A GENERALIZED RAMANUJAN-NAGELL EQUATION RELATED TO CERTAIN STRONGLY REGULAR GRAPHS

Transcription

1 #A35 INTEGERS 4 (204) A GENERALIZED RAMANUJAN-NAGELL EQUATION RELATED TO CERTAIN STRONGLY REGULAR GRAPHS B d Wgr Faculty of Mathmatics ad Computr Scic, Eidhov Uivrsity of Tchology, Eidhov, Th Nthrlads b.m.m.d.wgr@tu.l Rcivd: 0/24/3, Accptd: 4/29/4, Publishd: 8/5/4 Abstract A quadratic-xpotial Diophati quatio i 4 variabls, dscribig crtai strogly rgular graphs, is compltly solvd. Alog th way w coutr di rt typs of gralizd Ramauja-Nagll quatios whos complt solutio ca b foud i th litratur, ad w com across a problm o th ordr of th prim idal abov 2 i th class groups of crtai imagiary quadratic umbr filds, which is rlatd to th siz of th squarfr part of 2 ad to Wifrich prims, ad th solutio of which ca b basd o th abc-cojctur.. Itroductio Th qustio to dtrmi th strogly rgular graphs with paramtrs (v, k,, µ) with v = 2 ad = µ, was rctly posd by Natalia Tokarva 2. Somwhat latr Tokarva otd 3 that th problm had alrady b solvd by Brascoi, Codotti ad Vadrkam [2], but vrthlss w foud it, from a Diophati poit of viw, of som itrst to study a ramificatio of this problm. W ot th followig facts about strogly rgular graphs, s [5]. Thy satisfy (v k )µ = k(k ). With v = 2 ad = µ this bcoms 2 = +k(k )/µ. I this cas thir igvalus ar k ad ±t with t 2 = k µ, with t a itgr. From ths data Brascoi ad Codotti [] drivd th diophati quatio k 2 2 k + t 2 (2 ) = 0, which was subsqutly solvd i [2]. Th oly solutios turd out to b (k, t) = (0, 0), (, ), (2, ), (2, 0) for all, ad additioally (k, t) = (2 2 2, 2 2 ), ( , 2 2 ) for v. As a rsult, th oly otrivial strogly rgular graphs of th dsird typ (2, k, µ, µ) ar thos S [5] for th dfiitio of strogly rgular graphs with ths paramtrs. 2 Prsoal commuicatio to Adris Brouwr, March Prsoal commuicatio to BdW, April 203.

2 INTEGERS: 4 (204) 2 with v ad (k, µ) = (2 ± 2 2, 2 2 ± 2 2 ). Ths ar prcisly th graphs associatd to so-calld bt fuctios, s []. I studyig this diophati problm w tak a somwhat dviatig path 4. Without loss of grality w may assum that thr ar thr distict igvalus, i.., t ad k >. Th multiplicity of t th is (2 k/t)/2, so t k. It follows that also t µ. W writ k = at ad µ = bt. Th w fid t = a b ad 2 = (a 2 )t/b. Lt g = gcd(a, b) = gcd(b, t), ad writ a = cg, b = dg. It th follows that 2 is th product of th itgrs (a 2 )/d ad t/g, which thrfor ar both powrs of 2. Lt (a 2 )/d = 2 m. Th w hav m appl. Sic 2 = a(at )/b = a(a 2 ab )/b, th qustio ow has bcom to dtrmi th solutios i positiv itgrs, m, c, g of th diophati quatio 2 = c 2 m cg 2. () For th applicatio at had oly m is rlvat, but w will study < m as wll. With m thr obviously ar th four familis of Tabl. Our first, compltly lmtary, rsult is that thr ar o othrs. m c g [I] ay [II] 2 [III] v [IV] Tabl : Four familis of solutios of () with m. Thorm. All th solutios of () with m ar giv i Tabl. Proof. Not that c ad g ar odd, ad that cg 2 < 2 m. For m appl 2 th oly possibilitis for cg 2 < 2 m ar c = g =, ladig to m =, fittig i [I], ad for m = 2 also c = 3, g =, ladig to = 2, fittig i [II]. For m 3 w look at () modulo 2 m. Usig m w gt (cg) 2 (mod 2 m ), ad by m 3 this implis cg ± (mod 2 m ). So ithr c = g =, immdiatly ladig to m = ad thus to [I], or cg 2 m. Sic also cg 2 appl 2 m w gt g appl 2m 2 m < 3, hc g =. W ow hav c ± (mod 2m ) ad < c < 2 m, implyig c = 2 m or c = 2 m + or c = 2 m, ladig to xactly [III], [IV], [II] rspctivly. Not that this rsult implis th rsult of [2]. 4 I ow this ida to Adris Brouwr.

3 INTEGERS: 4 (204) 3 Wh m >, a fifth family ad sv isolatd solutios ar asily foud, s Tabl 2. For = 3 ad c = quatio () is prcisly th wll kow Ramauja- Nagll quatio [6]. m c g [V] ay [VI] [VII] Tabl 2: O family ad sv isolatd solutios of () with m >. I Sctios 2, 3 ad 4 w will prov th followig rsult, which is ot lmtary aymor, ad works for both cass m ad m > at oc. Thorm 2. All th solutios of () with m > ar giv i Tabl Small Th cass appl 2 ar lmtary. Proof of Thorms ad 2 wh appl 2. Clarly = lads to c = ad 2 m g 2 =, which for m 2 is impossibl modulo 4. So thr is oly th trivial solutio m = g =. Ad for = 2 w fid 3 = c 2 m cg 2, so c = or c = 3. With c = w hav 2 m g 2 = 3, which for m 3 is impossibl modulo 8. So w ar lft with th trivial m = 2, g = oly. Ad with c = 3 w hav 2 m 3g 2 =, which also for m 3 is impossibl modulo 8. So w ar lft with th trivial m = 2, g = oly. 3. Rcurrc Squcs From ow o w assum 3. Lt us writ D = 2. Lmma 3. For ay solutio (, m, c, g) of () thr xists a itgr h such that h 2 + Dg 2 = 2` c = 2m with ` = 2m 2, (2) ± h g 2. (3)

4 INTEGERS: 4 (204) 4 Proof. W viw quatio () as a quadratic quatio i c. Its discrimiat is 2 2m 4Dg 2, which must b a v squar, say 4h 2. This immdiatly givs th rsult. So ` is v, but wh studyig (2) w will also allow odd ` for th momt. Not th basic solutio (h, g, `) = (,, ) of (2). I th quadratic fild K = Q p D w thrfor look at = + p D, 2 which is a itgr of orm 2 2. Not that D is ot cssarily squarfr (.g. = 6 has D = 63 = 3 2 7), so th ordr O gratd by th basis {, }, big a subrig of th rig of itgrs (th maximal ordr of K), may b a propr subrig. Th discrimiat of K is th squarfr part of D, which, just lik D itslf, is cogrut to (mod 8). So i th rig of itgrs th prim 2 splits, say (2) = }}, ad without loss of grality w ca say ( ) = } 2. Not that it may happ that a smallr powr of } alrady is pricipal. Idd, for = 6 w hav } = 2 p 7 itslf alrady big pricipal, whr ( ) = 2 + p 63 = } 4. But ot that }, } 2, } 3 ar ot i th ordr O, ad it is th ordr which itrsts us. W hav th followig rsult. Lmma 4. Th smallst positiv s such that } s is a pricipal idal i O is s = 2. I a latr sctio w furthr commt o th ordr of } i th full class group for gral. I particular w gathr som vidc for th followig cojctur, showig (amog othr thigs) that it follows from (a ctiv vrsio of) th abccojctur (at last for larg ough ). Cojctur 5. For 6= 6 th smallst positiv s such that } s is a pricipal idal i th maximal ordr of K is s = 2. Proof of Lmma 4. Thr xists a miimal s > 0 such that } s is pricipal ad is i th ordr O. Lt 2 a + bp D b a grator of } s, th a, b ar coprim ad both odd, ad a 2 + Db 2 = 2 s+2. (4) Sic } 2 = ( ) is pricipal ad i O, w ow fid that s 2, ad a + b p k D = ±2 k + p D, with k = 2. (5) s Comparig imagiary parts i (5) givs that b 2 k, ad from th fact that b is odd it follows that b = ±. Equatio (4) th bcoms a 2 + D = 2 s+2, which is a 2 = 2 s This quatio, which is a gralizatio of th Ramauja-Nagll quatio that occurs for = 3, has, accordig to Szalay [8], oly th solutios giv i Tabl 3. Oly i cas [ii] w hav k = 2 itgral, ad this provs k =, s s = 2.

5 INTEGERS: 4 (204) 5 s a [i] ay [ii] 2 [iii] Tabl 3: Th solutios of a 2 = 2 s with a > 0. W xt show that th solutios h, g of (2) ar lmts of crtai biary rcurrc squcs. W dfi for k 0 h k = k + k, with h 0 = 2, h =, ad h k+ = h k 2 2 h k for k, k g k = k p, with g 0 = 0, g =, ad g k+ = g k 2 2 g k for k. D For v, say = 2r, w ca factor D as (2 r = 2 )(2 r + ). Now w dfi 2 r + + p D, µ = 2 r + p D, 2 satisfyig N( ) = 2 2r + 2 r ad N(µ) = 2 2r 2 r, µ = p D, µ = 2 r p D, 2 = (2 r + ), ad µ 2 = (2 r ). For = 2r ad appl 0 w dfi u appl = 2 r + v appl = appl + appl, with u 0 =, u = (2 r ), ad u appl+ = u appl 2 2 u appl for appl, 2 r (2 r ) µ appl+ + µ appl+, with v 0 =, v = 2 r +, ad v appl+ = v appl 2 2 v appl for appl. W prst a fw usful proprtis of ths rcurrc squcs. Lmma 6. (a) For ay 3 w hav g 2appl = g appl h appl for all appl 0. (b) For v = 2r w hav g 2appl+ = u appl v appl for all appl 0. (c) For ay ad v k = 2appl, w hav 2 ( 2)appl+ + h 2appl = h 2 appl, 2 ( 2)appl+ h 2appl = (2 )g 2 appl. (d) For ay v = 2r ad odd k = 2appl +, w hav 2 (r )(2appl+)+ + h 2appl+ = (2 r + )u 2 appl, 2 (r )(2appl+)+ h 2appl+ = (2 r )v 2 appl.

6 INTEGERS: 4 (204) 6 Proof. Trivial by writig out all quatios ad usig th mtiod proprtis of, µ. For curiosity oly, ot that (2 r + )u 2 appl + (2 r )v 2 appl = 2 (r )(2appl+)+2. Now that w hav itroducd th cssary biary rcurrc squcs, w ca stat th rlatio to th solutios of (2). Lmma 7. Lt (h, g, `) b a solutio of (2). (a) Thr xists a k 0 such that h = ±h k, g = ±g k ad ( 2)k = ` 2. (b) If ` is v ad quatio (3) holds with m = 2 ( 2)k +2 ad itgral c, th o of th four cass [A], [B], [C], [D] as show i Tabl 4 applis, accordig to k big v or odd, ad th ± i (3) big + or. k ± coditio c [A] ay 2appl + g appl = ± [B] h 2 appl 2 2 h 2 appl [C] 2r 2appl + + vappl 2 2 r + 2 r + vappl 2 [D] u 2 appl 2 r 2 r u 2 appl Tabl 4: Th four cass. Proof. (a) Equatio (2) implis that g, h ar coprim, so that 2 h ± gp D = }` 2. Lmma 4 th implis that 2 ` 2. W tak k = ` 2 ad thus hav 2 h ± gp D = k or k, ad th rsult follows. (b) Not that ` big v implis that at last o of, k is v. For v k = 2appl, (a) ad Lmma 6(a) say that g = ±g k = ±g appl h appl. If ± = + th quatio (3) ad Lmma 6(a,c) say that c = 2( 2)appl+ + h 2appl =. Th c big itgral implis g appl = ± ad c =. g 2 appl If ± = th quatio (3) ad Lmma 6(a,c) say that c = 2( 2)appl+ h 2appl = 2 h 2. Th c big itgral implis h 2 appl 2. appl 2 g 2 2appl g 2 2appl

7 INTEGERS: 4 (204) 7 For v = 2r ad odd k = 2appl +, (a) ad Lmma 6(b) say that g = ±g k = ±u appl v appl. If ± = + th quatio (3) ad Lmma 6(b,d) say that c = 2 (r )(2appl+)+ + h 2appl+ = 2r +. Th c big itgral implis vappl 2 2 r +. g 2 2appl+ v 2 appl If ± = th quatio (3) ad Lmma 6(b,d) say that c = 2 (r )(2appl+)+ h 2appl+ g2appl+ 2 = 2r u 2. Th c big itgral implis u 2 appl 2 r. appl Lt s trac th kow solutios. Familis [I] ad [II] hav k = 2, so appl =, ad c = or c = 2, so thy ar i cass [A] ad [B] with g = ad h =, rspctivly. Familis [III] ad [IV] hav k =, so appl = 0, ad c = 2 r or c = 2 r +, so thy ar i cass [D] ad [C] with u 0 = ad v 0 =, rspctivly. Family [V] has k = 4, so appl = 2, ad c = 2( 2)2+ + h 4 g4 2 = h2 2 g2 2 = h2 2 g2 2 =, so it is i cas [A]. Th kow solutios with = 3 ad v k = 2appl ar prstd Tabl 5, ad th kow solutios with = 4 ad v k = 2appl rsp. odd k = 2appl + ar prstd i Tabl 6. [A] [B] appl h appl g appl m c c 7 7 Tabl 5: Tracig th solutios with = 3 ad v k = 2appl to lmts i rcurrc squcs. 4. Solvig th Four Cass All four cass [A], [B], [C] ad [D] ca b rducd to diophati quatios kow from th litratur. Lmma 8. Cas [A] lads to oly th solutios from familis [I] ad [V], ad th thr isolatd solutios from [VI] with odd m.

8 INTEGERS: 4 (204) 8 appl appl h appl 2 7 u appl 5 9 g appl 0 3 v appl m m [A] c [C] c 5 5 [B] c 5 [D] c k = 2appl k = 2appl + Tabl 6: Tracig th solutios with = 4 ad v k = 2appl, rsp. odd k = 2appl +, to lmts i rcurrc squcs. Proof. Tabl 4 givs g k = ± ad c =. Th Equatio () bcoms th gralizd Ramauja-Nagll quatio g 2 = 2 m 2 +, which was compltly solvd by Szalay [8]. Lmma 9. Cas [B] lads to oly th solutios from family [II], ad th isolatd solutio from [VI] with m v. Proof. Not that w hav appl, ad th h appl (mod 2 2 ), so w hav ithr h appl = or h appl 2 2. I th lattr cas th coditio i Tabl 4 implis (2 2 ) 2 appl h 2 appl appl 2, ladig to appl 4. If = 3 w must hav h appl = ±. But h appl is vr cogrut to (mod 8), so h appl =. If = 4 th w must hav h appl =. Not that (wh appl ) w always hav h appl (mod 4). So w fid that h appl = always, ad it follows from Tabl 4 that c = 2, ad Equatio () ow bcoms g 2 = 2m 2. Hc m. Th quatio g2 = xt has b tratd by x Ljuggr [4], provig (amog othr rsults) that for v x always t appl 2. Hc ithr m =, g = ladig to family [II], or m = 2, i which cas 2 + must b a squar. This happs oly for = 3, ladig to m = 6, thus to th oly solutio from [VI] with v m. Lmma 0. Cass [C] ad [D] lad to oly th solutios from familis [III] ad [IV], ad th isolatd solutios [VII]. Proof. It is asy to s that u appl 2 r (mod 2 2r 2 ), v appl + 2 r (mod 2 2r 2 ) for all appl. If r 3 th it follows that v appl 2 r + ad u appl 2 r, so th coditio i Tabl 4 shows that i cas [C] (2 r + ) 2 appl 2 r + ad i cas [D] (2 r ) 2 appl 2 r, which both ar impossibl. Thus r = 2 or appl = 0. Th cas appl = 0 givs k =, so g =, ad m = 2 +, ad this givs xactly familis [III] ad [IV]. So w ar lft with r = 2 ad appl, so = 4. I cas [C] th coditio i Tabl 4 shows that vappl 2 appl 5, but also w alway hav v appl 3 (mod 4), lavig oly room for v appl =, c = 5. This lavs us with solvig 3 = 2 m 5g 2. This quatio is a spcial cas of th gralizd Ramauja-Nagll

9 INTEGERS: 4 (204) 9 quatio tratd i [9, Chaptr 7], from which it ca asily b dducd that th oly solutios ar (m, g) = (3, ), (7, 5) (solutios rs. 72 ad 223 i [9, Chaptr 7, Tabl I]). It might occur lswhr i th litratur as wll. I cas [D] th coditio i Tabl 4 shows that u 2 appl appl 3, but also always u appl 3 (mod 4), lavig oly room for u appl =, c = 3. This lavs us with solvig 5 = 2 m 3g 2. Agai this quatio is a spcial cas of th gralizd Ramauja- Nagll quatio tratd i [9, Chaptr 7], ad it ca asily b dducd that th oly solutios ar (m, g) = (3, ), (5, 3), (9, 3) (solutios rs. 43, 23 ad 257 i [9, Chaptr 7, Tabl I]). It might also occur lswhr i th litratur as wll. Proof of Thorms ad 2 wh 3. This is do i Lmmas 3, 7, 8, 9 ad Th Ordr of th Prim Idal Abov 2 i th Idal Class Group of p Q (2 ), ad Wifrich Prims W caot fully prov Cojctur 5, but w will idicat why w thik it is tru. W will dduc it from th abc-cojctur, ad w hav a partial rsult. Rcall that a Wifrich prim is a prim p for which 2 p (mod p 2 ). For ay odd prim p w itroduc w p,k as th ordr of 2 i th multiplicativ group Z p, k ad `p as th umbr of factors p i 2 p. Frmat s thorm shows that `p, ad Wifrich prims ar thos with `p 2. Thorm. Lt 3, 2 = D = 2 D 0 with D 0 squarfr ad. Lt } b a prim idal abov 2 i K = Q p D. (a) If < 2 /4 3/5 th th smallst positiv s such that } s is a pricipal idal i th maximal ordr of K is s = 2. (b) Th coditio < 2 /4 3/5 holds at last i th followig cass: () 6= 6 ad appl 200, (2) is ot a multipl of w p,2 for som Wifrich prim p. I particular Cojctur 5 is tru for all 6= 6 with 3 appl appl 200. Proof of Thorm. (a) W start as i th proof of Lmma 4. Thr xists a miimal s > 0 such that } s is pricipal i th rig of itgrs of K = Q p D 0. Lt 2 a + bp D 0 b a grator of this pricipal idal, th a, b ar both odd ad coprim, ad a 2 + D 0 b 2 = 2 s+2. Sic } 2 = ( ) (with = 2 + p D 0 ) is pricipal with orm 2 2, w ow fid that s 2. Lt us writ k = 2. s

10 INTEGERS: 4 (204) 0 Th coditio < 2 /4 3/5 implis D 0 > 2. As ks = 2 ad w 2/2 6/5 do t kow much about s w stimat k appl 2. W may howvr assum k 2, as k = is what w wat to prov. This mas that w gt s appl 2, ad from a 2 + D 0 b 2 = 2 s+2 w gt appl b appl 2/4+/2 /0 2/2 p < p. Ad this D 0 2 cotradicts 3. (b) W would lik to gt mor iformatio o how big ca bcom. To gt a ida of what happs w computd for all appl 200. Tabl 7 shows th cass with >. Not that i all ths cass, ad that i all of ths cass xcpt = 6 w hav < 2 /4 3/5, with for largr a ampl margi. This provs that coditio () is su cit Tabl 7: Th valus of > for all appl 200. Nxt lt coditio (2) hold, i.., is ot a multipl of w p,2 for som Wifrich prim p. W will prov that i this cas, as was alrady obsrvd i Tabl 7. This th is su cit, as implis appl, ad < 2 /4 3/5 is tru for 20, ad for 3 appl appl 9 with th xcptio of = 6 w alrady saw that < 2 /4 3/5. Th followig rsult is asy to prov: if p is a odd prim ad a (mod p t ) for som t but a 6 (mod p t+ ), th a p (mod p t+ ) but a p 6 (mod p t+2 ). By th obvious w p,`p p it ow follows that p - w p,`p, ad th abov rsult usd with iductio ow givs w p,k = w p,`pp k `p for k `p. Now assum that p is a prim factor of, ad p k but p k+ -. Th 2 (mod p 2k ), 2 p 6 (mod p`p+ ), ad w p,2k = w p,`pp 2k `p has w p,2k. Hc p 2k `p. Wh k `p for all p w fid that. But coditio (2) implis that `p = for all p, ad w r do. Extdig Tabl 7 soo bcoms computatioally challgig, as 2 has to b factord. Howvr, w ca asily comput a divisor of, ad thus a lowr boud, for may mor valus of, by simply tryig oly small prim factors. W computd for all prims up to 0 5 to which powr thy appar i 2 for all up to 2000.

11 INTEGERS: 4 (204) Tabl 8: Lowr bouds / cojcturd valus of for all appl 2000 for which -. W cojctur that th rsultig lowr bouds for ar th actual valus. I most cass w foud thm to b divisors of idd. But itrstigly w foud a fw xcptios. Th oly cass for whr w ar ot yt sur that th coditios of Thorm (b) ar fulfilld ar rlatd to Wifrich prims. Oly two such prims ar kow: 093 ad 35, with w 093,2 = 364, w 35,2 = 755. So th multipls of 364 ad 755 ar itrstig cass for. Idd, w foud that th valu for i thos cass dfiitly dos ot divid. S Tabl 8 for thos valus for appl Most probably 364 is th smallst for which th coditios of Thorm (b) do ot hold, but w ar ot tirly sur, as thr might xist a Wifrich prim p with xcptioally small w p,`p. If is divisibl by w p,2 for a Wifrich prim p, th th abov proof actually shows that wh is multiplid by at most p`p (for ach such p) it will bcom a multipl of. It sms quit saf to cojctur th followig. Cojctur 2. For all 7 w hav < 2 /4 3/5. Most probably a much sharpr boud is tru, probably a polyomial boud, mayb v < 2. Accordig to th Wifrich prim sarch 5, thr ar o othr Wifrich prims up to 0 7. A huristic stimat for th umbr of Wifrich prims up to x is 5 S

12 INTEGERS: 4 (204) 2 log log x, s [3]. This huristic is basd o th simpl xpctatio stimat X papplx for th umbr of p such that th scod p-ary digit from th right i 2 p is zro. A similar argumt for highr powrs of p idicats that th umbr of prims p X such that 2 p (mod p 3 ) (i.., `p 3) is fiit, probably at most, bcaus p This givs som idicatio that probably always divids tims p a ot too larg factor. Howvr, w p,`p might b much smallr tha p, ad thus a multiplicatio factor of p might alrady b larg compard to. W do ot kow how to fid a bttr lowr boud for w p,2 tha th trivial w p,2 > 2 log 2 p. p 6. Coctio to th abc-cojctur Millr 6 givs a argumt that a uppr boud for i trms of follows from th abc-cojctur. Th abc-cojctur stats that if a + b = c for coprim positiv itgrs, ad N is th product of th prim umbrs dividig a, b or c, th for vry > 0 thr ar oly fiitly may xcptios to c < N +. Idd, assumig 2 /4 3/5 for ifiitly may cotradicts th abc-cojctur, amly 2 = + 2 D 0 has c = 2 ad N appl 2D 0 = 2(2 )/ < 2 3/4+8/5, so that log c log N > 4/3 + 32/(5), which cotradicts th cojctur. Idd, assumig that th abc-cojctur is tru, thr is for vry > 0 a costat K = K( ) such that c < KN +, ad w gt < K /(+ ) 2 + /(+ ). This shows that ay < /3 will for su citly larg giv th truth of Cojctur 5 via Thorm (a). Robrt, Stwart ad Tbaum r [7] formulat a strog form of th abc-cojctur, log N implyig that log c < log N + C log log N for a costat C (asymptotically 4p 3). Usig c = s 2 ad N appl 2 + / appl 2 + w th obtai log 2 < ( + ) log 2 r ( + ) log 2 log + C, hc < xp C 0 for a costat C 0 log( + ) + log log 2 log slightly largr tha C, probably C 0 < 7.5. Not xactly polyomial, but this is a gral form of th abc-cojctur, ot usig th spcial form of our abc-xampl, ad it dos of cours imply Cojctur 5. Ev though Cojctur 5 follows from a ctiv vrsio of th abc-cojctur, it might b possibl to prov it i som othr way. Ackowldgmts Th author is gratful to Aart Blokhuis, Adris Brouwr, Victor Millr ad Natalia Tokarva for fruitful discussios. 6 R: Ordr of a idal i a class group, mssag to th NMBRTHRY mailig list, April 7, 203,

13 INTEGERS: 4 (204) 3 Rfrcs [] Aa Brascoi ad Bruo Codotti, Spctral aalysis of Boola fuctios as a graph igvalu problm, IEEE Tras. Computrs 48 (999), [2] Aa Brascoi, Bruo Codotti, ad J ry M. Vadrkam, A Charactrizatio of bt fuctios i trms of strogly rgular graphs, IEEE Tras. Computrs 50 (200), [3] Richard E. Cradall, Karl Dilchr, ad Carl Pomrac, A sarch for Wifrich ad Wilso prims, Math. Comp 66 (997), [4] W. Ljuggr, No Stigr om ubstmt likigr av form (x )/(x ) = y q, Norsk Mat. Tidsskr. 25 (943), [5] Jacobus H. va Lit ad Richard M. Wilso, A Cours i Combiatorics, Cambridg Uivrsity Prss, Cambridg, 992. [6] T. Nagll, Løsig Oppg. 2, 943, s.29, Norsk Mat. Tidsskr. 30 (948), [7] O. Robrt, C.L. Stwart, ad G. Tbaum, A rfimt of th abc cojctur, prprit, availabl at Grald.Tbaum/ PUBLIC/Prpublicatios t publicatios/abc.pdf. [8] László Szalay, Th quatios 2 N ± 2 M ± 2 L = z 2, Idag. Math. (N.S.) 3 (2002), [9] B d Wgr, Algorithms for Diophati Equatios, Ctrum voor Wiskud Iformatica, Amstrdam, 989.