Egyptian fractions, Sylvester s sequence, and the Erdős-Straus conjecture

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1 Egyptian fractions, Sylvester s sequence, and the Erdős-Straus conjecture Ji Hoon Chun Monday, August, 0 Egyptian fractions Many of these ideas are fro the Wikipedia entry Egyptian fractions.. Introduction An Egyptian fraction is a nuber of distinct unit fractions with positive denoinators added together. Exaple ( ) The restriction on distinct unit fractions is needed, because otherwise all fractions can easily be ade into Egyptian fractions siply by using a b b + + b. The Egyptians did not represent or 4 by unit fractions. (It can be though, + 6.). History.. Hieroglyphs Ancient Egyptians represented a reciprocal of a nuber by placing a hieroglyph above the nuber. The fractions,, and 4 had their own distinct sybols. (Clagett).. The Rhind Matheatical Papyrus The Rhind Matheatical Papyrus is a docuent ade around 650 BC (during the Second Interediate Period) in the Egyptian Middle Kingdo. It was later bought by Henry Rhind, and the papyrus was naed after hi. It was written by Ahes (Ahose). The papyrus is divided into three parts. The first part contains a list of Egyptian fraction representations of n for odd n fro n to n 0, as well as 40 probles. The second part is about geoetry, specifically volues, areas, and pyraids. The third book contains 4 additional probles. (Spalinger)

2 .. Other ancient texts Several other Egyptian papyri fro earlier ties have tables of Egyptian fractions. The Lahun Matheatical Papyri (around 850 BC) also had Egyptian fraction decopositions for n, the Egyptian Matheatical Leather Roll (around 900 BC) had Egyptian fraction decopositions for n fractions, and the Akhin wooden tablet (around 950 BC) also contains Egyptian fraction decopositions for n fractions.. Methods to convert a non-egyptian fraction into an Egyptian fraction.. Fractions in the for n n n+ + n(n+).. Fractions in the for n n n+ + n(n+) n n + n + n + 6n n n n + n n + 5 n n kn + kn n k + n k (n is a ultiple of ) (n is a ultiple of 5) + (k ) (Gardner 00) (k +n ) (Eves 95).. Fractions in the for n a n k + n k The greedy algorith +n (k a ) (Eves 95).4 The greedy algorith Soe of these ideas are fro the Wikipedia entry Greedy algorith for Egyptian fractions..4. Introduction The greedy algorith is a ethod that will convert any fraction into an Egyptian fraction. n n (od ) n (od ) n + n n. If nod, then repeat the ethod with n n as thew n until all fractions are unit fractions. (Sigler 00, Sylvester 880)

3 < n (od ) <. So, thuerator of the second fraction in the expansion continues to decrease until it reaches. n < n (assuing n > ), so the denoinator of the first fraction is saller than the denoinator of the original fraction. Also, n n > n (assuing n > ), so the denoinators of the fractions increase every step. No known algorith gives the ost concise Egyptian fraction representations of every fraction (for either eaning of concise, least nuber of ters or sallest denoinator). The greedy algorith ay give any ters with large denoinators when another ethod gives fewer ters and saller denoinators. The above fraction can be written ore concisely as , one of five representations of length..4. Maxiu-length expansions Depending on the fraction n, the greedy algorith ay give ters in the resulting Egyptian fraction, or fewer than x ters. n trivially has ter. n always has ters, as (let n n ) n + n + + (n +)(n +). Freitag and Phillips (999) give a necessary and sufficient condition for a fraction to have ters. Theore. For a fraction n, the greedy algorith for Egyptian fractions gives ters if n k! + (k N). Proof. Induction on is used. The stateent is true for. Consider soe. Then for k N, k!+ k( )!+ + k ( )!+ (where k k (k! + + )). Theore. For a fraction n, a necessary condition for its greedy algorith expansion to have x ters is for y k + (k N). Proof. n k + r (0 r < ). Then r. k+r k+ + r (k+)(k+r). When r, Fro these two theores it is possible to deduce thecessary and sufficient case. First of all there is an iportant definition. Definition. Let sets S ( ) be defined by the following rule: S {0} and s S iff 0 s < ( )! and s + ( + ) s t ( ) (od ( )!) (t S ). S 4 {0, 4} S 5 {0, 6,, 8} S 6 {0, 8, 0, 48, 60, 78, 90, 08}

4 Theore. A fraction n has ters in its greedy algorith expansion iff n k! + s + (s S, k 0, and k and s arot both 0). Proof. If n has ters in its greedy algorith expansion, fro Theore, n (k ( )! + s) + k!+s+ (0 s < ( )!). Then k!+s+ k( )!+s+ + (k( )!+s+)(k!+s+). Also, (k ( )! + s + ) (k! + s + ) (s + ) (s + ) (od ( )!). This forula is true for. Assue that it is true for soe. Then for s such that s ( + ) s t ( ) (od ( )!) (t S ), the forula is also true for. Since that equivalence is how S is defined, the forula is true for. The sallest n such that n : n! : n 0 4! : n 0 5! : n 0 6! Irrational nubers has ters in its greedy algorith expansion is: The greedy algorith can also be used for irrational nubers (fro the MathWorld entry Egyptian Fraction ), although then the series will be infinite (because an irrational nuber cannot be represented as the su of rational nubers). For instance, e π log Sylvester s sequence Soe of these ideas are fro the Wikipedia entry Sylvester s sequence..5. Introduction and relation to Egyptian fractions Sylvester s sequence is a sequence related to the greedy algorith for Egyptian fractions. One way to find it is to apply the greedy algorith for Egyptian fractions to, but at each step use the largest unit fraction that keeps the su of the unit fractions less than (change n n (od ) n + n n to n n (od ) n + + n ( n ). The sequence is coprised + ) of the denoinators of the resulting fractions , so the first 6 ters are,, 7, 4, 807, 644. (Sloane.) 4

5 .5. Foral (sequential) definition n Definition. e 0, + e i. i0 Corollary. e 0, ( ) +. Proof. An exercise. The sequence s values grow doubly exponentially, which eans that they grow at the rate of a bx. Specifically, E n+ + [, where E ( 6 exp i i log + (e i ) )] (Aho and Sloane 97, Vardi 99, Graha et al. 994.) Fact.. So, e i i0 Proof. Fro the recurrence equation, So, n i0 e i n i0.5. Relation to priubers ( ) + + ( ) + + ( ) + ( ) + + ( ) e i e i+ e 0 (n ). Theore. (Euclid) There are infinitely any priubers. Proof. For two ters e i and e j (i < j), e j (od e i ) since en(en )+ ( ) +. So, any two nubers in Sylvester s sequence are relatively prie. Any prie p divides no ore than onuber in Sylvester s sequence, because if it divides ore than one, thosubers would not be relatively prie. Since there are infinitely any nubers in Sylvester s sequence, there are also infinitely any priubers. Note. The second part of the first stateent assues and +. For and +, repeat the operation to get [ ( ) + ] ([ ( ) + ] ) + [ ( ) + ] ( ) + For the general case, and +k+, it s [+k (+k ) + ] ([+k (+k ) + ] ) + [+k (+k ) + ] (+k )+ 5

6 .5.4 Convergence of series According to Badea (99), given any sequence that grows such that e n + and e i E Q, there exists an N such that for all n > N the sequence is defined by e n +..6 The Erdős-Straus conjecture.6. Introduction Definition. A Diophantine equation is an equation where its solutions are restricted to integer values. Conjecture. (Erdős 950) The Diophantine equation 4 n a + b + c natural nuber n. can be solved for any The greedy algorith gives or fewer ters for ost cases of n. The greedy expansion gives 4 ters when n k! + s + (k N, 4, s {0, 4}), n 5, 49, 7,... or 7, 4, 65,... or 7 (od 4). 4 n n + n + + n ( ). n n When n (od ), + N, so this expansion + is an Egyptian fraction representation of 4 n. {n : n 7 (od 4)} {n : n (od )}, so that expansion also holds for the cas 7 (od 4). No siilar solution exists for the cas (od 4) (Mordell 967). It has been shown that given an interval [, N], the fraction of n in that interval that could be counterexaples to the conjecture 0 as N (Webb 970). This conjecture has been tested valid using coputer searches for n 0 4 (Swett)..6. Generalizations Conjecture. (Sierpiński 956) There exists soe N such that the Diophantine equation 5 n a + b + c can be solved for any natural nuber n N. Conjecture. (Schinzel) For any given N, there exists soe N such that the Diophantine equation n a + b + c can be solved for any natural nuber n N (Vaughan 970). References Aho, A. V. and Sloane, N. J. A. "Soe Doubly Exponential Sequences." Fib. Quart., 49-47, 97. Badea, Catalin (99). "A theore on irrationality of infinite series and applications". Acta Arithetica 6:. Clagett, Marshall Ancient Egyptian Science, A Source Book. Volue Three: Ancient Egyptian Matheatics (Meoirs of the Aerican Philosophical Society) Aerican Philosophical Society

7 Erdős, Paul (950), "Az /x + /x /x n a/b egyenlet egész száú egoldásairól (On a Diophantine Equation)", Mat. Lapok. : 9 0. Eves, Howard (95), An Introduction to the History of Matheatics, Holt, Reinhard, and Winston Freitag, H. T.; Phillips, G. M. (999). "Sylvester s algorith and Fibonacci nubers". Applications of Fibonacci nubers, Vol. 8 (Rochester, NY, 998). Dordrecht: Kluwer Acad. Publ.. pp Gardner, Milo (00), "The Egyptian Matheatical Leather Roll, attested short ter and long ter", in Gratton-Guiness, Ivor, History of the Matheatical Sciences, Hindustan Book Co, pp. 9 4 Graha, R. L.; Knuth, D. E.; and Patashnik, O. Research proble 4.65 in Concrete Matheatics: A Foundation for Coputer Science, nd ed. Reading, MA: Addison- Wesley, 994. Mordell, Louis J. (967), Diophantine Equations, Acadeic Press, pp Sierpiński, Wacław (956), "Sur les décopositions dobres rationnels en fractions priaires" (in French), Mathesis 65: 6. Sigler, Laurence E. (trans.) (00), Fibonacci s Liber Abaci, Springer-Verlag Sloane, N. J. A. Sequences A000058/M0865, A04546, and A0769 in "The On-Line Encyclopedia of Integer Sequences." Anthony Spalinger, The Rhind Matheatical Papyrus as a Historical Docuent, Studien zur Altägyptischen Kultur, Bd. 7 (990), pp. 95-, Helut Buske Verlag GbH Swett, Allan, The Erdos-Straus Conjecture, < retrieved Sylvester, J. J. (880). "On a point in the theory of vulgar fractions". Aerican Journal of Matheatics (4): 5. Vardi, I. "Are All Euclid Nubers Squarefree?" and "PowerMod to the Rescue." 5. and 5. in Coputational Recreations in Matheatica. Reading, MA: Addison-Wesley, pp. 8-89, 99. Vaughan, R. C. (970), "On a proble of Erdős, Straus and Schinzel", Matheatika 7 (0): Webb, Willia A. (970), "On 4/n /x + /y + /z", Proceedings of the Aerican Matheatical Society 5 ():