The Exact Number of Nonnegative Integer Solutions for a Linear Diophantine Inequality

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1 The Exact Numbe of Nonnegatve Intege Solutons fo a Lnea Dophantne Inequalty Rahm Mahmoudvand, Hossen Hassan 2, Abbas Fazaneh 3, Gaeth Howell 4 Abstact In ths pape, we pesent a smple and fast method fo countng the numbe of nonnegatve ntege solutons to the equalty a x +a 2x 2++a x n whee a, a 2,..., a and n ae postve nteges. As an applcaton, we use the method fo fndng the numbe of solutons of a Dophantne nequalty. Keywods: Countng, Nonnegatve ntege solutons, Dophantne nequalty. Intoducton Countng technques play an mpotant ole n computng pobabltes n andom expements of thowng dce, o classcal occupancy poblems. As a esult, they have come to fom a majo pat of the mathematcs cuculum n many statstcal publcatons. Fst we wll consde some mpotant applcatons of countng technques. Ross [3] showed that the numbe of ways fo placng n dentcal objects nto the dstnct cells s equvalent to the numbe of nonnegatve ntege solutons to the equaton x + x x n (wth x 0,,, ). () He also showed that the numbe of postve nteges solutons of () s ( n ). The numbe of nonnegatve ntege solutons of (), subject to the constant x b fo,, s ( n+ (b +b b ) ). Lettng x y + b fo each yelds the equaton y + + y n (b + b b ), (2) to be solved n nonnegatve nteges. The numbe of such solutons whee x b (,, ) can be obtaned usng the ncluson/excluson pncple (see, fo example, Rosen et al. []). Fo the latte stuaton, Muty [4] obtaned a smple method of countng the favoued numbe Rahm Mahmoudvand: Goup of Statstcs, Unvesty of Payame Noo, Toysekan, Ian, E-mal: mahmodvand@yahoo.com; 2 Hossen Hassan: Cente fo Optmsaton and Its Applcatons, School of Mathematcs, Cadff Unvesty, UK, CF24 4AG, Telephone: +44 (0) Fax: +44 (0) , E-mal: HassanH@cf.ac.uk; 3 Abbas Fazaneh: Goup of Compute, Islamc Azad Unvesty, Toysekan, Ian, E-mal: fazaneh.comp@gmal.com; 4 Gaeth Howell: Statstcs Goup, Cadff Unvesty, UK, CF24 4AG, E-mal: HowellGL@cf.ac.uk. of solutons. One genealzaton of () s the numbe of nonnegatve ntege solutons of the followng equaton, a x + a 2 x a x n. (3) Equaton (3) s well-known as a Lnea Dophantne Equaton. As s dscussed above fo the smple case, t s possble to obtan the numbe solutons of equaton () wth some bounds on x s fom () wthout any bounds on x s. It has been shown that the numbe solutons of (3) by some bounds on x s can be expessed as a functon of the numbe solutons of (3) wthout any bounds on x s (Esenbes et al. [5]). Theefoe, t s enough to estct ou effot to detemne the numbe solutons of (3) wthout any bounds on x s. Gven postve nteges a, a 2,, a that ae elatvely pme, t s well-known that fo all suffcently lage n the equaton (3) has a soluton wth nonnegatve nteges x (Tpath [2]). The geneatng functon of equaton (3) has the fom ϕ(t) [( t a )( t a2 ) ( t a )], and the numbe of non-negatve ntege solutons J(n) of equaton (2) s gven by the fomula: J(n) n! ϕn (0). (4) Calculaton of J(n) s dffcult n most stuatons. Antmov and Matvejevs, n [6] have dscussed seveal possble methods methods fo ts calculaton. Esenbes et al.(992) [5] pesented fast methods fo computng the exact o appoxmate numbe of solutons. In summay, thee ae two man poblem fo fndng the numbe of nonnegatve ntege soluton solutons of (3); the pesent methods, owng to the dffculty of the poblem, ae complcated, tme consumng, and encounte dffcultes when one wshes to extact a lst of such solutons. These ssues motvated us to obtan a smple method fo fndng the numbe of nonnegatve ntege solutons of (3) and povde a lst of the obtaned solutons. 2 New Method Among the two poblems consdeed,.e., computng the numbe solutons and geneatng the solutons, the fst one s by fa the most complex. Theefoe, t s vtal

2 to smplfy the poblem as much as possble n ode to obtan effcent computaton. Let us fst consde a fo 2,..., n (3). In ths case, we must fnd the numbe of nonnegatve ntege solutons fo a x + x x n. (5) Fo solvng (5), we can gve the possble values of x and efom (5) to fom (). Theefoe, [n/a ] w 0 n a w (6) s the numbe of nonnegatve ntege solutons fo equaton (5), whee [u] s the ntege pat of u and s a postve ntege and > 2. If 2 we must use [n/a ] I(a 2, w ) as the numbe of nonnegatve ntege solutons, whee I(a 2, w ) { a2 n a w 0 othewse (7) Now, let a fo 3,...,. In ths case, we must fnd the numbe of nonnegatve ntege solutons fo a x + a 2 x 2 + x x n. (8) Fo solvng (8), we can gve the possble values of x, x 2 and efom (8) to fom (). Theefoe, [n/a ] [(n a w )/a 2 ] w 2 0 n a w a 2 w (9) s the numbe of nonnegatve ntege solutons fo ths equaton. It should be noted that, the fomula s tue when s a postve ntege and > 3. Howeve, f 3 [n/a ] [(n a w )/a 2] we use I(a 3, w, w 2 ) as the numbe of nonnegatve ntege solutons, whee s(a,, a ; n) : [n/a ] w 0 whee [(n a w )/a 2] w 20 [(n a w... a 2 w 2 )/a ] I(a ; w,, w ) w 0 I(a ; w,, w ) () { a n a w...a w 0 othewse. (2) Note also that f a fo all, then s(a,, a ; n) s equal to n+, snce s(a,, a ; n) n n w w 2 0 n w n w 0 w 20 n n w w 0 w 20 n w... w 3 w 2 0 n+ w... w 3 w 20 n w... w 2 w 0 n w... w 2 + ( + w 2 ). (3) Now equalty s obtaned usng the fact that n m m + k n +. m m + k0 3 An applcaton Thee ae many poblems whch can be solved usng the poposed algothm. As a useful example, we use the algothm fo solvng the Dophantne nequalty a x + + a x n. (4) Let us now befly consde the chaactestc of the Dophantne Inequalty (fo moe nfomaton see, fo example, [6][7][8][9]). The man statement of the afoementoned theoem s n the language of lattces n numbe theoy. That s fo any convex set n the - dmensonal Eucldean space R symmetc wth espect to the ogn, and wth volume geate than 2, must contan a lattce pont othe than that of the ogn. In the language of lnea foms the poblem s estated as I(a 3, w, w 2 ) { a3 n a w a 2 w 2 0 othewse (0) a j L (X),, (5) Contnung the pocedue, we can get the followng fomula fo the numbe of nonnegatve ntege solutons of (3). wth eal coeffcents a j such that det(a j ) 0, supposng that thee exst postve eal numbes b, wth b det(a j ). Then thee exsts an ntege

3 vecto C such that L (C) b,, thus mplyng that a soluton exsts fo the above equatons and ndeed the mpled nequalty. The pape by Cheema, [3] suggests technques smla to the pogammng of ths eseach n ts wokng, and ndeed uses Mnkowsk s theoem to state that, whee denotes the dstance of a numbe fom ts neaest ntege, that thee always exsts a nonzeo ntege-vecto soluton X (x,, x ) to the nequaltes: L j (X) C, ( j ). (6) Anothe pactcal applcaton of the dscussed poblem s that of the Knapsack model, encounteed n many aeas wth a cleat explanaton offeed n [4]; the queston of how to fll a knapsack of lmted weght capacty wth dffeent tems whch best meet the needs of one s tp. Beged-Dov [4], fst ntoduced bounds on the numbe, N, of solutons to a x n wth the a s all beng natual-valued, as! n (n + a + + a ) N. (7) a! a These bounds wee obtaned n the followng way. Denote the ectangula box B(y,, y ) as the set of ponts Y (y,, y ) such that a x y (x + )a fo,, (8) whch has -dmensonal volume a. Secondly, defne the pyamd P (n) wth volume n!, whch denotes the set of ponts satsfyng y 0 fo,, and y n. The bounds ae obtaned as a consequence of the fact that each pont x as defned above [ belongs to a unque B, whch s the one wth x y a ], and f that x les n the pyamd P (n), then t necessaly obeys the lnea dophantne nequalty n queston. So the unon of the N boxes contans P (n). Ths somewhat smple topologcal agument allows the devaton of the above bounds. To add weght to Beged-Dov s agument n [4], some expemental esults ae calculated usng an algothm whch could be consdeed to be an ealy pecuso to the esults of ths pape. The tendency of the uppe and lowe bounds of the numbe of solutons to the lnea Dophantne nequalty to become close wth nceased numbe of vaables and ght hand sde s also touched upon. Padbeg and Lambe sought to espectvely mpove upon Beged-Dov s bounds. In the latte case an appoxmate numbe of solutons was eventually sought and found n [7]. Padbeg [2] consdeed the followng lowe bound (n + )! a N (9) Vey soon afte [4] was submtted, Padbeg took ts esult n [2] and shapened Beged-Dov s esult to the followng nequalty: ( (n + ) max! a, and ) + a N ( (n + N mn a j)! a, ) + a. (20) Hee a and a ae nteges satsfyng a n a j and a [ ] n a j fo all j,,. The ntal adjustment to the ognal esult s made by defnton of the new pyamd P (n + δ), whence x j n+δ, x j 0 fo j,, 0 δ <, (2) Then as above, takng a vecto ξ P (n + δ), then summng ove each element of the vecto we have (snce [x] x, x > 0): [ ] ξj ξj n + δ. (22) (n + δ) The lowe bound! a N s obtaned wth the j substtuton of P (n + δ) wth P (n) n the pevous poof, whch s then shapened by takng the lmt of ths bound as δ. The esult s mpoved futhe by makng the above substtuton fo a and a above, notng that a j a mn to obtan the bounds stated above. n a mn, (23) The pape of Padbeg also ntoduces the fomula fo the numbe of possble pattons exploed n ths pape, and quotes that anothe poof s mentoned n the book [5]. Lambe n hs pape [] of 974 ntoduced bounds whch n most cases wee bette stll, than what had been pevously dscussed: n + n + a N, (24) a a

4 Table : Compason between cuent methods and the new algothm. {a } n New (Exact) (7) (20) (24) (9) lowe uppe lowe uppe lowe uppe lowe 2, 3, , 3, , 3, , 3, ,, , 3, 4, 4, , 4, 4, 4, , 3, 4, 4, 5, , 3, 4, 5, 6, , 3, 5, 7, 9,, whee a a. Hs new bounds wee also able to show that the ato of uppe to lowe bounds tends to unty as and n gow lage. To attan the lowe bound, the nequaltes g a y + y n, (25) g wth the y s all nteges and g {, + } ae consdeed. The poof eques - whee P denotes the numbe of feasble (that s, nonnegatve) solutons to (25) - the povng of P g a g P g+, fo g,,. (26) The poof of the uppe bound s acheved usng the nequaltes g a y + y n + (x ), (27) g g and eques the asseton - whee Q denotes the numbe of feasble solutons to (25) and (27) - that Q g a g Q g+, fo g,,. (28) Both ae acheved n smla fashon. As mentoned above, Lambe n [7], dscoveed uppe and lowe bounds fo ths numbe. Howeve, the algothm poposed hee s able to compute the exact numbe of solutons. To do ths, we convet (4) to (3) by addng an exta nonnegatve ntege vaable x to (4). Then we need to solve a x + + a x + x n and usng the algothm the numbe of nonnegatve ntege solutons to (4) s: s(a,, a, ; n) [n/a ] w 0 [(n a x )/a 2 ] w 20 [(n a x... a 2x 2)/a ] w 0. (29) It should be noted that n the educed fom of nequalty we have a. Theefoe I(a ; w,, w ) fo all w,, w. Let us fst consde an smple example. Suppose we ae nteested n fndng the numbe of nonnegatve solutons to 0x + x 2 + x 3 2. (30) The lowe and uppe bounds on the numbe of solutons to ths nequalty, 4 and 455 espectvely, ae obtaned fom the algothm of (20), whlst we know the exact numbe of soluton s 97. It can be seen easly that these bounds epesent a wde devaton fom the actual numbe of solutons. Let us now use the poposed algothm fo solvng (30). As we mentoned above, fst we need to efom (30) to 0x + x 2 + x 3 + x 4 2. Thus, the soluton s as follows [2/0] [(2 0w )/] 2 w w 2 w 2 0 w 3 0 [(2 0w w 2 )/] + w w 2 w 2 0 w (3) Table shows the esultng lowe and uppe bounds gven fo the numbe of solutons to the nequalty wth coeffcents a and elevant n. The thd column shows the exact numbe of solutons gven by the method of ths note. Refeences [] H. K. Rosen, G. J. Mchaels, L. J. Goss, W. J. Gossman, R. D. She, Handbook of Dscete and Combnatoal Mathematcs, CRC Pess, New Yok, [2] A. Tpath, On a Lnea Dophantne Poblem of Fobenus, Integes: Electonc Jounal of Combnatoal Numbe Theoy, 6, no. A4, pp. -6, [3] S. M. Ross, A Fst Couse n Pobablty, CRC Pess, New Yok, 976.

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