Leonrdo Journl of Sciences ISSN 58-0 Issue, Jnury-June 0 p. -0 An Alterntive Approch to Estimting the Bounds of the Denomintors of Egyptin Frctions School of Humn Life Sciences, University of Tsmni, Locked Bg 0, Lunceston, Tsmni 750, Austrli. E-mil: Simon.Brown@uts.edu.u Phone: +6 645400; Fx: +6 64995 Astrct Mny frctions (/) cn e expressed s the sum of three unit frctions. However, it remins to e shown (i) how the denomintors of the unit frctions re to e clculted explicitly nd (ii) how to determine whether three unit frction is sufficient to express prticulr frction. I show tht the rnge of ech of the denomintors cn e estimted using their sum (S) nd product (P) nd provide ounds on oth S nd P. The nlysis presented lso provides mens of identifying some of those cses for which more thn three unit frctions is required. Keywords Denomintor; Egyptin frction; Inequlity. Introduction Any frction /, where 0 < < re integers, cn e expressed s sum of unit frctions. This pproch hd een used in ncient Egypt [, ] nd re known s Egyptin frctions. How the Egyptins clculted the denomintors of the unit frctions remins mtter of discussion [, 4]. Mny frctions cn e written s sum of just three unit frctions [5-0] http://ljs.cdemicdirect.org/
An Alterntive Approch to Estimting the Bounds of the Denomintors of Egyptin Frctions = + +, () p q r where 0 < p < q < r re integers. For exmple, only two unit frctions re needed if = [8], ut these cn e expressed s the sum of three unit frctions ecuse ny unit frction cn e expnded using /n = (/(n + )) + (/n(n + )). If = only three terms re needed [0] nd it hs een conjectured tht this is lso the cse for = 4 [], = 5 [5,6] nd = 6,7 [,]. However, some / cn not e expressed s the sum of three unit frctions [4], for exmple 0/ cn not e represented in this wy [], so the generl question remins n unsolved prolem. Nevertheless, lmost ll frctions cn e expressed s the sum of three unit frctions [7] nd for ny, the numer of frctions required is less thn (log ) ½ [9]. While / cn often e expressed s the sum of three unit frctions (), there re usully solutions for severl vlues of p nd there re often severl (q, r) for prticulr p [5]. The ounds on the denomintors re nd where < p < + + + 9 <, () p p < q < p p p pq < r p p q ( p + q ) + p = + nd q = + + (), (4) [5]. Of course, ()-(4) provide spce within which to seek solutions of (), ut they do not represent simple mens of clculting the denomintors. Moreover, since some /, such s 0/, cn not e expressed s the sum of three unit frctions [,4], it would lso e useful to e le to identify those /. Here I provide mens of clculting the denomintors nd mke some progress towrds identifying those / tht cn not e expressed s the sum of three unit frctions.
Leonrdo Journl of Sciences ISSN 58-0 Issue, Jnury-June 0 p. -0 Clculting the denomintors The sum nd product of the integer denomintors re S = p + q + r nd P = p q r, (5) respectively. Ech (S, P) corresponds to single /, which is ovious, if S = p + q + r S = p + q + r nd P = pqr P = p q r, ut consider two other cses: (i) S = S, in which cse p + q + r = p + (q + m) + (r m), ut P = p(q + m)(r m) P. (ii) P = P, in which cse pqr = p(mq)(r/m), ut S = p + mq + r/m nd S S if m nd mq r. Using (5), () cn e written in terms of S nd P from which ( S p) P + p = (6) pp p = p + P (7) p S (Figure ). As S nd P re integers, (7) implies tht p = m =,,..., (8) gcd P ( p,p ) m where gcd(x, y) is the gretest common divisor of x nd y, nd m is n integer in rnge to e determined from the ounds of P estimted elow. Eliminting one denomintor (sy r) from S nd P yields qudrtic in the other two ( S p) pq + P 0 pq =. (9) Treting one root s known (sy p), the roots of (9) give vlues for the other two denomintors (q nd r in this cse) ( q, r) = S p ± ( S p) p 4pP p from which S cn e eliminted using (7) (0)
An Alterntive Approch to Estimting the Bounds of the Denomintors of Egyptin Frctions ( q,r) p p = P ± P 4pP. () p p p Figure. The reltionship etween S nd P for / = 8/9 nd selected vlues of p. The points represent specific (p, q, r) for p = 5 ( ), 6 ( ) nd 8 ( ). The solid curves re given y (7) for p = 5, 6 nd 8, s indicted. The ovious prolems with (0) nd () re tht (i) specific vlues of S nd P re required nd (ii) it is not known whether three terms in () re sufficient. Moreover, when () pplies there cn e severl (q, r)s for ny given p, for exmple, for 8/9 there re 40 different solutions for p = 5 nd 7 solutions for p = 6 (Figure ), ut none for p =. This necessittes some insight into the rnges within which to serch for the denomintors, S nd P. Bounds on S nd P It is ovious from (5) tht p < S < r nd p < P < r. Alterntively, one could simply sustitute the ounds of the denomintors (-4) into (5) to estimte the ounds of S nd P. However, etter pproch is sed on the men inequlity, since () is relted to the hrmonic men (H) y H = /, which implies S P 7 () 7 4
Leonrdo Journl of Sciences ISSN 58-0 Issue, Jnury-June 0 p. -0 These ounds cn e improved using cuic with roots p, q nd r f ( x) ( x p)( x q)( x r) = x ( p + q + r) x + ( pq + pr + qr) x pqr = () which cn e expressed in terms of S nd P f = (4) ( x) x Sx + Px P The roots of f(x) re positive integers, which implies tht the roots of f (x) must lso e rel so S (/)P > 0, from which n upper ound on P cn e otined, nd since q nd r must e rel, n improved lower ound for P cn e otined from (), so S 4p > P > (5) ( p ) which is shrper thn (). A lower ound for S is otined using (7) in (5) ( p + ) p S > (6) p nd n upper ound for S cn e otined from the Schweitzer inequlity [6] 9 ( p q + r) ( p + r) + + + p q r (7) 4pr which, using () nd rerrnging, yields ( p + r) 9 S. (8) 4pr However, p/r < nd r/p < r/ (), so (8) ecomes 9 r S < + (9) 4 nd, for lgeric convenience, sustituting Nicklls [7] ound on the roots of (4) into (9) gives r < S + S P (0) 9 S < S + 4 S P +. () 5
An Alterntive Approch to Estimting the Bounds of the Denomintors of Egyptin Frctions Rerrnging () yields P < ( 5S + 7)( 7S 7), () 08 which is similr to (5) since the leding term is 5S /08, ut, fter sustituting (6), the RHS of () ecomes 08 ( p + 9) ( ) ( )( ) 5p ( p + 9) ( ) ( 5p 7 p 6p + 7 < 7 p + 7 ) p p, Figure. The region within which lie the solutions of () for / = 8/9. The points represent specific (p, q, r) for p = 5, 6,..., 0 (there re no solutions to () for p > 0). The dshed curves re given y (7) for p = 5 or p = 0, s indicted. The coordintes of the solid curves re (P min, S min ) nd (P mx, S mx ), which re given y the lower nd upper ounds, respectively, of () nd (4) for p = 5, 6,..., 0. so 4 p ( p ) 5p < P < ( p + 9) ( ) ( 7 p + 7 ) p where the rnge of p is given y (). Similrly, the ounds of S re ( p + ) 5( p + 9) p p < S < p + p ( ) ( 7 p + 7 ) p, (). (4) Together, (), (4) nd (7) define n re within which the (S, P)s re locted (Figure ). As is pprent from Figure, the numer of solutions of () is smll compred with the numer of integer (S, P) coordintes within the re defined y (7), () nd (4). One might e tempted to speculte tht there is correltion etween the numer of solutions of () nd the numer of coordintes, which corresponds pproximtely to the re of the region, 6
Leonrdo Journl of Sciences ISSN 58-0 Issue, Jnury-June 0 p. -0 ut there is no necessry correltion. For exmple, the res defined decline in the sequence =, 4, 5 for = 6 (Figure A). Of these, only 4/6 cn e expressed s the sum of three unit frctions [], lthough there re only two (p, q, r) = (5, 4, 070) nd (6, 6, 98) s shown in Figure A. Similrly, the res defined for = 7, 7 nd 7 for = 7 re similr (Figure B), ut neither 7/7 nor 7/7 hs ny solutions [], wheres 7/7 hs the 4 solutions shown in Figure B. Figure. The regions defined y (7), () nd (4) for =, 4, 5 nd = 6 (A) nd = 7 nd =7, 7, 7 (B). The curves in (A) nd (B), respectively, re = or = 7 ( ), = 4 or = 7 ( ), nd = 5 or = 7 ( ). The points represent the only solutions for 4/6 (A) or 7/74 (B) nd there re no solutions to () for the other cses. Where pproprite, it ws ssumed tht p vried in the rnge defined y (). Applictions Equtions (7-8), (0-) nd (-4) provide systemtic mens of estimting p, q nd r, either numericlly or nlyticlly. For exmple, if = 8, = 9 nd p = 8, then p = 05 nd p = 996, so (8) is P = 8m for m =,,... However, () yields 4449088/05 < P < 6697480400/5644800 so m rnges from 4449088/(05 8) 9 to 6697480400/(5644800 8) 68 nd S = 8 + (05/996) 8m = 8 + 5m for m = 9, 0,..., 68. Sustituting these into (0) yields = (5) ( q,r) ( 5m ± 5m 904m ) m = 9,0, K, 68 from which the four solutions shown in Figure for p = 8 re otined (Tle ). 7
An Alterntive Approch to Estimting the Bounds of the Denomintors of Egyptin Frctions Tle. Properties of the solutions of () for / = 8/9 nd p = 8. The four solutions given re those for which the specified m gives integrl solutions (q nd r) to (5). m Sum of Product of Clculted denomintors denomintors (S) denomintors (P) q r 9 6 995 5 84 5 40 00 5 40 4 58 455 4 476 56 868 4968 808 Sierpiński [5] showed tht = + 6n + n + + ( n + )( 4n + ) ( 4n + )( 6n + ) for n =,,... This expression yields the solution with the smllest S nd P for the lest p, ut there re mny others [5]. For exmple, (6) = + 6n + n + gives nother (q, r) for the sme p, wheres + ( n + )( n + ) ( n + )( n + )( 6n + ) (7) = + + 6n + n + 6n + ( n + )( 6n + ) is solution for lrger p. Nturlly, (6) nd (7) hve different (S, P) coordintes, ut they re oth locted on line (7) through S = n +, wheres (8) is not. All three lie (6-8) within the region defined y (7), () nd (4) (Figure 4). It remins to otin generl expression for the fmily of solutions of which (6-8) form smll prt (Figure 4). Tking (8) s n nlyticl exmple, =, = 6n + nd p = n +, so p = n + nd p = (n + ) (6n + ). As gcd(n +, (n + ) (6n + )) =, P = p m = (n + ) (6n + )m (8) nd S = n + + (n + )m (7). Compring the ounds of P () with P = (n + ) (6n + )m, yields expressions corresponding to 8 m 995, sed on their prtil frction expnsions. Sustituting S nd P into (0) yields n explicit expression for (q, r) in m nd n ( ) q,r = ( n + ) m ± ( n + ) m 4( n + )( 6n + ) m (8), (9) which implies the Diophntine eqution ( n ) m 4( n + )( 6n + ) m y = 0 positive solution is ( m, y) ( 6n +,n ( 6n + ) ) +. The only =, nd sustituting this into (9) yields the (q, r) 8
Leonrdo Journl of Sciences ISSN 58-0 Issue, Jnury-June 0 p. -0 given in (8). The Diophntine eqution ( S p) p 4pP y = 0 implied in (0), or its equivlent in (), my provide n indiction s to whether prticulr / cn e expressed s (): if there is no integer y, there cn e no solution to (). Figure 4. The region within which lie the solutions of () for / = /. The points ( ) represent specific (p, q, r) for p = 4, 4,..., 66 (there re no solutions to () for p > 66). The three filled points ( ) correspond to p = n + (6-7) nd p = n + (8) for n = 0. The ounds re given y (7), for p = 4 or p = 66, nd () nd (4) for p = 4, 4,..., 66. Conclusions The sum (S) nd product (P) of the denomintors of () is linerly relted to (7) nd cn e used to clculte the denomintors (0-). The ounds of P () nd S (4) nd systemtic mens of clculting P (8) fcilitte the efficient estimtion of denomintors. The method is useful in oth numericl nd nlyticl prolems. References. Neugeuer O., The exct sciences in ntiquity. New York, Dover Pulictions, Inc., 969.. Clgett M., Ancient Egyptin science: source ook. III. Ancient Egyptin mthemtics. Phildelphi, Americn Philosophicl Society, 999. 9
An Alterntive Approch to Estimting the Bounds of the Denomintors of Egyptin Frctions. Adulziz A. A., On the Egyptin method of decomposing /n into unit frctions, Histori Mthemtic, 008, 5, p. -8. 4. Dorsett C., A solution for the Rhind ppyrus unit frction decompositions, Texs College Mthemtics Journl, 008, 5(), p. -4. 5. Sierpinski W., Sur les décompositions de nomres rtionnels en frctions primires, Mthesis, 956, 65, p. 6-. 6. Sierpinski W., Elementry theory of numers. Wrsw, Pnstwowe Wydwn Nukowe, 964. 7. Vughn R. C., On prolem of Erdös, Strus nd Schinzel, Mtemtik, 970, 7, p. 9-98. 8. Culpin D., Griffiths D., Egyptin frctions, Mthemticl Gzette, 979, 6(4), p. 49-5. 9. Vose M. D., Egyptin frctions, Bulletin of the London Mthemticl Society, 985, 7, p. -4. 0. Hgedorn T. R., A proof of conjecture on Egyptin frctions, Americn Mthemticl Monthly, 000, 07(), p. 6-6.. Erdős P., Az /x + /x +... + /x n = / egyenlet egész számú megoldásiról, Mtemtiki Lpok, 950, 4, p. 9-0.. Aigner A., Brüche ls Summe von Stmmrûchen, Journl für die reine und ngewndte Mthemtik, 964, 4/5, p. 74-79.. We W. A., Rtionls not expressile s sum of three unit frctions, Elemente der Mthemtik, 974, 9(), p. -6. 4. Ymmoto K., On conjecture of Erdös, Memoirs of the Fculty of Science, Kyushu University, 964, 8(), p. 66-67. 5. Brown S., Bounds of the denomintors of Egyptin frctions, World Applied Progrmming, 0, (9), p. 45-40. 6. Schweitzer P., Egy egyenlőtkenség z ritmetíki kőzépértékről, Mthemtiki és Physiki Lpok, 94,, p. 57-6. 7. Nicklls R. W. D., A new ound for polynomils when ll the roots re rel, Mthemticl Gzette, 0, 95(54), p. 50-56. 0