sometimes, rather inaccurately, attributed to James Joseph Sylvester .

• The “many divisors” method: Following his exposition of the greedy algorithm, Fibonacci proposed yet another approach. This involved expanding a fraction $a/b$ by seeking out a number $c$ that possessed a multitude of divisors, with the condition that $b/2 < c < b$. The original fraction $a/b$ would then be transformed into $ac/bc$, and $ac$ would subsequently be expanded as a sum of divisors of $bc$. This method bears a striking resemblance to the technique proposed by Hultsch and Bruins, which they suggested as an explanation for some of the expansions found in the Rhind papyrus . It seems that even centuries apart, mathematicians often stumble upon similar solutions to similar problems.

Modern number theory

Though Egyptian fractions have largely receded from the forefront of practical mathematical applications—a sensible development, one might argue—modern number theorists continue to find them a surprisingly fertile ground for study. Their investigations span a diverse range of problems, including establishing bounds for the length or the maximum denominator in Egyptian fraction representations, discovering expansions with specific structural forms or where denominators belong to particular types, analyzing the termination properties of various expansion algorithms, and demonstrating the existence of such expansions for sufficiently dense sets of adequately smooth numbers . It seems some mathematicians just enjoy making things complicated again.

• One of the earliest published works by the prolific Paul Erdős demonstrated that it is fundamentally impossible for a harmonic progression to form an Egyptian fraction representation of an integer . The underlying reason is rather elegant: such a progression must, by necessity, contain at least one denominator divisible by a prime number that does not divide any other denominator in the sequence. Decades later, almost two decades after his passing, Erdős’s final publication (co-authored posthumously) proved that every integer can be represented in a form where all denominators are products of precisely three primes. A fitting, if slightly obscure, capstone to a remarkable career.

• The Erdős–Graham conjecture , a challenging problem in combinatorial number theory , posits that if the integers greater than 1 are partitioned into a finite number of subsets, then at least one of these subsets must contain a finite sub-collection whose reciprocals sum to exactly one. More formally, for any $r > 0$, and for any $r$-coloring of the integers greater than one, there exists a finite monochromatic subset $S$ of these integers such that $\sum_{n\in S} 1/n = 1$. This intriguing conjecture was definitively proven in 2003 by Ernest S. Croot III .

• Znám’s problem and the concept of primary pseudoperfect numbers are intimately connected to the existence of Egyptian fractions that take a specific form: $\sum (1/x_i) + \prod (1/x_i) = 1$. For example, the primary pseudoperfect number 1806, which is the product of the primes 2, 3, 7, and 43, gives rise to the Egyptian fraction $1 = 1/2 + 1/3 + 1/7 + 1/43 + 1/1806$. It’s almost as if these numbers were designed for such demonstrations.

• While Egyptian fractions are conventionally defined by the strict requirement that all denominators must be distinct, this condition can, in some contexts, be relaxed to allow for repeated denominators. However, this relaxed formulation does not actually provide a shortcut to representing numbers with fewer fractions. Any expansion containing repeated fractions can always be converted into an Egyptian fraction of equal or even smaller length through the repeated application of certain replacement rules. If $k$ is an odd number, the replacement is $1/k + 1/k = 2/(k+1) + 2/(k(k+1))$. If $k$ is even, the simplification is even more direct: $1/k + 1/k$ simply becomes $2/k$. This result was first rigorously demonstrated by Takenouchi in 1921.

• Graham and Jewett further expanded on this idea, proving that it is indeed possible to transform expansions with repeated denominators into (albeit longer) Egyptian fractions using the replacement: $${\frac {1}{k}}+{\frac {1}{k}}={\frac {1}{k}}+{\frac {1}{k+1}}+{\frac {1}{k(k+1)}}$$ This particular method, however, can lead to expansions that are remarkably lengthy and feature astonishingly large denominators. Consider, for instance, the expansion of $4/5$: $${\frac {4}{5}}={\frac {1}{5}}+{\frac {1}{6}}+{\frac {1}{7}}+{\frac {1}{8}}+{\frac {1}{30}}+{\frac {1}{31}}+{\frac {1}{32}}+{\frac {1}{42}}+{\frac {1}{43}}+{\frac {1}{56}}+{\frac {1}{930}}+{\frac {1}{931}}+{\frac {1}{992}}+{\frac {1}{1806}}+{\frac {1}{865,830}}$$ Botts (1967) originally employed this very replacement technique to establish that any rational number can possess Egyptian fraction representations with arbitrarily large minimum denominators. Because apparently, some problems thrive on extreme values.

• Any given fraction $x/y$ possesses an Egyptian fraction representation where the maximum denominator is bounded by a rather intimidating expression: $O\left(y\log y\left(\log \log y\right)^{4}\left(\log \log \log y\right)^{2}\right)$. Moreover, it can be represented with at most $O\left({\sqrt {\log y}}\right)$ terms. However, the number of terms must sometimes be at least proportional to $\log \log y$; this holds true for the fractions in the sequence $1/2, 2/3, 6/7, 42/43, 1806/1807, \ldots$, whose denominators are formed by Sylvester’s sequence . It has been conjectured, with a certain hopeful optimism, that $O(\log \log y)$ terms are always sufficient. Furthermore, it is also feasible to discover representations where both the maximum denominator and the total number of terms are kept relatively small.

• Graham (1964) provided a definitive characterization of the numbers that can be represented by Egyptian fractions where all denominators are $n$-th powers. Specifically, a rational number $q$ can be expressed as an Egyptian fraction with square denominators if and only if $q$ falls within one of two half-open intervals: $\left[0,{\frac {\pi ^{2}}{6}}-1\right)\cup \left[1,{\frac {\pi ^{2}}{6}}\right)$.

• Martin (1999) demonstrated that any rational number can be expanded into very dense Egyptian fraction representations, utilizing a constant fraction of the denominators up to $N$ for any sufficiently large $N$.

• An Engel expansion , sometimes referred to as an “Egyptian product” (a rather grand title for a mere expansion), represents a specialized form of Egyptian fraction expansion. In this particular structure, each successive denominator is a multiple of the preceding one: $$x={\frac {1}{a_{1}}}+{\frac {1}{a_{1}a_{2}}}+{\frac {1}{a_{1}a_{2}a_{3}}}+\cdots .$$ An additional constraint dictates that the sequence of multipliers $a_i$ must be nondecreasing. Every rational number conveniently possesses a finite Engel expansion , while irrational numbers are relegated to having an infinite one.

• Anshel & Goldfeld (1991) delved into the intriguing problem of numbers that possess multiple distinct Egyptian fraction representations, all sharing the same number of terms and the same product of denominators. For instance, they offered the example: $${\frac {5}{12}}={\frac {1}{4}}+{\frac {1}{10}}+{\frac {1}{15}}={\frac {1}{5}}+{\frac {1}{6}}+{\frac {1}{20}}$$ Unlike the ancient Egyptians, these researchers permitted denominators to be repeated in their expansions. They applied their findings from this problem to the characterization of free products of Abelian groups using a small set of numerical parameters: specifically, the rank of the commutator subgroup , the number of terms in the free product, and the product of the orders of the factors.

• The total count of distinct $n$-term Egyptian fraction representations for the number one is bounded both above and below by rather rapidly growing double exponential functions of $n$. This suggests a surprisingly complex underlying structure for what seems, on the surface, a simple concept.

• More specifically, the number of distinct Egyptian fraction representations of the number one, constrained such that the maximum denominator is at most $n$, is bounded above and below by a function of the form $2^{cn+o(n)}$, where $c \approx 0.91117$. More generally, for any rational number $x$, a similar bound exists, albeit with a different constant $c$, which naturally depends on $x$.

Open problems

Despite the considerable intellectual effort mathematicians have poured into them, some rather notable problems concerning Egyptian fractions stubbornly remain unsolved. It seems that even the most determined minds cannot always conquer every mathematical peak.

• The Erdős–Straus conjecture remains a particularly vexing challenge. It concerns the length of the shortest expansion for a fraction of the form $4/n$. The conjecture asks: does an expansion $4/n = 1/x + 1/y + 1/z$ exist for every integer $n$? While this has been confirmed for all $n < 10^{17}$, and for all but a vanishingly small fraction of possible $n$ values, the general truth of the conjecture—its applicability to all $n$—remains tantalizingly unknown. A persistent mathematical enigma.

• It is equally unknown whether an odd greedy expansion exists for every fraction that possesses an odd denominator. If Fibonacci’s greedy method were to be modified such that it consistently selects the smallest possible odd denominator, under what precise conditions would this adapted algorithm produce a finite expansion? An obvious prerequisite is that the initial fraction $x/y$ must itself have an odd denominator $y$. While it is conjectured that this condition is also sufficient, this remains unproven. What is known, however, is that every $x/y$ with an odd $y$ does indeed possess an expansion into distinct odd unit fractions, though this construction relies on a method different from the greedy algorithm.

• While brute-force search algorithms can be employed to uncover the Egyptian fraction representation of a given number with the absolute fewest possible terms, or to minimize the largest denominator involved, such algorithms are often notoriously inefficient. The existence of polynomial time algorithms for these specific problems, or more broadly, the precise computational complexity of such problems, continues to evade a definitive answer. Some mysteries, it seems, prefer to remain veiled.

Guy (2004) delves into these problems with greater specificity and, with a touch of mathematical masochism, enumerates numerous additional open problems in this field.

See also

• List of sums of reciprocals
• 17-animal inheritance puzzle

Notes

• ^ Dick & Ogle (2018); Koshaleva & Kreinovich (2021)
• ^ Wilson et al. (2011).
• ^ Winkler (2004).
• ^ Ritter (2002). See also Katz (2007) and Robson & Stedall (2009).
• ^ Hultsch (1895); Bruins (1957)
• ^ Gillings (1982); Gardner (2002)
• ^ Knorr (1982).
• ^ Eves (1953).
• ^ Struik (1967).
• ^ Kusuba (2004).
• ^ Sigler (2002), chapter II.7
• ^ Erdős (1932); Graham (2013)
• ^ Butler, Erdős & Graham (2015).
• ^ See Wagon (1999) and Beeckmans (1993)
• ^ Yokota (1988).
• ^ Vose (1985).
• ^ a b Erdős (1950).
• ^ Tenenbaum & Yokota (1990).
• ^ Konyagin (2014).
• ^ Conlon et al. (2025).
• ^ Breusch (1954); Stewart (1954)
• ^ Stewart (1992).