EGYPTIAN FRACTION EXPANSIONS FOR RATIONAL NUMBERS BETWEEN 0 AND OBTAINED WITH ENGEL SERIES ELVIA NIDIA GONZÁLEZ AND JULIA BERGNER, PHD DEPARTMENT OF MATHEMATICS Abstract. The aciet Egyptias epressed ratioal umbers as the fiite sum of distict uit fractios. Were the Egyptias limited by this otatio? I fact, they were ot as every ratioal umber ca be writte as a fiite sum of distict uit fractios. Moreover, these epasios are ot uique. There eist several di eret algorithms for computig Egyptia fractio epasios, all of which produce di eret represetatios of the same ratioal umber. Oe such algorithm, called a Egel series, produces a fiite icreasig sequece of itegers for every ratioal umber. This sequece is the used to obtai a Egyptia fractio epasio. Motivated by the work of M. Mays, this project aims to ivestigate properties of atural umber deomiators that produce legth Egyptia fractio epasios usig Egel series for betwee 0 ad. While computig Egel epasios usig Mathematica, a helpful patter emerged: ratioal umbers which produce legth Egyptia fractio epasios were those whose deomiator mius oe was divisible by every atural umber betwee (ad icludig) ad. We cojecture ad prove that this will always hold for legth Egel series for betwee 0 ad where = k lcm(,,...,, ) + for k N. Furthermore, we cojecture that = lcm(,,...,, ) + is the smallest such that produces a legth Egyptia fractio epasio. A proof that = k lcm(,,...,, ) + works is icluded alog with a ivestigatio of whether or ot = lcm(,,...,, ) + the least such that does.. Itroductio The Rhid papyrus is resposible for preservig the mathematical methods employed by the aciet Egyptias. It is clear from this documet that the Egyptias had a uituitive way of epressig ratioal umbers. Ulike the curret precedet, i which oe iteger is writte over aother iteger, they would write out ratioal umbers as a sum of distict uit fractios. For eample, would be writte as + 8 istead. Additioally, the Rhid papyrus icluded etesive tables, oe of which icluded di eret represetatios of for odd betwee ad 0 usig the followig idetity [] = + + ( + ). However impractical these epasios may seem, they make certai real-life problems much easier to solve. Oe such real-life applicatio is the sharig problem. This problem focuses o sharig whole items (like loaves of bread) with multiple people. For eample, how ca five loaves be split evely amogst eight people? The solutio ca be writte as the proper fractio, 8. This meas each perso gets five eighths of a loaf. Now, applyig this is ot as easy as writig out the solutio as a fractio. How would someoe cut five-eighths from eight loaves? Each loaf could be cut i half, the those halves ito quarters, the those quarters ito eighths. From here, each of the eight persos would receive five small pieces of bread. This solutio, although correct, is ot very practical. If we represet 8 as + 8, this sharig problem becomes sigificatly easier to put ito practice. Each perso receives half a loaf first, which esures that four loaves are distributed evely amogst all eight people. Key words ad phrases. Egyptia fractios, Egel Series, Number Theory. This work is fuded through UC LEADS at the Uiversity of Califoria, Berkeley.
ELVIA NIDIA GONZÁLEZ AND JULIA BERGNER, PHD DEPARTMENT OF MATHEMATICS Net, the fial loaf ca be cut ito eighths ad each perso would receive oe of those pieces. Alog with beig easier to distribute the bread, this solutio allows each perso to have oe larger piece (the half loaf) alog with the smaller piece (the eighth loaf). These represetatios ispired may questios. Some have bee aswered. For eample, we kow that Egyptia fractio epasios are ot uique. Di eret algorithms may produce di eret represetatios of the same fractio. Our eample of 8 ca also be epressed as + 0 + 0. Table lists several more eamples. Table. Epasios of various ratioal umbers usig di eret algorithms. Egel Series Greedy Algorithm Cotiued Fractio + + + 8 + + 0 + + + 0 + 0 + 80 + 8 + 8 + 8 + 0 + 0 8 + 8 + 8 + + 0 + Aother questio that arose from Egyptia fractio epasios iquired whether or ot the Egyptias were limited by the use of this otatio. That is, could they epress ay ratioal umber as a sum of distict uit fractio? The Egyptias, as it turs out, were ot at all limited. Ay ratioal umber ca be epressed as the fiite sum of distict uity fractios. Furthermore, eve irratioal umbers ca be epressed as a ifiite sum of distict uit fractios. The focus of this project is Egyptia fractio epasios for ratioal umbers betwee 0 ad (but ot icludig 0 ad ) obtaied usig a Egel series. Some eamples of these epasios ca be see o the secod colum of Table.. Egel Epasios A Egel series, for ay real umber, is sequece of icreasig itegers []. Ay real umber, say z, ca be epressed as a Egyptia fractio epasio usig a Egel series i a uique way. This represetatio for ay real z ca be computed by defiig u = z the lettig a = d u e.from here, each subsequet a i+ ad u i+ is obtaied by first computig u i+ = u i a i the computig a i+ = d u i e,wheredeis the ceilig fuctio defied as dre = s if ad oly if s <rapple s [, Page 9, (.)]. If z is a ratioal umber, this process will halt wheever u k = 0 ad the Egel series is {a,a,...,a k }. The Egyptia fractio epasio for a ratioal umber is derived from the series as follows: z = kx i= = + + + a a a i a a a a a a k. Otherwise, if z is irratioal, this algorithm cotiues ad we ed up with a ifiite Egel series {a,a,a,...}. This ifiite Egel series ca be used to obtai a Egyptia fractio epasio with the followig sum []: X. a a a i i= Ay Egyptia fractio epasio derived usig a Egel series will be referred to as a Egel epasio from ow o. Below we prove that Egel epasios are fiite for all ratioal umbers. Theorem.. Egel epasios are fiite for all ratioal umbers.
EGYPTIAN FRACTION EXPANSIONS USING ENGEL SERIES Proof. Suppose m Q. Them = for, Z ad = 0. Begi to compute the Egel epasio by first lettig Net, compute u usig a ad u to obtai u = ad a = d e. u = u a =( a ) = a. By the defiitio of the ceilig fuctio, d u e = d e = a if ad oly if a < apple a.thusa < if ad oly if a <. Subtractig from both sides of the iequality ad addig to both leaves us with a <.The umerator for u is a. From the iequalities above, a <which is the umerator for u. We ca ow fid aother positive iteger a such that a ( a ) < a. The left had side of this iequality is precisely the umerator for u ad the right had side is precisely the umerator for u. So o matter which u i we start with, the umerator for u i+ will be less tha the umerator for u i. Sice the umerators for the u i s are strictly decreasig, evetually we will ed up with zero i the umerator for some u i (say u k has a umerator of 0) ad this process halts with which is a fiite Egel epasio as desired. kx i= a a a i. Results The work withi this project was ispired by a paper by Michael E. Mays titled, A Worst Case of the Fiboacci-Sylvester Epasio []. Mays eplored Egyptia fractio epasios of ratioal umbers usig the greedy algorithm. More specifically, his paper ivestigated the properties of fractio epasios which had legths that matched the umerator usig the greedy algorithm. I a similar maer, this paper ivestigates fractio epasios whose legths match their umerators usig Egel epasios. Iitially, this project bega by simply lookig at tables of Egel epasios. Tables ad o er a few eamples of Egel epasios whose legths match their umerators. Each table begis with a deomiator that is oe greater tha the umerator ad eds whe a Egel epasio of legth equal to the umerator is obtaied. The first colum is simply the ratioal umber beig ivestigated, the secod colum is the Egel series, ad the third colum is the Egel epasio. It was through tables such as these that a patter first emerged. First, we oticed every iteger betwee (ad icludig) ad had to divide oe mius the deomiator (that is, ). This observatio led us to cojecture that as log as = lcm(,,...,, )+, the Egel series produced for will always be of legth. Net, we oticed that = lcm(,,...,, ) + seems to be the least value for the deomiator of that does produce the desired epasio. We do prove that = lcm(,,...,, ) + does ideed always produce a legth Egel epasio. We also cojecture (but do ot prove) that this is the least such with this property. Istead, we logically demostrate that = lcm(,,...,, ) + is the least deomiator that produces a si term Egel epasio for. Fially, we observed that deomiators other tha = lcm(,,...,, ) + produced a epasio of legth. These deomiators seemed to icrease i some predictable fashio. Table demostrates that ay deomiator equal to plus ay multiple of lcm(,,...,, ) will produce a legth epasio for. The appedi cotais similar tables with more details ad for larger umerators. Sice the deomiators ca get uruly fairly quickly (see the last row of Table for a good demostratio of this) the majority of the tables will iclude the Egel series rather tha the Egel epasio for ratioal umbers.
ELVIA NIDIA GONZÁLEZ AND JULIA BERGNER, PHD DEPARTMENT OF MATHEMATICS Table. Egel epasios for Egel Series Egel Epasio {, } + {, } + 0 = {} {,, } + + 8 Table. Egel epasios for Egel Series Egel Epasio {,, } + + 0 {, } + {, } + 8 = {} 9 {, } + 9 {, } + {, } + = {} {,,, } + 0 + 0 + 80 Table. Legth Egel series ad Egel epasios for. Egel Series Egel Epasio {, } + {,, } + + 8 {,,, } + 0 + 0 + 80 {,,,, } + 08 +,8 +,08 + 8,9,888
EGYPTIAN FRACTION EXPANSIONS USING ENGEL SERIES Table. Other possible deomiators that produce a legth Egel Epasio for whe = Egel Series = lcm(, ) + {,, } = lcm(, ) + {,, } 9 9 = lcm(, ) + {, 0, 9} = lcm(, ) + {9,, } = lcm(, ) + {,, } = lcm(, ) + { 9 }. Theorems ad Proofs This sectio will iclude proofs of cojectures made withi the previous sectio. = k lcm(,,...,, ) + will produce a legth Egel epasio. Theorem.. Suppose is a positive iteger ad = k lcm(, legth Egel epasio. I order to prove Theorem., we eed a lemma first. Lemma.. [, Page 9, (.)] dr + se = dre + sorbr + sc = brc + sfors Z.,...,, ) +. The First we prove that will have a Proof Of Theorem.. Suppose = k lcm(,,...,, ) + ad (0, ) where N. To obtai the Egel epasio of begi by lettig u =. By assumptio, = k lcm(,,...,, ) + hece u = k lcm(,,...,, ) +. Net, a = d u e which ca be rewritte as a = d k lcm(,,...,, ) + k lcm(,,...,, ) e = d + e. The result of dividig k lcm(,,...,, ) by will be a atural umber so by the lemma, a ca be rewritte agai as k lcm(,,...,, ) a = + d e. Sice N, will always be a positive uit fractio ad d e will always equal. Agai the, a ca be rewritte as k lcm(,,...,, ) k lcm(,,...,, ) + a = +=.
ELVIA NIDIA GONZÁLEZ AND JULIA BERGNER, PHD DEPARTMENT OF MATHEMATICS Cotiuig to fid the Egel epasio requires computig u = u a which becomes u = k lcm(,,...,, ) + k lcm(,,...,, ) + = k lcm(,,...,, ) + k lcm(,,...,, ) k lcm(,,...,, ) + = k lcm(,,...,, ) +. Notice that the umerator of u = ad the umerator of u =. The umerator of u is oe less tha the umerator of u. Ideed, u will have a umerator of, u will have a umerator of, ad so o. Evetually, The u + term will have a umerator of 0 ad this process halts with the Egyptia fractio epasio = a + a a + + a a a as desired.. Cojecture Cojecture.. The from Theorem. is the least such which produces a legth Egel epasio. The tables icluded i the appedi do make a strog case for = lcm(,,...,, ) + beig the least deomiator that produces the desired legth epasio for. Below is a eample that proves it, logically, for a small fied value of. First we eed these additioal lemmas alog with Lemma.. Lemma.. [, Page 9, Eercise ] Lemma.. [, Page 8] d r s e = br + s c s dre = r, r Z,brc = r Eample. Let =. The by Theorem. = lcm(,,,, ) + =. We will show that this is the smallest possible value for that will produce a legth Egel epasio... Case : =p. Suppose =p. The = p = p. This is a oe-term Egel epasio. Hece the followig umbers are elimiated from cosideratio for are, 8,, 0,,, 8,, 0... Case : =p. Suppose =p. The = p = p.... Subcase : p is eve. If p is eve, the p the p ca be writte as p = p 0 for p N. Thus as i case. = p = p 0 = p 0 = p
EGYPTIAN FRACTION EXPANSIONS USING ENGEL SERIES... Subcase : p is odd. If p is odd the p =p 0 +. Now apply Egel s algorithm. u = = p = p = p 0 + a = d p0 + e By Lemma., a ca be rewritte usig the floor fuctio: a = b (p0 + ) + c = b p0 + c = bp 0 +c. Net, use Lemma. to separate the from the floor fuctio: a = bp 0 c +. Fially, sice p 0 N Z by Lemma., bp 0 c = p 0 ad a = p 0 +. Fid u by cotiuig the algorithm for Egel series: u = a u =(p 0 + ) p 0 = + p 0 + To fid a take the ceilig of the reciprocal of u : a = dp 0 +e = dp 0 e +=p 0 +. Repeat the algorithm oce more to obtai u = a u = p 0 + (p0 + ) = 0. Sice u = 0, we halt ad the Egel epasio is = p = (p 0 + ) = p 0 + = p 0 + + (p 0 + )(p 0 + ). This is still ot a si-term epasio. Now elimiated as cadidates for are 9,,,,, 9,,, ad... Case : =p. Suppose =p. The = p = p.... Subcase : p is a multiple of. If p is a multiple of, I ca write p =p 0 for p 0 N ad = p 0 = p 0 which has already bee covered by case.... Subcase : p mod. If p mod the p = p 0 + ad = p = ( p 0 + ) = p 0 + = u so a = d p0 + e = b (p0 + ) + c = b p0 + c = bp 0 + c. Sice p N Z, by Lemma., a i ca be rewritte as: a = b c + p0 = p 0 +. Net, cotiue by fidig u ad a : u = u a = p 0 + (p0 + ) = p 0 + ad a = dp 0 +e = dp 0 e +=p 0 +. Fially, u = u a = p 0 + (p0 + ) = 0 ad the algorithm halts. The Egel epasio for this case becomes: = (p 0 + ) = p 0 + + (p 0 + ) (p 0 + ),
8 ELVIA NIDIA GONZÁLEZ AND JULIA BERGNER, PHD DEPARTMENT OF MATHEMATICS which is less tha si terms log. Now elimiated as possibilities for are 0,,, 8,, 0,,, ad 8.... Subcase : p mod. Notice that if p mod the p = p 0 +. The followig two cases will eplore the possible Egel epasio produced for eve p or odd p.... Sub-Subcase A: p 0 is odd. If p is odd the p =p 0 + for some p 0 N ad u = = p = p = p 0 +. Net, use u to fid a : a = d p0 + e = b (p0 + ) + c = b p0 + c = bp 0 +c = bp 0 c +=p 0 +. Cotiue by usig a ad u to fid: u = p 0 + (p0 + ) = p 0 + ad a = d p0 + e Because p 0 is odd ad positive it ca be rewritte as p 0 =p 00 + (agai p 00 N). I ca ow cotiue to rewrite a as follows: a = d (p00 + ) + e = d p00 + e = dp 00 +e = dp 00 e +=p 00 +. Sice p 0 =p 00 +, this ca be rewritte as p 00 = p0. This allows a to be rewritte oe last time i terms of p 0 : a =p 00 + = ( p0 + )+= p0 +. The et iteratio will produce u = p 0 + p0 + = 0 ad the algorithm halts. For this sub case, the Egel epasio is: = p = p = p 0 + + (p 0 + ) (p 00 + ). This is still ot a si term Egel epasio so 8, 0,,, ad have bee elimiated as possible optios for the deomiator of.... Sub-Subcase B: p 0 is eve. Simillarly to.., both u ad a remai the same, so begi by computig: ad u = p 0 + (p0 + ) = p0 + p 0 p 0 = + p 0 + a = d p0 + e. I order to simplify a further, use the fact that p 0 is eve so it ca be rewritte as p 0 =p 00 for some p 00 N. Thus we get: a = d p00 + e = b (p00 + ) + c = b p00 + c = bp 00 +c = bp 00 c +=p 00 +. Furthermore, sice p 0 =p 00 we ca rewrite this as p 00 = p0. This allows us to rewrite a as: a = p0 += p0 +.
EGYPTIAN FRACTION EXPANSIONS USING ENGEL SERIES 9 The cotiue with the algorithm to fid: ad u = p 0 + p0 + = a = d u e =p 0 +. p 0 + Fially u = p 0 + p0 + = 0 ad we halt. This produces the followig three-term Egel epasio: = p = p = p 0 + + (p 0 + ) (p 00 + ) + (p 0 + ) (p 00 + ) (p 0 + ). Sice this is still ot a legth Egel epasio,,, 8, ad 0 have also bee elimiated... The remaiig <0. We ow oly have to elimiate,,,, 9,,, 9,,,,,,, 9,, ad 9 as possible choices for.... mod. Sice mod, =p + for some p N. First compute: u = = p + ad Net, ad a = d p + (p + ) + e = b c = bp + 0 c = b0 c + p = p +. u = (p + ) = p + p + p + a = d e = dpe +=p +. Fially, u = p+ (p + ) = 0. This produces the followig Egel epasio: = p + + (p + )(p + ) which is still ot the desired legth. Thus,,, 9,,,, ad 9 have bee elimiated as possible deomiators for. Table. Values of that do ot yield legth Egel epasios for. Egel Series {,,,, } {,,, } 9 {,, 9} {, } {,, 8, } {, 8,, 9, } {8, 9,, } 9 {9, 0, 9} {0, }
0 ELVIA NIDIA GONZÁLEZ AND JULIA BERGNER, PHD DEPARTMENT OF MATHEMATICS... mod. The remaiig ie possible deomiators are elimiated by brute force i Table. Fially, the smallest possible such that produces a legth Egel epasio is = lcm(,,,, ) + as desired.. Coclusio We have show a method for choosig a deomiator that will always produce Egel epasio whose legth is equal to its deomiator. This method is the relatioship betwee the umerator ad the deomiator give by = k lcm(,,...,, ) +. With this, we ca obtai etire families of legth Egel epasios for fied. We have also cojectured that = lcm(,,...,, ) + is the least deomiator to produce our desired legth epasio. I the future, we would like to prove that this is ideed the smallest deomiator with this property. I additio, we are iterested i the best case scearios (Egel epasios of legth two). We would like to employ similar methods used i this project to obtai a patter for these legth epasios the prove this patter holds.
EGYPTIAN FRACTION EXPANSIONS USING ENGEL SERIES. Appedi Egel Series Egel Series Egel Series {, } {,, } {8, } {, } {, } {8, } = {} = {} = 8 {8} {,, } {, 8} {9,, } 8 {, 8} {, } {9, } 9 = {} 8 = {} = 9 {9} 0 {, } 9 {, 0, 9} 8 {0, } {, } 0 {, 0} 9 {0, 9} = {} = {} 0 = 0 {0} Egel Series Egel Series Egel Series {,, } = {} {, } = {, } {,, } 8 = {} {, } 8 = 9 {, 9} 9 {8, 0, 9} 8 = {} 9 {, 9} 0 = {8, } 9 {, } 0 = {} {8, } 0 = {, } {, } = 8 {9} {, } = {, } {9, } = {} {, } = {9, } {,,, } = {} {9, } = {, } {, 9,, } = 9 {9} {, } = {, } {0,, 9, }
ELVIA NIDIA GONZÁLEZ AND JULIA BERGNER, PHD DEPARTMENT OF MATHEMATICS Egel Series Egel Series Egel Series {,, } = {} {9, } {,,, } {,, } = 9 {9} 8 {, } {, 9} {0,, } 9 {, 9} 8 {, } {0,, } 0 = {} 9 {, 9} 8 {0, } {,, } 0 = {} 9 {0, 9} {, } {, 8, } 0 = 0 {0} {,, } {,, } {,, } {, } {,, } {, 8, } = {} {, } {,, } {, } = {} {, } {,, } {8, 9} = {} 8 {, 9} {8,, 9, } {, } 9 {, 9} 8 {8, 9} {, 9} 0 = {} 9 {8, 9} 8 {, 9} {,, } 0 = 8 {8} 9 {, 99} {, 8, } {9,,, } 0 = {} {,, } {9, } {,,,, } {, } {9,, }
EGYPTIAN FRACTION EXPANSIONS USING ENGEL SERIES Refereces. Erdos, Shallit. New bouds o the legth of fiite Pierce ad Egel Series. Sémiaire de Théorie des Nombres Bordeau (99), -.. Graham, Kuth, Patashik. Cocrete Mathematics. AddisoWesley,Massachusetts,st Editio (989), Chapter.. V. Laohakosol, T. Chaichaa, J. Rattaamoog, ad N.R. Kaasri. Egel Series ad Cohe-Egyptia Fractio Epasios. Hidawi Publishig Corporatio. Iteratioal Joural of Mathematics ad Mathematical Scieces. Article ID 80 (009), pages.. Mays, Michael E. A Worst Case of the Fiboacci-Sylvester Epasio. The Joural of Combiatorial Mathematics ad Combiatorial Computig. (98), -8.. O Reilly, Decla Creatig Egyptia Fractios Mathematics i School. Vol., No. (Nov., 99), Page.. Reyi, A. A New Approach to the Theory of Egel s Series. Aales Uiversitatis Scietiarum Budapestiesis de Rolado Eötvös Nomiatae. Volume (9), -.. Weisstei, Eric W. Egyptia Fractio. From MathWorld A Wolfram Web Resource. http://mathworld.wolfram.com/egyptiafractio.html