On the number of sums of three unit fractions

Transcription

1 Notes o Number Theory ad Discrete Mathematics Vol. 9, 0, No., 8 O the umber of sums of three uit fractios Simo Brow School of Huma Life Scieces, Uiversity of Tasmaia, Locked Bag 0, Laucesto, Tasmaia 70, Australia Simo.Brow@utas.edu.au Abstract: Ratioal fractios ca ofte be expressed as the sum of three uit fractios ad, geerally, such a fractio ca be expaded i several ways. A estimate of the maximum umber of possible solutios is give. A expressio for five possible solutios is give ad this is used to obtai several geeral expasios. Keywords: Polyomials, Uit fractios. AMS Classificatio: D8. Itroductio May fractios (a/b, where 0 < a < b are itegers) ca be expressed as the sum of three uit fractios a 0 < x y z, () b x y z where the deomiators are itegers [, 7]. It is ormal to require that the deomiators are distict, but I relax this costrait for reasos that will become apparet. Of course, some fractios ca ot be writte i this form [8, 9], but where () does apply, there are likely to be solutios for more tha oe x for each of which there may be more tha oe (y, z) [, ]. This prompts oe to ask how may solutios there might be ad how to estimate the deomiators, ad, perhaps, what determies whether or ot there are solutios to (). Upper limit of the umber of solutios There ca be may solutios to () ad some of them are made clearer i this form (( ax b) y bx)( ( ax b) z bx ) b x. () Five possible solutios ca be obtaied directly from (), correspodig to (( ax b) y bx, ( ax b) z bx) ( b, bx ), ( x, b x),( bx, bx), ( x, b ), (, b x ) () 8

2 if x < b, ad there are more uless both b ad x are prime. The solutio correspodig to (bx, bx) must yield y z, which ecessitates the relaxatio of the usual requiremet that x, y ad z be distict (). Sice the divisors are employed i complemetary pairs, the umber of possible solutios is N x d α α α ( b x p p... p ) ( αi ) i, () where the p i are the prime factors of b x ad the α i are itegers. For example, if x ad b, b x ad the divisors are the five i () (( b, bx ), ( x, b x), ( bx, bx), ( x, b ) ad (, b x )) ad three others ((, ), (, ), (, )). The umber of possible solutios ca vary cosiderably. For example, if b, N x 8 for x (listed above), but for x (b x 7 ) N x. However, some of these possibilities may ot yield iteger solutios. For example, if a, b ad x, all of the 8 possibilities yield iteger solutios, but for x, oly of the possibilities are solutios. Of course, N x is a estimate of the umber of possible solutios for a specific x ad may differet values of x may be possible, so ( b x ) b a N d. x b a For example, for a ad b, there are solutios for x ragig from to []. Determiig the deomiators The five possible solutios give i () yield ( x ) bx( x ) x( b ) bx( b ) b bx bx ( y, z) ax b ax b ax b ax b ax b ax b. () b( x b) x( x b) bx bx( bx), ax b ax b ax b ax b If either b or x is ot prime the other solutios are easily determied i the same way. b / a x b / a it is simple, Obviously, all of the solutios deped o kowig x, but sice if potetially tedious, to idetify appropriate values of x. I geeral, whether or ot b or x is prime, the solutios of () are bx p bx bx p ( y, z), ax b p ax b where p is product of some of the prime factors of bx. There is o solutio if (ax p)ł(bx p) or płbx which might occur, for example, if p x (). The five aalytical solutios i () ca be used to obtai expasios of the form a F 0 () ( ) F ( ) F ( ) F ( ) 9

3 0 where the F i () are polyomials i iteger with itegral coefficiets. As for (), it is ot possible to express every fractio i the form of () ad it is ot possible to write a geeral expressio i this form []. However, there are may specific examples of these [, ], but as a example the explicit expasios of /( ) obtaied usig the five solutios i () are ( )( ) ( )( ) ( )( ) ( )( ) ( )( )( ) (7) as is easily cofirmed. Of these, Schizel [] credited the fourth to Sierpiski ad I have reported the secod previously []. All of these expasios have the smallest x, but it is also possible to use the first solutio, for example, i () to obtai expasios with larger x values ( )( ) ( )( ) of which I have previously reported the first [] ad the last is trivial. Similarly, () also provides a easy meas of geeratig expasios related to (7): ( ) ( ) ( ) ( )( )

4 ( )( ). More geeral expasios Equatio () ca be used to geerate much more geeral expressios. For example, (( ax b) y bx, ( ax b) z bx) ( b, bx ) yields A A A A ( A )( ) A A A A ( A )( ) A A A A A A A A A A A A A A ( A )( ) ( A ) for iteger A ad, which ca be summarised as {m Z: A m > }. ad the more geeral expressio for iteger B a. (0) a Ba B ( B )( a Ba ) ( B)( B )( a Ba ) ( A )( A ) ( A )( A ) A m A m A m m (, z bx) ( x, b x) Usig ( ax b) y bx ( ax b) ( A m )( A ) a related geeral solutio is A m A m A m m ( A m)( A ) Fially, expressios i which a is ot fixed ca be obtaied from () a a a a ( )( a a ) ( )( )( a ) a a a a ( )( a a ) ( )( )( a ) a a a a ( )( a a ) ( )( )( a ) (8). (9)

5 Refereces [] Brow, S., Bouds of the deomiators of Egyptia fractios, World Applied Programmig, Vol., 0, 0. [] Brow, S., A alterative approach to estimatig the bouds of the deomiators of Egyptia fractios, Leoardo Joural of Scieces, submitted, 0. [] Schizel, A., Sur quelques propriétés des ombres / et /, où est u ombre impair, Mathesis, Vol., 9, 9-. [] Schizel, A., O sums of three uit fractios with polyomial deomiators, Fuctioes et Approximatio, Vol. 8, 000, 89. [] Sierpiski, W., Sur les décompositios de ombres ratioels e fractios primaires, Mathesis, Vol., 9,. [] Vaugha, R. C., O a problem of Erdös, Straus ad Schizel, Matematika, Vol. 7, 970, 998. [7] Vose, M. D., Egyptia fractios, Bulleti of the Lodo Mathematical Society, Vol. 7, 98,. [8] Webb, W. A., Ratioals ot expressible as a sum of three uit fractios, Elemete der Mathematik, Vol. 9, 97,. [9] Yamamoto, K., O the Diophatie equatio / /x /y /z, Memoirs of the Faculty of Sciece, Kyushu Uiversity, Vol. 9, 9, 77.