Practical solution of the diophantine equation X nr +Y n = q

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1 Elem. Math ) /11/ DOI /EM/165 c Swiss Mathematical Society, 2011 Elemete der Mathematik Practical solutio of the diophatie equatio X r +Y = q Kostatios A. Draziotis Kostatios Draziotis received his Ph.D. from the Aristotle Uiversity of Thessaloiki, Greece, i After a five years visitig professorship at the Techological ad Educatioal Istitute of Kavala, Greece, he is ow workig as a teacher i secodary educatio. His research iterests are diophatie geometry ad cryptography. 1 Itroductio The biomial theorem is a fudametal result of elemetary algebra, which describes the algebraic expasio of powers of a biomial a + b) α,whereα is a complex umber. It asserts that if x < 1adα is a complex umber, the 1 + x) α = k=0 ) α x k. k This seemigly simple theorem allows us to study the diophatie equatio X r + Y = q, 1.1) with positive itegers, r, q,where isassumedtobeoddad 3.. Das Fide gazzahliger Lösuge polyomialer Gleichuge ist i der Regel eie schwierige Aufgabe, wie beispielsweise die Vermutugvo Fermatzeigt. I dem achfolgede Beitrag utersucht der Autor für atürliche Zahle > 2 ugerade) ud r > 0 sowie gaze Zahle q = 0 die diophatische Gleichug X r + Y = q. Erbeweist, dass die gazzahlige Lösuge x, y) der zur Diskussio stehede Gleichug der Abschätzug x q 1/r geüge. Da diese Abschätzug iteressaterweise uabhägig vo ist, fidet ma, dass die Gleichug X r + Y = 1 ur die offesichtliche triviale gazzahlige Lösuge besitzt. Für seie Beweis beötigt der Autor im wesetliche ur de biomische Lehrsatz sowie eiige elemetare Eigeschafte der Eulersche Ɣ-Fuktio.

2 20 K.A. Draziotis We shall prove the followig theorem: Theorem 1.1 If x, y) Z 2 is a solutio to the equatio X + Y = q, the x q. If x, y) is a solutio to the equatio 1.1), the x r, y) is a solutio to the equatio X + Y = q. Thus applyig Theorem 1.1 to equatio 1.1) we get Corollary 1.2 If x, y) Z 2 is a solutio to the equatio X r +Y = q, the x r q. Our proof of Theorem 1.1 is etirely elemetary, the basic tool beig the biomial theorem which will fially provide us with a represetatio of the iteger solutios of the equatio X + Y = q i terms of the gamma fuctio. Note that the boud o x i the theorem does ot deped o the expoet. Applyig this to the corollary, we see that the umber of iteger solutios of equatio 1.1) is bouded i terms of oly q ad r. Observe that, if x r q, the y 2 q. LetX, Y, x, y be ukows ad c, r fixed positive itegers. We cosider a expoetial equatio of the form 1 X x ± Y y = c with x = ry ad y 3, odd. 1.2) Corollary 1.2 reduces the study of a expoetial diophatie equatio of the form 1.2) to studyig a bouded umber of simpler) expoetial diophatie equatios of the form a x ± b y = c, 1.3) where a, b) takes values from a fiite list of pairs of itegers. Ideed, if we fix x, y uder the restrictio x = ry ad y 3 odd), the Corollary 1.2 yields X r c ad Y y 2 c < 2 c. 1.4) So it is eough to solve a x ±b y = c, for the fiitely may a, b) satisfyig the iequalities 1.4). Sice this holds for every x, y with the previous restrictio) this reduces equatio 1.2) to fiitely may equatios of the form 1.3). Also, i the special case where X, Y are fixed, say X, Y ) = a, b), the LeVeque, i [6], proved that the equatio a x b y = 1has at most oe solutio i x, y). 2 We coclude the that the umber of solutios to equatio 1.2) with x = ry ad y 3, odd) is 2 c 2. Specializig further to the case c = 1, we get the equatio X x Y y = 1, which is related with the well-kow Catala cojecture [3] proved160 years after its first appearaceby Mihăilescu [8]. This cojecture ow a theorem) asserts that the oly two cosecutive positive itegers which are perfect powers are 8 ad 9, i.e., the equatio X x Y y = 1 has o other o-trivial solutio i positive itegers, except = 1. The rich history of this problem is traced i paper [7] ad also gives a brief summary of the proof of P. Mihăilescu. If y is odd ad 3, the from Corollary 1.2 we get X 1, so X = 0 1 This is really a diophatie equatio i X, Y, y, sicer = x/y is fixed as i 1.1). 2 Except whe a = 3, b = 2, i which case oe fids the two solutios x, y) = 1, 1), 2, 3).

3 Practical solutio of the diophatie equatio X r + Y = q 21 or 1 the case X = 1 is ot possible sice X > 0); thus i the first case we derive the cotradictio Y y = 1 < 0 ad the secod case gives the trivial solutio X, Y ) = 1, 0). If y is eve, the x = ry is eve too. Factorizig the equatio X x Y y = 1weget X, Y ) = 1, 0). Thus, Corollary 1.3 If r is a fixed positive iteger, the the diophatie equatio X yr Y y = 1 admits o o-trivial iteger solutio i X, Y, y) with y 2 ad X, Y > 0. If we fix x, y at r ad, respectively, the we get the iitial equatio 1.1), which ca be treated by what is kow as Ruge s method. Results of this sort have bee established for istace i [1, 4, 5, 9, 10]. This method, wheever it ca be applied, provides a polyomial boud for x, with respect to the absolute values of the coefficiets of the defiig polyomial ad the degree, which i our case is r. Thus, these bouds are ot useful if we wat to study the correspodig expoetial equatio. Here is a brief outlie of the paper. I Sectio 2 we give the proof of Theorem 1.1. I Sectio 3 we obtai a algorithm for the computatio of the iteger solutios of equatio 1.1). Fially, the method is illustrated by some examples. 2 Solutios of the equatio X + Y = q Let x, y) be a iteger solutio of X + Y = q. The the biomial theorem gives q x ) 1 = 1) j ) 1 ) j! 1... j 1) q j x 1 j. j 0 Note that the biomial series is coverget whe x > q. Applyig repeatedly the fuctioal equatio of the gamma fuctio zɣz) = Ɣz + 1),we see that j 1 1 ) i = 1) j Ɣ j 1 ) Ɣ 1 ). Thus i=0 q x ) 1 1) j+1 1) j Ɣ j = 1 ) j! Ɣ 1 ) q j x 1 j j 0 = 1 Ɣ 1 ) j 0 Ɣ j 1 ) q j x 1 j. j! We set a j = Ɣ j 1 ), j!

4 22 K.A. Draziotis so that q x ) 1 = x ) q ) j a j. 2.1) Ɣ 1 x j 0 All these equalities are valid if x > q. Recall that a fuctio f x) is called completely mootoic c.m.) o a iterval I, if 1) f ) x) 0 for every o-egative iteger ad every x I. Lemma 2.1 i) Let a + 1 b > a, α = max a, c), ad a b Ɣx + b) gx; a, b, c) = x + c) x >α). Ɣx + a) The, 1/gx; a, b, c) is c.m. o the iterval b, ), if c a. ii) We have k 1 a j = b k,where j=0 ) Ɣ k 1 b k =. k 1)! iii) We have lim b k = 0. k Proof. i) See [2, Theorem 3 ii)]. ii) This follows via iductio o k from the fuctioal equatio Ɣ1 + z) = zɣz) z C \ Z 0 ). iii) Usig the otatio of part i) of our lemma, we set a = 1/, b = 0. The a + 1 b > a. Let a b Ɣx + b) gx) = x + c) Ɣx + a), the 1/gx) is c.m. o 0, ) for c 1/. Thus, ) 1 Ɣ x 1 gx) = x + c) 1 Ɣx) is decreasig o 0, ), forsomefixedc > 0. The same holds true, if x = k Z >0. Thus, ) Ɣ k 1 r k = k + c) 1 Ɣk) is a decreasig sequece. Therefore r k < r 2,fork > 2. So ) Ɣ k 1 k + c) 1 Ɣk) < r 2,

5 Practical solutio of the diophatie equatio X r + Y = q 23 hece 0 ) Ɣ k 1 Ɣk) = b k < r 2 k + c) 1 0, whe k. The result follows. Remark. Istead of deducig iii) from part i) of the lemma, oe may for istace apply Stirlig s formula for the gamma fuctio. Proof of Theorem 1.1. We proved i Lemma 2.1 that a j = 0, so a 0 = Ɣ 1 ) = j=0 a j. Let x, y) be a iteger solutio of the equatio X + Y = q. Relatio 2.1) gives ) 1 Ɣ y = Ɣ 1 j=1 ) q x ) 1 = a0 x x a j q x ) j, thus ) 1 Ɣ y + x q j a j x j 1 < q j 1 a j x j 1. Suppose that x > q. The all the previous iequalities are valid sice the series are coverget. Thus, ) 1 Ɣ y + x < a j. Sice a j > 0for j > 0, we get a j = ) a j = a 0 = a 0 = 1 Ɣ. So ) 1 Ɣ y + x < ) a j = a 0 = 1 Ɣ. It follows that y + x < 1, thus y + x =0. So y = x. O the other had x + y = q, thus replacig y with x, wegetx + 1) x = q. Sice is odd, we get the cotradictio q = 0. We coclude therefore that x q.

6 24 K.A. Draziotis 3 A algorithm for the solutio of the equatio X r + Y = q As before, let x, y) Z 2 with x r + y = q. The oly iterestig case is xy < 0. Let x > 0ady < 0. We set y = z,wherez > 0. The we get x r z = q, thus x r z)px, z) = q, where Px, z) = x r r + x r 2r z x r z 2 + z 1. Hece x r z) q. Sowegetz = x r h for some divisor h of q. Substitutig this ito Px, z), we the compute the iteger roots of the equatio Thus, we get Px, x r h) = q h. x r r x r z 2 + h 1 = q h, so x r h 1 q ) = h q. h h The same holds true, if x < 0ady > 0. So we get the followig algorithm: Iput., r, q positive itegers with 3, odd. Output. The iteger solutios of the equatio 1.1). 1. Compute the divisors of q. 2. For each divisor h of q compute the ratioal umber k h = h q)/h. 3. Compute the set S h of the divisors of k h. 4. Compute the set S h of elemets of S h which are r q. 5. The iteger solutios of 1.1) are {x, y) Z 2 x r S h with x r + y = q}, where h rus through the set of divisors of q. Below we give some examples. Here the values of q have bee chose experimetally, usig Maple, i order to give o-trivial solutios to the diophatie equatio 1.1). For, r, q) = 3, 2, ),wegetx, y) = ±12, 1), ±1, 144). For, r, q) = 3, 3, ),wegetx, y) = 13, 2). For, r, q) = 3, 1, 3 383),wegetx, y) = 15, 2), 2, 15). For, r, q) = 5, 2, ),wegetx, y) = ±15, 2). For, r, q) = 5, 1, ),wegetx, y) = 2, 40), 40, 2). For, r, q) = 15, 1, 1 453) ad, r, q) = 15, 1, 2 141), there is o iteger solutio. I all these examples it took a few secods to fid the results o a Petium 2.6 GHz PC.

7 Practical solutio of the diophatie equatio X r + Y = q 25 Ackowledgemets The author is idebted to the referee for valuable remarks. Refereces [1] Ayad, M.: Sur le théorème de Ruge. Acta Arith ), [2] Bustoz, J.; Ismail, M.E.H.: O gamma fuctio iequalities. Math. Comp ) 176, [3] Catala, E.: Note extraite d ue lettre adressée a l éditeur. J. Reie Agew. Math ), 192. [4] Grytczuk, A.; Schizel, A.: O Ruge s Theorem about Diophatie equatios. Colloq. Math. Soc. Jáos Bolyai ), [5] Hilliker, D.L.; Straus, E.G.: Determiatio of bouds for the solutios to those biary Diophatie equatios that satisfy the hypotheses of Ruge s theorem. Tras. Amer. Math. Soc ) 2, [6] LeVeque, Wm.J.: O the equatio a x b y = 1. Amer. J. Math ), [7] Metsäkylä, T.: Catala s cojecture: aother old Diophatie problem solved. Bull. Amer. Math. Soc. N.S.) ) 1, [8] Mihăilescu, P.: Primary cyclotomic uits ad a proof of Catala s cojecture. J. Reie Agew. Math ), [9] Schizel, A.: A improvemet of Ruge s theorem o diophatie equatios. Commet. Potificia Acad. Sci ), 1 9. [10] Tegely, Sz.: O the Diophatie equatio Fx) = Gy). Acta Arith ) 2, Kostatios A. Draziotis Kromis Thessaloiki, Greece drazioti@gmail.com