High order Fuchsian equations for the square lattice Ising model: $chi^{(6)}$

High order F uc hsian equat ions for the square latt ice Ising mo del: χ (6) S. Boukraa † , S. Hassani § , I. Jensen ‡ , J.-M. Maill ar d || and N. Zenine § † LPTHIRM and D ´ epartemen t d’A´ e ronautique, Universit ´ e de Blida, Algeria § Cen tre de Rech erche Nucl ´ eaire d’Alger, 2 Bd. F rant z F anon, BP 399 , 16000 Alger, Algeria ‡ AR C Cen tre of Excellence for Mathematics and Statistics of Complex S ystems Departmen t of Mathematics and Stat istics, The Univ ersity o f M e lb ourne, Victoria 3010, Australia || LPTMC, Universit ´ e de Paris 6, T our 24, 4` eme ´ etage, case 121, 4 Place Jussieu, 75252 Paris Cedex 05, F ranc e E-mail: maillard@ lptmc.jussieu .fr, I.Jensen @ms.unimelb.e d u .au, njzenine @yahoo.com, boukraa@ mail.univ-bli da.dz Abstract. This pap er deals with ˜ χ (6) , the si x-particle c ontribution to the magnetic susceptibility of the square lattice Ising mo de l. W e ha ve generated, modulo a prime, series coefficients for ˜ χ (6) . The lengt h of the seri es is sufficien t to produce the corresponding F uchsian li near differen tial equation (modulo a pr ime). W e obtain the F uchsian linear differen tial equation that annihilates the “depleted” series Φ (6) = ˜ χ (6) − 2 3 ˜ χ (4) + 2 45 ˜ χ (2) . The factorization of the corresponding differen tial oper at or is p erformed using a method of f act orization mo dulo a prime int ro duced in a previous paper. The “depleted” differen tial op erato r is sho wn to ha ve a structure simi lar to the corresponding operator for ˜ χ (5) . It s plits i n to factors of sm aller orders , wi th the left-most factor of order six being equiv alent to the symmetric fifth pow er of the li near di ffe rential op erator corresp onding to the elliptic int egral E . The right-most factor has a direct sum structure, and using series calculated modulo several primes, all the factors in the direct sum ha v e been r ec onstructed in exact ar ithmet ics. P A CS : 0 5 .50.+q, 05 .10.-a, 02.3 0.Hq, 02.3 0.Gp, 02.40 .Xx AMS Cl a ss i fica tion scheme n um b ers : 34 M55, 4 7E05, 81Qxx, 32 G3 4, 3 4 Lxx, 34Mxx, 14 Kxx Key-w ords : Susceptibilit y of the Is ing mo del, long series expansions, F uc hsian linear differential equations, indicial expo nen ts, mo dular formal calculations, singular behavior, diff-Pad´ e series analysis , apparent singula rities, natural boundar y , modular factorization of differential op erators, rational reconstructio n. 1. In tro duction and recalls The magnetic susceptibilit y (high temper ature χ + and lo w tempera tur e χ − ) of the square lattice Ising mo del is given by [1] χ + ( w ) = X χ (2 n +1) ( w ) = 1 s · (1 − s 4 ) 1 4 · X ˜ χ (2 n +1) ( w ) (1) High or der F u ch sian ODE 2 and χ − ( w ) = X χ (2 n ) ( w ) = (1 − 1 /s 4 ) 1 4 · X ˜ χ (2 n ) ( w ) . (2) in terms of the self-dual tempe rature v ariable w = 1 2 s/ (1 + s 2 ), with s = sinh(2 J / kT ). The n -particle contributions ˜ χ ( n ) are given by n − 1 dimensional int egr als [2, 3, 4, 5], ˜ χ ( n ) ( w ) = 1 n ! ·  n − 1 Y j =1 Z 2 π 0 dφ j 2 π  n Y j =1 y j  · R ( n ) ·  G ( n )  2 , (3) where § G ( n ) = Y 1 ≤ i < j ≤ n h ij , h ij = 2 sin (( φ i − φ j ) / 2) · √ x i x j 1 − x i x j , (4) and R ( n ) = 1 + Q n i =1 x i 1 − Q n i =1 x i , (5) with x i = 2 w 1 − 2 w cos( φ i ) + q (1 − 2 w cos( φ i )) 2 − 4 w 2 , (6) y i = 2 w q (1 − 2 w cos( φ i )) 2 − 4 w 2 , n X j =1 φ j = 0 (7) As n grows the series generation in the v aria ble w of the integrals (3) b ecomes very time consuming . In [6] calculations mo dulo a prime were p erformed on simplified int egr als Φ ( n ) H and this work demo nstrated that most o f the p ertinen t informa tio n (singularities, critical exp onen ts, ...) can b e obtained from linea r O DE s known mo dulo a prime corr e sponding to the integrals Φ ( n ) H . In order to go b eyond ˜ χ (4) this strateg y was a dopted pre v iously for the 5-particle co n tribution ˜ χ (5) [7, 8] and here for the 6-particle contribution ˜ χ 6) . In a previous pap er [7 ] mass iv e computer calcula tions were p erformed on ˜ χ (5) , ˜ χ (6) and χ (in e x act arithmetics and/or mo dulo a prime). Thes e calculations c onfirmed previously co nj ectured sing ularities for the linea r ODEs of the ˜ χ ( n ) ’s as well as their critical exp onents, and shed some light o n imp ortant physical problems such as the existence of a natural b oundary for the susceptibility o f the squa r e Ising mo del and the subtle r esummation of logar ithm ic b eha viours of the n - pa rticle contributions ˜ χ ( n ) to give r ise to the p ow er laws of the full susceptibility χ . As fa r as χ (5) is concer ned, the linear ODE for ˜ χ (5) was found mo dulo a single pr ime [7] and it is of minimal order 33. 1.1. R esults on ˜ χ (5) In [8] the linear differential op erator for ˜ χ (5) was carefully analysed. In particular it was found tha t the minimal order linear differe ntial o perator for ˜ χ (5) can b e r educed to a minimal order linear differential op erator L 29 of orde r 29 for the linear combination Φ (5) = ˜ χ (5) − 1 2 ˜ χ (3) + 1 120 ˜ χ (1) . (8) § The F ermionic term G ( n ) has several representat ions [3]. High or der F u ch sian ODE 3 W e shall us e the term “ depleted” series for a series obtained by substracting fro m ˜ χ ( n ) a definite amount of the low er n -particle co n tributions ˜ χ ( n − 2 k ) , k = 1 , 2 , · · · , as in (8), such that the differential op erator a nnihilating the de ple ted series is of low er order. Since the depleted series is annihilated by an ODE of low er o r der, it fo llows that in the ODE for the origina l ser ie s , w e must have the o ccurrence of a direct sum structure. It was fo und [8] that the linear differential op erator L 29 , ca n b e factor ised as a pro duct of an or der five, a n order tw elve, an order one, and an o rder ele v en linear differential oper ator L 29 = L 5 · L 12 · ˜ L 1 · L 11 , (9) where the or der ele ven linear differential oper ator ha s a direct-sum decomp osition L 11 = ( Z 2 · N 1 ) ⊕ V 2 ⊕ ( F 3 · F 2 · L s 1 ) . (10) Z 2 is a second order op erator also o ccurring in the factorization of the linear differential o perator [9] a ssocia ted with ˜ χ (3) and it c orresp o nds to a mo dular form of weight one [10]. V 2 is a s econd order op erator equiv alen t to the second order op erator asso ciated with ˜ χ (2) (or eq uiv alently to the complete elliptic in tegr al E ). F 2 and F 3 are remar k able second and third or der glob al ly nilp otent linea r differential op erators [8, 10]. The first or de r linear differential op erator ˜ L 1 quite remar k ably has a p olynomial solution . The fifth order linea r differential op erator L 5 was shown to b e equiv alen t to the symmetric fourth p o wer of (the second or der op erator) L E corres p onding to the complete elliptic in tegral E . The complete and detailed a nalysis of L 12 , the order t welv e oper ator in (9) is b eyond our c ur ren t computiona l r e ssources (see [8] for deta ils). It is imp ortant to note that these factorization res ults a re exact and have b een obtained from ser ies and O DEs obtained mo dulo a single prime . F or the r econstruction in exact ar ithmetics o f the factors o ccurring in the differential op erator L 11 , we had to obtain the ser ies and ODEs for mor e than one prime. The length of the series necessary to obtain the underlying ODE is initially unknown, except p erhaps for s ome rough estimates. Once the first non- minima l order O DE s hav e b een o btained mo dulo a prime, the m inimu m length of the series necessa ry to obtain non-minimal order ODEs for a n y other primes is known exactly . This knowledge comes from a r elation we rep orted in [7] a nd that we called the ”O DE formula”. B e y ond understanding the terms o ccurring in the ”ODE formula” a nd the light they shed on the ODEs underlying the pr oblem, the for m ula has b een of mo st imp ortance in terms of ga ins in the computational effor t. F or instance, w e initially g enerated, mo dulo a prime, 100 00 terms for ˜ χ (5) and we found tha t we can obtain non-minimal O DEs using only so me 7400 terms, while non-minimal order ODE s for Φ (5) can b e o btained using some 6 200 terms, repr esen ting a great reduction in the required computatio nal effor t. 1.2. The O DE formula Let us denote by Q the order of the ODE w e are looking for and b y D the degr ee of the p olynomials in front of the der iv atives (w e write the ODE in the homog eneous deriv ativ e x d dx ). W e must then ha ve ( Q + 1)( D + 1 ) terms in the se ries in or der to determine the unknown p olynomial co efficien ts. If an ODE exists, it app ears that the nu mber of terms actually neces sary for the ODE to b e obtained is given b y N = ( Q + 1 )( D + 1) − f , (11) where f is a p ositive integer and indicates the num b er of ODE- solutions to the linear system of equa tions for the p olynomial co efficien ts. High or der F u ch sian ODE 4 F rom empirica l observ ation, we have seen [7] that N is a lso given, linearly in terms of Q a nd D , b y N = d · Q + q · D − C. (12) While Q a nd D are the o rder and the degre e, res pectively , of any non-minimal or der ODE that we choos e to lo ok for, the pa rameters d , q a nd C dep end on the series we are w orking with. In a ll the cases we hav e co nsidered, we have fo und that q is the order of the minimal or de r ODE a nd d is the num ber of singula r ities (counted with m ultiplicity) excluding any appa ren t singularities and the singular p oin t x = 0. The parameter C was shown in [8] to b e in an exa c t r elationship with the degree D app of the appare nt poly nomial o f the minimal order ODE D app = ( d − 1)( q − 1) − C − 1 . (13) Note tha t there are many ODEs that annihilate a given ser ie s. Among all these ODEs, ther e is a unique one o f minimal order. In our ca lculations we ha ve seen that it is easier to pr oduce ODEs, which a re not of minimal order [11], in the se ns e that few er terms are needed to obtain these O DEs compared to what is required to obtain the minimal o r der ODE . Even more impor tan tly for computational purp oses, there is a non-minima l order ODE that r equires the minimum num b er o f terms in or der to be obtained. Next w e demonstr a te how we use the ODE formula to optimize our calculations, i.e. genera te just the neces sary num ber of terms in the series. F rom (11-13), the parameter D is given as: D = d − 1 + D app + f Q − q + 1 (14) and this m ust b e a p ositiv e integer. The parameters f and Q ar e int ege r s with the constraints f ≥ 1 and Q ≥ q . It is a simple calculation to r un through the integers f and Q resulting in a p ositive integer D . F or each such triplet ( Q 0 , D 0 , f 0 ) the num b e r N 0 = ( Q 0 + 1)( D 0 + 1) − f 0 is the num b er of ter ms in the ser ies req uir ed to obtain f 0 ODEs o f o rder Q 0 and degr ee D 0 . Among all these N 0 there is a minimum . W e call the corr esponding ODE the ”optimal ODE” . T o obtain the ODE fo r o ther primes , it is thus o nly necessary to genera te the minimum num b er of series ter ms. F or instance, for ˜ χ (5) , the ODE fo r m ula reads N = 72 · Q + 33 · D − 887 = ( Q + 1 )( D + 1) − f . (15) The o ptimal ODE, i.e., the ODE that requires the minim um num b er of terms in the series has the triplet ( Q 0 , D 0 , f 0 ) = (56 , 129 , 8 ) which corresp onds to the minim um nu mber N 0 = 7402. Note tha t the minimal o r der O DE has the triplet (3 3 , 1456 , 1) and requir es 495 37 series terms. The minim um num b er of terms N 0 is implicitly given by the ODE formula (12). Plugging the par ameter D given in (14) in N = ( Q + 1)( D + 1) − f , one obtains N = ( Q + 1 ) d + D app + ( D app + f ) q Q − q + 1 . (16) W e can view N as a c ontinuous function of Q and f and we find that it has tw o extremums when d N / d Q = 0. F o r the p ositiv e extremum one has Q 0 = q − 1 + 1 d q ( D app + f ) q d, (17) D 0 = d − 1 + 1 q q ( D app + f ) q d, (18) N 0 = q d + D app + 2 q ( D app + f ) q d (19) High or der F u ch sian ODE 5 F or the e x ample o f ˜ χ (5) considered ab o ve, o ne obta ins (with f = 1 ) Q 0 ≃ 57 . 20 , D 0 ≃ 125 . 9 7 , N 0 ≃ 73 88 . 09 . (20) The gain in the num b er of terms is already very significant for Q = q + 1 and ca n be measur e d by the discrete deriv ative of the hyperb ola N ( Q ) given in (16). Since w e should compute ov er the integers, it is easier to c o mpute the difference of ( D + 1) ( Q + 1) ev aluated at the points Q = q and Q = q + 1. At the o rder Q = q , from (14) one obtains D ( Q = q ) = d − 1 + D app + f 1 , wher e f 1 is a po sitiv e int eger . At the order Q = q + 1 , o ne has D ( Q = q + 1) = d − 1 + ( D app + f 2 ) / 2, where f 2 is a p ositiv e int eger with same par it y as D app . The gain in the num b er of terms is ∆ N ( q, q + 1) = − d + q f 1 + 1 2 ( D app − f 2 ) q + ( f 1 − f 2 ) . (21) F or χ (5) , and with the v alues f 1 = 1 and f 2 = 2 (since D app = 1384 is even) , the “saving” in the num ber of ter ms is 2273 6 to b e compared with the 4953 7 ter ms neede d to o btain the minimal or der ODE (i.e. Q = q ). As Q increa ses, one appr oches the minim um of the hyperb ola (16) which is N 0 = 7388 . 09 (with f = 1). Over the int eger s the minim um is 740 2 o bta ined with f = 8. This pr o cess can b e rep eated by computing ∆ N ( q , q + 2) and in this case D app + f 3 should b e m ultiple of 3. As can b e seen from the “discrete” deriv ative (21), the degree of the a ppa ren t po lynomial is cr ucial. F or ODEs with no appar en t singularities the minimal order ODE is the optimal ODE. In this ca s e, the hyper bola N ( Q ) can still hav e a minimum that is no t in the integers. Note that we may define a minimal de gr e e ODE, i.e. the ODE that ha s D = d meaning that there is no singular ities other than the “true” singula rities of the minimal order ODE (no appar en t and no spurious singularities k ). The order o f this minimal de gr e e ODE is (see (14)) Q = q + D app + f − 1 , (22) giving for ˜ χ (5) , the order Q = 1417 and 1035 1 3 as the num be r of terms (the minimum f being 1). N ote that this m inimal de gr e e ODE is useless for our computational purp oses. In this paper all of thes e types o f mo dular calculatio ns and appr o ac hes have bee n applied to ˜ χ (6) . Section 2 shows the computational details (timing, ...) for the generation o f the fir st se r ies and the first ODE s, mo dulo a pr ime, from which we infer the optimal leng th of the s eries to be gener ated for other primes. In Section 3, we rep ort on the ODE annihilating ˜ χ (6) and on the ODE annihilating the cor responding “depleted” series. The sing ularities and lo cal exp onen ts confirm the results obtained from a diff-Pad ´ e analysis a nd given in a previo us pap er [7]. In Section 4, the pro gram of factor ization develop ed for ˜ χ (5) is used to factorize as far as p ossible the differential op erator corresp onding to the ODE of ˜ χ (6) . W e will s ee that o ur conjecture [8, 11] on the factoriza tion structure of the ˜ χ ( n ) holds for n = 6 . Some right factors in the differential op erator for ˜ χ (6) are obtained in exac t arithmetics. Section 5 is the conclusion. k If we denote by L q the minimal or der differen tial op erator, the non-minimal or der different ial operator L Q,D (with Q > q and D > d ) has D − d singularities whi ch are spurious wi th resp ect to L q . The spurious singularities are the ones of the operator L Q − q occurri ng i n the factorization L Q,D = L Q − q · L q . High or der F u ch sian ODE 6 2. The series of ˜ χ (6) mo dulo a prim e As shown in [7] the c a lculation of a series for ˜ χ (6) is a pr o blem with computatio nal complexity O ( N 4 ln N ). Note that ˜ χ (2 n ) is an ev en function in w and w e therefore generally work with a series in the v ariable x = w 2 , though the series for ˜ χ (6) is still calculated in the w v ariable. In T able 1 we have listed a summary of res ults for the formula (11) for v arious series with new r esults for ˜ χ (6) added. In [7] w e g a ve a rough estimate of the num ber o f terms required to o bta in the ODE for ˜ χ (6) and thought this b ey ond our computational reso ur ces. Ho wev er, upon clos er insp ection of T able 1 one obser v es that the minimum num be r of terms required to find the ODE in x for ˜ χ (2 n ) , n = 1 , 2 (or Φ (2 n ) H ) is a lw ays sma ller than the num b er of terms r e quired for ˜ χ (2 n − 1) (or Φ (2 n − 1) H ). This a lso holds for the combination 6 ˜ χ ( n ) − ( n − 2) ˜ χ ( n − 2) . It is reasona ble to exp ect tha t this w ould b e true for ˜ χ (6) as w ell. In pa rticular this would mean that the n umber of terms required to find the ODE for 6 ˜ χ (6) − 4 ˜ χ (4) should be smaller tha n the 64 00 or so terms needed to find the ODE for 6 ˜ χ (5) − 3 ˜ χ (3) . There is of course no wa y of kno wing whether or not this line of rea s oning is correct. In particular we w ould have liked to further re duce the num b er of terms to be calculated (one can fo r instance no te that the num b er of terms required to find the optimal ODE for ˜ χ (2 n ) or Φ (2 n ) H is some 10-2 0% less tha n the num b er of terms requir ed to find the optimal ODE fo r ˜ χ (2 n − 1) or Φ (2 n − 1) H , resp ectiv ely), but s ince finding the ODE for the first time is a hit-or-miss pr o position we na turally wan ted to ensur e, to the gr eatest extent p ossible, that we had eno ugh terms to find the O DE for 6 ˜ χ (6) − 4 ˜ χ (4) . F or this reason it was decided to genera te a serie s to order 650 0 in x (13 000 in w ) for ˜ χ (6) with the firm hop e that this would s uffice to find the optimal ODE for at least 6 ˜ χ (6) − 4 ˜ χ (4) (in fact it is also enough ter ms to find the optimal O DE for ˜ χ (6) itself ). T able 1. Summary of r esults for v arious series. The last three columns are the data f or the optimal ODE. The Φ ( n ) H series are the mo de l i ntegrals [6]. Series N = d · Q + q · D − C Q 0 D 0 ( Q 0 + 1)( D 0 + 1) ˜ χ (1) 1 Q + 1 D + 1 1 1 4 ˜ χ (2) 1 Q + 2 D + 1 2 1 6 ˜ χ (3) 12 Q + 7 D − 37 11 17 216 ˜ χ (4) 7 Q + 10 D − 36 15 9 160 ˜ χ (5) 72 Q + 33 D − 887 56 129 7410 ˜ χ (6) 43 Q + 5 2 D − 1 121 84 73 6290 6 ˜ χ (3) − ˜ χ (1) 12 Q + 6 D − 26 10 17 198 6 ˜ χ (4) − 2 ˜ χ (2) 6 Q + 8 D − 17 13 8 126 6 ˜ χ (5) − 3 ˜ χ (3) 68 Q + 30 D − 732 52 120 6413 6 ˜ χ (6) − 4 ˜ χ (4) 40 Q + 48 D − 945 80 66 5427 Φ (3) H 10 Q + 5 D − 21 8 13 12 6 Φ (4) H 5 Q + 6 D − 12 9 6 70 Φ (5) H 45 Q + 17 D − 277 28 80 2349 Φ (6) H 26 Q + 27 D − 342 48 39 1960 Φ (7) H 145 Q + 49 D − 194 3 92 257 23994 High or der F u ch sian ODE 7 In [7 ] the calculatio n of ˜ χ (5) to 10000 terms re quired some 17 000 CPU ho urs on an SGI Altrix cluster with 1.6GHz Itanium2 proces s ors. Given that the alg orithms for ˜ χ (5) and ˜ χ (6) has the same computational complexity this would indicate that the time required to calculate the series for ˜ χ (6) to 130 00 terms in w would b e at least 50000 CPU hours (the algor ithm for ˜ χ (6) has a slightly larger pre-factor than that for ˜ χ (5) ). In fact it tur ned out that almos t 65 000 CP U hour s was req uired a nd this calculation was per formed over a six mo n ths per io d. The series to order 65 00 was calcula ted mo dulo the prime 32749. A s in [8 ] we wan t to factorise v arious differential ope rators and reconstruct the rig h t-most factors exactly using the results from se veral primes. W e thus need to r educe a s m uch a s po ssible the length of the series by iden tifying some rig h t factors. As we detail in the following s ection the optimal ODE for ˜ χ (6) can be obta ined with less than 6 3 00 terms while the optimal ODE for the combination 6 ˜ χ (6) − 4 ˜ χ (4) requires ‘just’ ov er 540 0 terms. F urthermore w e find (using our series mo dulo a single prime) that ˜ χ (2) is a solutio n of this ODE and tha t one can simplify further b y considering the linear combination Φ (6) = ˜ χ (6) − 2 3 ˜ χ (4) + 2 45 ˜ χ (2) whose o ptim al ODE requires a little more than 5 100 terms. The ODE for ˜ χ (6) has d 2 d x 2 as the lower der iv ative, meaning that c 1 + c 2 x is a solution ( c 1 and c 2 are co nstan ts). Checking that c 1 + c 2 x is still a solution of the ODE for Φ (6) and pro ducing the series d 2 d x 2 Φ (6) ( x ), we arr iv e at a ser ies whose minimal ODE requires a little less than 5000 terms. W e therefor e calculated a further tw o series to o rder 5000 mo dulo the primes 32719 a nd 327 17. These calculations requir e d an additiona l 4 5000 hours of CPU time. Using the fac to risation pro cedure detailed in Section 4, we found a factor of o rder 3, X 3 , which rig h t divides the differential o p erator for d 2 d x 2 Φ (6) ( x ), and we manag ed to reconstruct X 3 in exa ct arithmetic us ing 3 primes ¶ . Applying X 3 , that is form the series X 3  d 2 d x 2 Φ (6)  , results in a series w ho se optimal ODE requires less than 4800 terms. At ab out the s ame time as these de velopments to ok place a new system was installed by National Computational Infra s tructrure (NCI) whose Na tio nal F acility provides the na tional p eak computing facilit y for Australia n resea rc hers. This new system is an SGI XE cluster using quad-c o re 3.0GHz In tel Harp erto wn cpus. Our co de runs ab out 4 0% faster (takes ab out 0 . 6 times the time) o n this facility when compared to the Altix cluster a nd a calculation of a ser ies to o rder 4800 takes ab out 11000 CPU hour s p er prime. W e ca lc ula ted series to this order for a further 6 pr imes, namely , 32 7 13, 3 2707, 326 9 3, 32 687, 32653 and 3 2647 (so me o f these ca lculations were per formed on the facilities of the Victoria n Partnership fo r Adv anced Computing using a cluster with AMD Barcelo na 2.3GHz qua d co re pr ocesso rs). 3. F uc hsian diff eren tial equation for ˜ χ (6) F rom the ˜ χ (6) series mo dulo a prime, w e obtained v arious ODEs which hav e the ODE formula N = 43 Q + 52 D − 112 1 = ( Q + 1)( D + 1) − f , (23) th us showing that the ODE for ˜ χ (6) is o f minimal order 5 2. W e denote by L 52 the corres p onding linear differential o perator . ¶ X 3 is equiv alent to the different ial op erator L 3 give n i n this pap er. High or der F u ch sian ODE 8 The p olynomial in fro n t of the highest deriv ative and carrying the singularities of L 52 (i.e. the ODE o f ˜ χ (6) ) reads (1 − 1 6 x ) 30 · (1 − 25 x ) · (1 − 9 x ) · (1 − x ) · (1 − 4 x ) 5 · (1 − 8 x ) × (1 − x + 16 x 2 ) · (1 − 10 x + 29 x 2 ) · P app , (24) where P app is a p olynomial whose ro ots ar e appa ren t singula r ities. Even though we hav e not computed the minimal o rder ODE, fro m (13), we can infer that the degree of P app is D app = 1020. All the s ingularities agr ee with the ones found in [7] from a diff-Pad ´ e ana ly sis and we have confirmatio n that (1 − 8 x ) is the only singula rit y not predicted [6] by the simplified in tegr als Φ (6) H . F urthermor e, using the exact (mo dulo a prime) ODE, we can confirm the lo cal exp onen ts computed fr om a diff-Pad ´ e analys is in [7] for a ll singularities except those at x = 0 , 1 / 16 a nd x = ∞ , which are correct but incomplete. The complete set of lo cal e xponents † a t these latter p oin ts read: x = 0 , ρ = − 1 , − 1 / 2 , 0 3 , 1 / 2 , 1 5 , 3 / 2 , 2 5 , 3 4 , 4 4 , 5 4 , 6 4 , 7 3 , 8 3 , 9 3 , 10 3 , 11 , · · · , 17 , x = 1 / 16 , ρ = − 2 , − 7 / 4 , − 3 / 2 , − 5 / 4 , − 1 3 , − 1 / 2 , 0 6 , 1 / 2 , 1 4 , 2 4 , 3 3 , 4 3 , 5 3 , 6 2 , 7 2 , 8 2 , 9 2 , 10 , · · · , 21 , x = ∞ , ρ = − 1 2 , − 1 / 2 , 0 3 , 1 / 2 6 , 1 2 , 3 / 2 5 , 2 , 5 / 2 3 , 3 , 7 / 2 3 , 4 , 9 / 2 2 , 11 / 2 2 , 13 / 2 2 , 15 / 2 2 , 17 / 2 2 , 19 / 2 2 , 21 / 2 2 , 23 / 2 2 , 25 / 2 , 2 7 / 2 , 29 / 2 , 3 1 / 2 , 33 / 2 , 35 / 2 , 37 / 2 , 19 . Having obtained the ODE formula (23), one c a n s e e that the minimal or der ODE requires 5639 1 terms (plug Q = q = 52 , d = 4 3, D app = 10 20 and f = 1 into (16)). And it is a simple calculation, (see parag raph a ft er (14)) to obtain the n umber of terms necessary for the o ptimal ODE . This c orresp o nds to Q = 84, D = 73 , f = 3 and N = 6287 terms. If we had to pro duce the optimal ODE for χ (6) for o ther primes it is 6 290 series co efficients that should b e generated. As mentionned in the previous se c tion, our conjecture that the ˜ χ ( n ) satisfy (with α n − 2 = ( n − 2) / 6) ˜ χ ( n ) = α n − 2 · ˜ χ ( n − 2) + β n − 4 · ˜ χ ( n − 4) + · · · + Φ ( n ) , (25) is also verified. F or the series Φ (6) = ˜ χ (6) − 2 3 ˜ χ (4) + 2 45 ˜ χ (2) , (26) we obtain non-minimal order ODE s from which w e infer the ODE formula 39 Q + 46 D − 861 = ( Q + 1 )( D + 1) − f , (27) showing that the minimal or der is 4 6 with an appar en t p olynomial (see (13)) of degree D app = 84 8. The minimal o rder ODE for Φ (6) requires the genera tion of 41736 co efficien ts series, while the optimal ODE r e quires 51 20 ter ms corr esponding to Q = 79, D = 63. It is interesting to see that the req uired num b er of terms decreases sharply from 41 736 (for the minimal order ODE Q = q = 4 6) to 2227 2 for the non- minimal o rder ODE Q = q + 1 = 47 . The gain ∆ N (46 , 47) = 19464 terms is given by (2 1 ) for q = 46, d = 3 9 f 1 = 1 and f 2 = 2 since D app = 848 is even. The gain ∆ N (46 , 48) = 259 58. † The notation is 0 3 for 0 , 0 , 0 and 9 / 2 2 for 9 / 2 , 9 / 2, etc. High or der F u ch sian ODE 9 Denoting by L 46 the differential op erator corr esponding to Φ (6) and recalling [1 1 ] the differential op erator L 10 corres p onding to ˜ χ (4) , o ne sees from (26) that the differential oper ator for ˜ χ (6) has the “dir e ct sum ¶ decomp osition” L 52 = L 10 ⊕ L 46 . (28) The sum of the or ders of the differ en tial o perator s L 46 and L 10 is larger than 52, indicating that a c o mmon factor, namely an order four differen tial oper ator, o ccurs at the rig h t of b oth L 46 and L 10 . The solutions o f this order four ODE hav e b een given in eqs. (31-3 3) a nd eq. (43) of[11]. The differential op erator (that we denote by L (4) 4 ) is given in eq. (42 ) of [11] as a pro duct o f four order one differe ntial op erators. Since the expr essions for these differential op erators were not written in [11], we give, for the sake o f completeness, in App endix A the full factorization of the differential op erator L (4) 4 . F urthermor e, we note that in the ODE for ˜ χ (6) the deriv atives of order zero and one are missing (the c o rrespo nding different ial op erator has D 2 x as the low est deriv ativ e ‡ ). The consta n t a nd the degree one poly nomial x are solutions of L 52 . The constant is a solution of the common factor L (4) 4 , but the deg ree one poly no mial x is not a so lutio n of L 10 and thus s hould occ ur in L 46 . W e thus hav e an order five differ en tial op erator that right divides L 46 ˜ L 5 = D 2 x ⊕ L (4) 4 =  D x − 1 x  ⊕ L (4) 4 . (29) W e now tur n to the factoriz a tion mo dulo a prime of the differential op erator L 46 keeping in mind that ˜ L 5 is a rig h t factor. 4. F actorization mo dulo a prime of the di fferen tial op erator L 46 The lo cal exp onen ts at the singularities of the ODE of Φ (6) allows us to e a sily track the facto rs c a rrying the v arious singula r b ehaviours. What we mean is the following. The lo cal exp onen ts for the ODE of Φ (5) , a t for instance w = 0, are all integers. Pro ducing the series having the highest exp onen t, w e obtain either the full O DE o r a right factor. If the se ries with the highest expo nen t yields the full ODE then in o rder to obtain a right factor we hav e to lo ok at the ODEs cor responding to combinations of series inv olving b oth the highest and the next highes t ex ponent as explained and done in [8]. F or Φ (6) and a t x = 0, w e ha ve tw o types of lo cal e x ponents, integer a nd ha lf- int eger ones. W e thus have a “ partition” of the solutions to the full ODE. In other words we have “ tw o highest exp onent s” § and it is ther efore more likely tha t we can av oid using combination se ries. The ODE for Φ (6) corres p onding to L 46 has at x = 0 the lo cal expo ne nts ρ = − 1 2 , − 1 / 2 , 0 3 , 1 / 2 , 1 5 , 3 / 2 , 2 5 , 3 4 , 4 4 , 5 4 , 6 3 , 7 3 , 8 3 , 9 2 , 10 2 , 11 , 12 , 13 (30) we have then tw o “highes t exp onents”, ρ = 13 and ρ = 3 / 2. This means that we can pro duce b oth the serie s x ρ (1 + · · · ) a nd se e whether o r not either o f these gives r ise to ¶ Recall [ 11] that the differential op erator for ˜ χ (2) is a factor in the direct sum of L 10 . ‡ The notation D x is d dx . § Note that for ˜ χ (5) , other singularities than w = 0 ha ve half- and fourth-intege rs exp onents. There wa s no need in [ 8] to use the pro ced ure presented here. High or der F u ch sian ODE 10 a right factor. If s o w e may not need to resor t to the combination metho d presented in Section 4 of [8]. A t the singularity x = 1 / 1 6, the lo cal expo nen ts are ρ = − 2 , − 7 / 4 , − 3 / 2 , − 5 / 4 , − 1 3 , − 1 / 2 , 0 6 , 1 / 2 , 1 4 , 2 3 , 3 3 , 4 3 , 5 2 , 6 2 , 7 2 , 8 , 9 , · · · , 19 and we ha ve three “highest exp onents”, ρ = − 5 / 4, ρ = 1 / 2 and ρ = 19. A t the singula rit y x = ∞ , there a re tw o “highest exp onen ts”, ρ = 4 and ρ = 33 / 2 since the lo cal exp onen ts ar e ρ = − 1 2 , − 1 / 2 2 , 0 3 , 1 / 2 6 , 1 2 , 3 / 2 5 , 2 , 5 / 2 2 , 3 , 7 / 2 2 , 4 , 9 / 2 2 , 11 / 2 2 , 13 / 2 2 , 15 / 2 2 , 17 / 2 2 , 19 / 2 2 , 21 / 2 , · · · , 33 / 2 Before we pro ceed, we introduce the notation L 46 = O n 2 · O n 3 , with 46 = n 2 + n 3 , which we use to indicate that the op erator L 46 factorizes into tw o op erators of orders n 2 and n 3 , r espectively . Only when a differential op erator is definitive do we give it a lab el other than the O . Let us b egin by the co njectur e [8] that L 46 has a left-mos t op erator of orde r six which is the symmetric fifth p ow er of L E . Solutions to the symmetric p o wer o f L E are po lynomials of homogeneo us degrees in the elliptic integrals with the co efficien ts of the combination b e ing ratio na ls. The solutio ns carr ying the half-integer e xponents should therefore b e those of an op erator o ccurring necess arily at the rig ht o f L 46 . So fro m the t wo “ highest exp onents” ρ = 13 and ρ = 3 / 2 at x = 0, we need only obtain the ODE of the unique ser ies x 3 / 2 (1 + · · · ). Indeed, ac ting by L 46 on the series x 3 / 2 · (1 + · · · ) pro duces a series annihilated by an order 40 ODE , leading to the facto rization L 46 = L 6 · O 40 . (31) When we shift L 46 to x = 1 / 16 and act on t − 5 / 4 · (1 + · · · ), with t = x − 1 / 16 , we obtain an order five ODE leading to: L 46 = O 41 · O 5 . (32) Shifting L 46 to x = 1 / 16 and acting o n t 1 / 2 · (1 + · · · ) pro duces: L 46 = O 36 · O 10 . (33) Shifting L 46 to x = ∞ and acting on t 4 · (1 + · · · ), with t = 1 / x , gives: L 46 = O 33 · O 13 . (34) Some factor s are common to these thr e e fac to rizations. Shifting the ODE ba c k to x = 0 and car rying out our fa c torization pro cedure [8], one obtains (some final lab elling is given) L 46 = O 41 · O 5 = O 41 · ˜ L 3 · L 2 , (35) L 46 = O 36 · O 10 = O 36 · O 1 · L 4 · ˜ L 3 · L 2 , (36) L 46 = O 33 · O 13 = O 33 · O 1 · O 3 · L 4 · ˜ L 3 · L 2 . (37) The o rder one differential op erator O 1 in the last fac to rization is equiv alent to an order o ne differe ntial op erator o ccurring in the ˜ L 5 of (29). The pro duct O 3 · L 4 · ˜ L 3 · L 2 can b e expressed as a dire ct sum O 3 · L 4 · ˜ L 3 · L 2 = L 3 ⊕  L 4 · ˜ L 3 · L 2  . (38) High or der F u ch sian ODE 11 Collecting the re s ults given in the factoriza tio ns (31) and (37) with (3 8), and keeping in mind the rig h t factor (29), one obtains L 46 = L 6 · L 23 · L 17 (39) with L 17 = ˜ L 5 ⊕ L 3 ⊕  L 4 · ˜ L 3 · L 2  , (40) ˜ L 5 =  D x − 1 x  ⊕ L (4) 4 . (41) Having obtained all these differential o perator s, a final check is p erformed by acting on Φ (6) by the corre s ponding O DEs in the order g iv en in (39) and doing this we do indeed get zero . 4.1. The differ ential op er ator L 6 The sixth order linear differ en tial op erator L 6 is the o ne that we co nj ectured [8] sho uld annihilate a homog eneous p olynomial of the complete elliptic integrals E and K of (homogeneous) degr ee five. It sho uld then b e irreducible. The lo cal expo nen ts at the origin of the linea r O DE co r responding to L 6 are x = 0 , ρ = − 12 , − 11 , − 8 , − 5 , − 4 , 0 (42) Plugging a g eneric series P c n x n int o the linear ODE fixes all the co e fficie n ts with the exception of the co efficient c 0 . The “ surviv al” of a s ing le co efficien t is a particular feature o f an irreducible fa c to r with o ne non-log arithmic so lution. The differe ntial op erator L 6 being a symmetric power of L E means that its so lution is a p olynomial in E a nd K defined as K = 2 F 1 ([1 / 2 , 1 / 2] , [1] , 16 x ) , E = 2 F 1 ([1 / 2 , − 1 / 2] , [1] , 16 x ) . (43) The ODE cor r esponding to L 6 should only hav e singularities at x = 0 , 1 / 16 and x = ∞ , a nd this is indeed the case. The lo cal exponents at x = 1 / 16 a r e x = 1 / 1 6 , ρ = − 4 8 2 , − 47 , − 44 , − 40 , 0 . (44) The lo cal exp onent s a t x = 0 a nd x = 1 / 16 suggest the following ansatz to b e plugged into the linear ODE (of L 6 ): 1 x 12 · (1 − 1 6 x ) 48 · 5 X i =0 P 5 − i,i ( x ) · K 5 − i E i . (45) The po lynomials P 5 − i,i ( x ) can b e determined numerically and the s olution (analytical at x = 0) of the ODE cor responding to L 6 is 1 x 12 · (1 − 1 6 x ) 48 ·  (1 − 16 x ) 4 P 5 , 0 · K 5 + (1 − 16 x ) 3 P 4 , 1 · K 4 E + (1 − 16 x ) 2 P 3 , 2 · K 3 E 2 + (1 − 16 x ) P 2 , 3 · K 2 E 3 + P 1 , 4 · K E 4 + P 0 , 5 · E 5  . The p olynomials P 5 − i,i ( x ) with coe fficie nts known mo dulo a prime, are of deg ree resp ectiv ely , 1 11, 112 , 113 , 113 , 113 and 11 3. As conjectured the linear differen tial op erator L 6 is thus e quivalent to the symmet ric fifth p ower of L E . High or der F u ch sian ODE 12 4.2. The differ ential op er ator L 17 The differ e ntial op erator L 17 has in its deco mposition the differential o perator ˜ L 5 which is known exactly . The s olutions of ˜ L 5 are the degree o ne p olynomial x and the four so lutions of L (4) 4 given in [1 1 ]. As fo r the o ther fac to rs of L 17 , i.e. L 2 , L 3 , ˜ L 3 and L 4 , we have b een able to expres s all o f them in exact a r ithmetics. T o expr ess a differential op erator in ex act ar ithmet ics the stra igh tforward approach is to ra tionally r econstruct the differential opera tor using several mo dulo prime calcula tions. How ever, an alternative would b e to reconstr uct the so lutions to the differen tial op erator if they are kno wn. This is what we hav e do ne for L 2 and L 3 . The singular ities of the ODEs co rrespo nding to L 2 and L 3 are only x = 0, x = 1 / 1 6 and x = ∞ . It is therefore reaso nable to a s sume tha t the solutions can be express ed as p olynomials in K ( x ) and E ( x ). F or the ODE corr esponding to L 2 , the solution (analytical a t x = 0) wr itten in terms of ˜ χ (2) , is: sol( L 2 ) =  x d d x − 2  ˜ χ (2) . (46) W ritten in this wa y , it is easy to recog nize the co efficien ts in exact arithmetics with only tw o primes. The differential op erator L 2 is thus: L 2 = D 2 x − 2 (1 + 8 x ) x · (1 − 16 x ) D x + 4 x · (1 − 16 x ) . (47) F or the third or der differential op erator L 3 , we as sumed that it is equiv alent to a symmetric s q uare of L E . Indeed, the solution (analytical at x = 0) written also in terms of  ˜ χ (2)  2 , app e ars as: sol( L 3 ) = 1 x ·  x (1 − 1 6 x ) 2 (16 x − 3) · d 2 d x 2 + (1 − 16 x ) (64 x 2 − 44 x + 9) · d d x − 8 (1 − 8 x ) (16 x + 9)   ˜ χ (2)  2 . (48) Here also, tw o pr imes ar e mor e than sufficient to reco gnize the co efficien ts. The differential oper ator L 3 , in exa ct ar ithm etics, r e a ds L 3 = D 3 x + p 2 p 3 D 2 x + p 1 p 3 D x + p 0 p 3 , (49) with: p 3 = x 2 · (1 − 1 6 x ) 2  − 81 + 1986 x − 1705 6 x 2 + 34 304 x 3 + 81 92 x 4  , p 2 = 2 x 2 · (1 − 16 x ) (224 7 − 464 96 x + 35788 8 x 2 − 56 5248 x 3 − 65 536 x 4 ) , p 1 = 6  27 − 942 x + 1115 2 x 2 − 10 1632 x 3 + 37 2736 x 4 − 65 536 x 5  , p 0 = 12 (9 − 308 x − 6208 x 2 − 10 1376 x 3 − 49 152 x 4 ) . W e ha ve not been able to find the solution o f the ODE c o responding to ˜ L 3 . The r ational reconstruction has b een do ne o n the differential op erator itself (see Appendix B). Rationa lly re constructed, the differ e ntial op erator ˜ L 3 reads ˜ L 3 = D 3 x + q 2 q 3 D 2 x + q 1 q 3 D x + q 0 q 3 , (50) High or der F u ch sian ODE 13 with: q 3 = x 2 · (1 − 4 x ) (1 − 1 6 x ) 3 Q 3 , Q 3 = − 8 + 252 x − 167 8 x 2 + 36 07 x 3 + 43 52 x 4 , q 2 = 2 x · (1 − 16 x ) 2  − 12 + 1172 x − 3049 9 x 2 + 25 2146 x 3 − 87 2579 x 4 + 77 0128 x 5 + 11 83744 x 6  , q 1 = 4 (1 − 16 x )  6 + 185 x − 28373 x 2 + 68 9440 x 3 − 51 28290 x 4 + 16 119599 x 5 − 13 139200 x 6 − 17 825792 x 7  , q 0 = 4  − 294 + 946 9 x + 844 80 x 2 − 46 52220 x 3 + 33 948640 x 4 − 97 687536 x 5 + 89 128960 x 7 + 74 981376 x 6  . All the calcula tio ns on the pr evious differential op erators hav e b een done with the t wo primes 327 49 and 3 2719. F or the differential op erator L 4 we need more primes. The differential oper ator L 4 has the form L 4 = x 3 · (1 − 1 6 x ) 4 (1 − 4 x ) (1 − 8 x ) Q 4 3 P (26) 4 · D 4 x + x 2 · (1 − 16 x ) 3 Q 3 3 P (33) 3 · D 3 x + x (1 − 1 6 x ) 2 Q 2 3 P (38) 2 · D 2 x + (1 − 16 x ) Q 3 P (43) 1 · D x + P (47) 0 , (51) where Q 3 is the appa ren t po lynomial of ˜ L 3 in (50) and P ( n ) j are p olynomials in x of degree n . T o p erform the r ational reconstruction of the p olynomials P ( n ) j , we had to g enerate the series fo r Φ (6) for ano ther seven primes, then obtain the optimal ODEs a nd factor ize the differential op erators L 46 for each prime. After the rational reconstructio n was completed successfully the r esulting differential op erator L 4 was chec ked aga inst the lo cal e x ponents a nd the conditions on the appar en t singularities. The p olynomials P ( n ) j are given in exact arithmetics in Appendix C. Note that we ha ve also chec ked that these rationally reco ns tructed differ en tial op erators are globally nilp oten t as they should b e. 4.3. The differ ential op er ator L 23 The differential oper ator L 23 has the ODE for m ula 21 Q + 23 D + 1360 = ( Q + 1)( D + 1) − f , (52) and at x = 0, the lo cal exp onen ts read: ρ = − 25 , − 2 4 , − 23 2 , − 22 2 , − 21 2 , − 20 2 , − 19 , − 1 8 , − 17 2 , − 16 , 1 , 2 , 3 , 4 , 5 , 6 , − 4 7 / 2 , − 45 / 2 W e c a n use the same metho d as b efore in or de r to facto rize L 23 . By pro ducing the series with the highest lo cal exp onen ts ρ = 6 and ρ = − 45 / 2, we obtained the full ODE for each series, i.e. an ODE formula compatible with the minimal order 23. The singular ities of the linear ODE co rrespo nding to L 23 are (bes ide s x = 0 ): (1 − 16 x ) (1 − 4 x ) (1 − x ) (1 − 9 x ) (1 − 25 x ) (1 − 10 x + 29 x 2 ) (1 − x + 16 x 2 ) W e may then s hif t the ODE cor responding to L 23 to a singular po in t other than x = 0 , pro duce the series o f the highest e xponent a nd s e e whether this g ives an ODE of or der High or der F u ch sian ODE 14 less than 23. At x = 1 / 16, the series of the highest exp onent ρ = 11 pr oduced the full ODE. Lik ewise, a t other points and ex ponents such as ( x = 1 / 4 , ρ = − 41 / 2), ( x = 1 / 9 , ρ = − 47 / 2), ( x = 1 / 25 , ρ = − 63 / 2), ( x = 1 , ρ = − 47 / 2 ) and ( x = ∞ , ρ = − 38 , − 47 / 2), the serie s give rise to the full O DE. Next we show how the lo c al stru ctur e of solutions a ppear around x = 0. W e int ro duce the notation [ x p ] to mean tha t the ser ies b egins as x p · ( const. + · · · ). The results of our computations a r e the following. Tw o sets of five solutions ca n be written as (with k = 1 , 2 ) [ x k ] ln( x ) 4 + [ x − 21 ] ln( x ) 3 + [ x − 22 ] ln( x ) 2 + [ x − 23 ] ln( x ) + [ x − 25 ] , [ x k ] ln( x ) 3 + [ x − 21 ] ln( x ) 2 + [ x − 22 ] ln( x ) + [ x − 24 ] , [ x k ] ln( x ) 2 + [ x − 21 ] ln( x ) + [ x − 24 ] , [ x k ] ln( x ) + [ x − 21 ] and [ x k ] . (53) Three sets o f three solutions can b e written as (with k = 3 , 4 , 5) [ x k ] ln( x ) 2 + [ x − 21 ] ln( x ) + [ x − 24 ] , [ x k ] ln( x ) + [ x − 21 ] and [ x k ] . (54) Two so lutions ca n b e written as [ x 6 ] ln( x ) + [ x ] and [ x 6 ] (55) Finally there are t wo non- logarithmic s olutions be having as x − 47 / 2 · (1 + · · · ) a nd x − 45 / 2 · (1 + · · · ). Besides the series x ρ · (1 + · · · ) with ( ρ = 6 a nd ρ = − 45 / 2 ) that hav e given the full ODE, we may even try the non ambiguous solutions such [ x 2 ] in fr on t o f ln( x ) 4 and [ x 5 ] in fro n t of ln( x ) 2 . B ut these series pro duce the full O DE. As is the ca se with the tw elfth order differential op erator L 12 o ccurring in ˜ χ (5) , we hav e no fina l co nclusion as to whether or not L 23 is re ducible , a nd without p erforming the factor ization ba sed on the combination metho d pres en ted in Sectio n 4 of [8] we do not e x pect to b e able to reach any such conclusio n. The representativ e o ptimal ODE of L 23 used in the calculations is of order 6 7, making the computational time obstruction more s ev ere tha n what we faced with the tw elfth order differential op erator o ccurring [8] in ˜ χ (5) . 4.4. Sum m a ry Let us now summarize our r e sults. The linea r differential op erator L 46 , corr esponding to Φ (6) = ˜ χ (6) − 2 3 ˜ χ (4) + 2 45 ˜ χ (2) can b e written as L 46 = L 6 · L 23 · L 17 , (56) with L 17 = L (4) 4 ⊕  D x − 1 x  ⊕ L 3 ⊕  L 4 · ˜ L 3 · L 2  (57) The o rder seven teen linear differential op erator L 17 contains only the singular ities of the linea r ODE cor responding to L 10 (the op erator for ˜ χ (4) ) plus the “new” § sing ula rit y x = 1 / 8 . The singular it y x = 1 / 8 o ccurs o nly in the four th o rder linear differential § It is “new” with resp ect to what we obtained from the Φ (6) H int egrals [6] and our Landau si ng ulari ty analysis [7]. High or der F u ch sian ODE 15 op erator L 4 . The third order differen tial op erator ˜ L 3 is r esp onsible for the ρ = − 5 / 4 , ρ = − 7 / 4 singular b ehavior aro und the (a n ti-)ferromagnetic p oin t x = 1 / 16. Comparing the r esults of ˜ χ (6) with those of ˜ χ (3) , ˜ χ (4) and ˜ χ (5) we note tha t our conjecture still ho lds: for a given ˜ χ ( n ) there is an or de r n differential oper ator equiv alen t to the ( n − 1)-th s ymmetric p o wer of L E at left of the depleted differential op erators, cor responding to the linear combinations ˜ χ (3) − 1 6 ˜ χ (1) , ˜ χ (4) − 2 6 ˜ χ (2) , ˜ χ (5) − 3 6 ˜ χ (3) + 1 120 ˜ χ (1) and now ˜ χ (6) − 4 6 ˜ χ (4) + 2 45 ˜ χ (2) . F or a given ˜ χ ( n ) and once the “ con tributions” of low er terms ( ˜ χ ( n − 2 k ) , k = 1 , 2 , · · · ) hav e b een substracted, the ODE of the “depleted” series still contains s ome factors occur ring in the ODE of the lo wer terms ( ˜ χ ( n − 2 k ) ). F or ˜ χ (5) , we hav e that the differential op erator Z 2 · N 1 , which o ccurs in the ODE of ˜ χ (3) , contin ues to be a right factor in the ODE of ˜ χ (5) − 3 6 ˜ χ (3) + 1 120 ˜ χ (1) . F or ˜ χ (6) , we have that the differ en tial op erator L (4) 4 , which o ccurs in the ODE of ˜ χ (4) , contin ues to b e a r igh t facto r in the ODE of ˜ χ (6) − 4 6 ˜ χ (4) + 2 45 ˜ χ (2) . As was the case for ˜ χ (5) with the differential op erators o f o r der tw o and three ( F 2 and F 3 ), we simila r ly have for ˜ χ (6) , the emerg ence o f tw o differential op erators o f o rder three a nd fo ur ( ˜ L 3 and L 4 ), whic h are globally nilp otent a nd for which we have no solutions. W e may imagine that a ll these ODEs hav e solutions in ter ms (of symmetric power) of hyperge ometric functions (with pull-back) a s we s uc c eeded to show [10] for Z 2 . P ro viding these solutions in terms of mo dular forms is clea r ly a challenge. Similarly to the t welfth order differ en tial op erator L 12 o ccurring in ˜ χ (5) , we faced with the differential op erator L 23 the same obstruction to its p oten tial factoriza tion, namely pro hibitiv e co mputational times . 5. Conclusio n W e have ca lculated, mo dulo a prime, a long series for the s ix-particle con tribution ˜ χ (6) to the magnetic susceptibility of the square la ttice Ising mo del. This s eries has bee n us ed to obtain the F uschian differential eq ua tion that annihilates ˜ χ (6) . The metho d of factor ization [8] prev io usly used for ˜ χ (5) is a pplied to the differential op erator L 52 of ˜ χ (6) . With the ODE k nown modulo a single prime, we hav e b een a ble to g o, as far a s the co mput atio na l r essources allow, in the factoriza tion of the co rrespo nding differential op erator. W e hav e found s ev eral remark able results. The factor ization structure of L 52 generalizes what we have found for the linear differential op erators o f ˜ χ (3) , ˜ χ (4) and ˜ χ (5) . In particula r, we found in ˜ χ (6) the o ccurrence of the term ˜ χ (4) but also the low er term ˜ χ (2) , le a ding to the differential opera tor L 46 corres p onding to the “depleted” series Φ (6) = ˜ χ (6) − 2 3 ˜ χ (4) + 2 45 ˜ χ (2) . The left-most factor L 6 of L 46 is a sixth or der op erator equiv alent to the symmetric fifth p ow er o f the second or der op erator L E corres p onding to complete elliptic integrals o f the firs t (or second) kind. W e exp ect that this happ ens for all ˜ χ ( n ) , i.e. we conjecture the o ccurrence in ˜ χ ( n ) of terms prop ortional to ˜ χ ( n − 2 k ) meaning a direct sum structure, and the o ccurrence of a n − th order differential op erator that left divide s the differential o perator cor responding to the “depleted” ser ies (25) of ˜ χ ( n ) . Some right factors of small order app ear in the factoriz ation of L 46 . W e hav e used the previously rep orted “O DE formula” to optimize our calculations. W e have generated other s e r ies of the minimum numb er of terms , mo dulo eight o ther primes, and have obtained the corresp onding ODEs and the cor responding facto rizations. High or der F u ch sian ODE 16 These nine factorizatio ns hav e b een used to p erform a rationa l r econstruction and obtain in exac t a r ithmetics the right facto rs o ccurring in L 46 . Our analysis is lacking the factoriz a tion of L 23 for whic h, and similarly to L 12 o ccurring in ˜ χ (5) , we hav e no conclusion on whether they are reducible. Even if these differential op erators ar e known in exact arithmetics, their fa c to rization remains a challenge for the metho ds implemen ted in v arious pac k ages of symbolic calcula tion. The massive ca lculations p erformed on ˜ χ (5) and ˜ χ (6) are a t the limit of our computational ressour ces and the next step, namely ˜ χ (7) and/or ˜ χ (8) seems to b e really out of reach. A motiv a tion for obta ining these very hig h or de r F uchsian op erators is to understand hidden mathematical structures fro m the fac to rs of these op erators. In this r espect, the ma in r esults we have obtained on ˜ χ (6) are the order three and four o perator s ( ˜ L 3 and L 4 ) that we succe eded to get in exact ar ithm etics and which a re waiting for an el liptic cu rve mathematica l interpretation. Providing a mathematica l in terpretatio n for all these differential op erators in terms of mo dular forms is clea rly our nex t challenge. The series and differential op erators studied in this pap er can b e found at [12]. Ac kno wledgme n ts W e are gr a teful to A. Bos tan for ch ecking the global nilp otence of the ra tio nally reconstructed differential op erators ˜ L 3 and L 4 . IJ is supp orted by the Australian Research Council under g r an t DP07 70705. The calcula tions would no t hav e b e en po ssible without a generous g ran t fro m the National Co mput atio na l Infrastructure (NCI) whose Na tio nal F acility pr o vides the national p eak computing fa c ilit y for Australian resear c hers. W e als o made use o f the facilities of the Victorian Partnership for Adv anced Computing (VP AC). This work has b een per formed witho ut a n y s upp ort of the ANR, the ERC, the MAE. App endix A. The order four di fferen tial op erator L (4) 4 The order four differential o p erator L (4) 4 is a rig h t facto r in L 10 the differential op erator for ˜ χ (4) . It is a pro duct of an order one differential op erator and an order three differential oper ator that ca n b e written as a direct s um: L (4) 4 = L (4) 1 , 3 ·  L (4) 1 , 2 ⊕ L (4) 1 , 1 ⊕ D x  (A.1) In terms of the v ariable x = w 2 they are: L (4) 1 , 1 = D x + 768 x 2 (1 − 16 x ) (1 − 24 x + 96 x 2 ) , (A.2) L (4) 1 , 2 = D x + 1 + 384 x 2 + 20 48 x 3 2 x · (1 − 16 x ) (1 − 48 x + 128 x 2 ) , (A.3) and L (4) 1 , 3 = D x + 2 p 0 p 1 . (A.4) with: p 1 = x · (1 − 1 6 x ) (1 − 4 x ) (80 x + 7) ( − 7 + 96 x − 115 2 x 2 + 10 240 x 3 ) , p 0 = 65 536000 x 6 − 36 536320 x 5 + 48 1280 x 4 + 25 4592 x 3 − 24 800 x 2 + 21 49 x − 49 . High or der F u ch sian ODE 17 App endix B. Reconstruction in exact arithme tics of the di ff eren tial op erator ˜ L 3 The ODE corr e s ponding to ˜ L 3 app ears as (where the singular ities ar e easily recognized): ˜ L 3 = x 3 · ( x − 1 16 ) 3 ( x − 1 4 ) P 3 · D 3 x + x 2 · ( x − 1 16 ) 2 P 2 · D 2 x + x · ( x − 1 16 ) P 1 · D x + x · P 0 (B.1) The p olynomials P 3 , · · · , P 0 are o f deg rees, resp ectiv ely , 4, 6, 7 and 7 in x . W e hav e 27 co efficien ts (not c o un ting the ov era ll one) to reconstruct. F or e asy lab eling, these p olynomials ar e denoted as ( P 3 is the p olynomial whose ro ots ar e a ppa ren t singularities) P 3 = x 4 + 3 X k =0 a k x k , P 2 = 6 X k =0 b k x k , P 1 = 7 X k =0 c k x k , P 0 = 7 X k =0 d k x k . The indicia l expo nen ts obtained with b oth ODEs (with the tw o primes 32749 and 32719 ) are x = 0 , ρ = − 2 , 0 , 2 , x = ∞ , ρ = 1 , 2 , 5 / 2 , x = 1 / 16 , ρ = − 15 / 4 , − 13 / 4 , − 1 , x = 1 / 4 , ρ = 0 , 1 , 7 / 2 , P 3 ( α ) = 0 , ρ = 0 , 1 , 3 . By demanding that the ODE co rrespo nding to the almost gener ic ˜ L 3 gives the a bov e indicial exp onen ts, lea ds to some conditions on the unknown co efficien ts a k , b k , c k and d k . The or der of the ODE b eing 3, we obtain for each singularity a maximum of three conditions. This is a ma xim um, b ecause some exp onen ts are b y construction automatically satisfied. F or instance, at x = 1 / 4, we obtain only one condition rela ted to the ex p onent ρ = 7 / 2. A t the s ingularit y x = 0 , the indicial equa tion of ˜ L 3 gives ρ = 0 as a r oot automatically sa tisfied a nd a po lynomial in ρ 2 depe nding on so me o f the unknown co efficien ts o f ˜ L 3 . By requiring ρ = − 2 and ρ = 2 a s ro ots of this po ly nomial, we obtain b 0 = 3 64 a 0 , c 0 = 3 1024 a 0 (B.2) With these v alues a ssigned, w e r equire that ρ = 1 , 2 , 5 / 2 b e r o ots of the indicial equation at the singularity x = ∞ . O ne then g ets b 6 = 17 2 , c 7 = 16 , d 7 = 5 . (B.3) Similarly , the indicia l equations ev aluated at the lo cal exp onents for the sing ula rities x = 1 / 16 and x = 1 / 4 g iv e four equa tions, fixing (e.g .) the co efficien ts b 4 , b 5 , c 6 and d 6 in terms of other co efficien ts. Next we turn to the apparent sing ularies. These are g iv en b y the ro ots of P 3 . Calling α a ro ot of P 3 (with unknown a k ), the indicia l equa tio n app ears with ρ = 0 and ρ = 1 a s auto matically s atisfied r o ots. Requiring ρ = 3 as ro o t of the indicial equation, gives a p olynomial in α of degr ee three . Zero ing each term gives 22 solutions. High or der F u ch sian ODE 18 Discarding a ll the solutions where a co efficien t from ˜ L 3 is zer o, one is left with fiv e solutions. F rom these solutio ns , there is only o ne solution which is a cceptable, b ecause it matches with the actual v alues of the co efficien ts known in prime. This fixes three co efficien ts in terms of the others. A t this p oin t, we hav e fixed 12 co efficients among the 27 using only the knowledge ab out the lo cal exp onen ts. The c o ndition o n the lo cal exp onen ts at the appar en t singularities is only neces sary , the sufficient co ndition is the absence of loga rithmic solutions aro und the s ingularit y x = α . The conditions on the non-o ccurrence of logarithmic solutions at the a pparen t singularities can b e imp osed either b y requiring the conditions of eq. (A.8) in [7] to be fulfilled or equiv alen tly by zer oing the co efficients in front of the lo g ’s in the for mal solutions of ˜ L 3 at α . With a ge ne r ic appar en t p olynomial, the calculations can b e cum b ersome. So let us fix some co efficient s. One finds that the ra tio − 2 a 1 /a 0 app ears with b oth primes 327 4 9 and 3 2719 as the n umber 6 3. Also for both primes one obtains 4 a 2 /a 0 = 839 , − 8 a 3 /a 0 = 360 7 and 2 14 d 0 /a 0 = 14 7. F urthermor e, one may co mpute the (analytica l a t x = 0) series a t b oth pr imes in the hop e that some co efficien ts will b e “ simple” enough to b e recog nized. The series with the prime 3274 9 g iv es x 2 + 48 x 3 + 15 27 x 4 + 75 41 x 5 + 31 99 x 6 + · · · (B.4) while with the prime 32 719, it rea ds x 2 + 48 x 3 + 15 27 x 4 + 75 71 x 5 + 40 69 x 6 + · · · (B.5) W e note that the same v alues o ccur a t order s 3 and 4 . These num ber s ar e therefore likely to b e exact. Also the difference b et ween the v alues at o rder 5 is a multip le of the difference 3274 9 − 3271 9, a nd simila rly at o rder 6. It is eas y to “g uess” these v alues as r espectively , 48, 1527, 40 290 and 9529 20. Comparing with the series solution of ˜ L 3 fixes four co efficients. W e hav e then tw elve co efficien ts fixed exactly a nd nine co efficien ts fixed by reconstructio n. The formal solutions of ˜ L 3 at the appare n t singularity α g iv e tw o logarithmic so lutions, with leading term, each C α k ( x − α ) 3 ln( x − α ) , k = 0 , · · · , 3 (B.6) where C dep ends on the remaining non fixed co efficien ts of ˜ L 3 . W e hav e then eigh t (non-linear) equations for six unkno wns to solve. This c an b e done b y rational reconstructio n and chec k. App endix C. The differential op erator L 4 in exact arithmeti cs The degre e n p olynomials P ( n ) j ( x ) o ccurring in the differential op e rator L 4 read: P (26) 4 = 2 8000 − 7854 000 x + 8 73083400 x 2 − 54 03701212 0 x 3 + 20 99285510 560 x 4 − 5258 29546902 98 x 5 + 76 64183841 73454 x 6 − 13 05110830 870633 x 7 − 2512 64549473 230968 x 8 + 77 27889974 481947660 x 9 − 1486 05250583 921845896 x 10 + 22 52938824 290334087840 x 11 − 2964 54756711 83771992224 x 12 + 35 44468037 92968575565792 x 13 − 3850 02396038 4577768909952 x 14 + 36 76155274 0911534545901568 x 15 High or der F u ch sian ODE 19 − 2963 38746597 146803591135232 x 16 + 19 53967934 450852091348254720 x 17 − 1033 28925663 59614848157876224 x 18 + 43 34542461 7004971574289235968 x 19 − 1428 07225508 285034141616963584 x 20 + 35 95058204 12663945726355570688 x 21 − 6360 26962079 787427490890252288 x 22 + 61 67971925 23902897669611192320 x 23 +4508 17698728 30521912080728064 x 24 − 72 44453247 75545659452335063040 x 25 +5216 86412421 099571093753036800 x 26 , P (33) 3 = − 4480000 + 156 9568000 x − 2 38072889 600 x 2 + 21 28184847 1520 x 3 − 1268 59510153 7120 x 4 + 53 55523061 0961720 x 5 − 16 40958998 875092768 x 6 +3603 21811807 27162732 x 7 − 51 14285626 75996247108 x 8 +1919 88541926 0765103140 x 9 + 12 90054571 27386313373184 x 10 − 4541 11374725 9527374959592 x 11 + 96 03568975 5227434986877112 x 12 − 1580 42146870 8164786235613784 x 13 + 22 08789769 1588508601005658336 x 14 − 2742 69909442 085751262554453856 x 15 + 30 87338965 228238905750107987648 x 16 − 3147 49226131 66692487806647824256 x 17 +2862 92076483 602608978320943481344 x 18 − 2277 95273374 0370146287983798312960 x 19 +1557 14208585 21621122719931928608768 x 20 − 9014 73105967 50652057735905075527680 x 21 +4370 37767842 717994841190340774330368 x 22 − 1755 68604455 9298411692425577783885824 x 23 +5764 64660724 9819312839063743970148352 x 24 − 1509 96442560 08129321411266837095645184 x 25 +2998 86580445 90195137583404663925899264 x 26 − 3974 50909348 62435542362760545321353216 x 27 +1939 81672176 99147074111209484113149952 x 28 +4044 72570762 17292533523320942836580352 x 29 − 8406 00427917 75646091063152898350252032 x 30 +4072 48409875 87942318458896159738953728 x 31 +3408 88013041 11660197683822288919592960 x 32 − 3487 30255389 17765121024203000119296000 x 33 , P (33) 3 = − 4480000 + 156 9568000 x − 2 38072889 600 x 2 + 21 28184847 1520 x 3 − 1268 59510153 7120 x 4 + 53 55523061 0961720 x 5 − 16 40958998 875092768 x 6 +3603 21811807 27162732 x 7 − 51 14285626 75996247108 x 8 +1919 88541926 0765103140 x 9 + 12 90054571 27386313373184 x 10 − 4541 11374725 9527374959592 x 11 + 96 03568975 5227434986877112 x 12 − 1580 42146870 8164786235613784 x 13 + 22 08789769 1588508601005658336 x 14 − 2742 69909442 085751262554453856 x 15 + 30 87338965 228238905750107987648 x 16 − 3147 49226131 66692487806647824256 x 17 High or der F u ch sian ODE 20 +2862 92076483 602608978320943481344 x 18 − 2277 95273374 0370146287983798312960 x 19 +1557 14208585 21621122719931928608768 x 20 − 9014 73105967 50652057735905075527680 x 21 +4370 37767842 717994841190340774330368 x 22 − 1755 68604455 9298411692425577783885824 x 23 +5764 64660724 9819312839063743970148352 x 24 − 1509 96442560 08129321411266837095645184 x 25 +2998 86580445 90195137583404663925899264 x 26 − 3974 50909348 62435542362760545321353216 x 27 +1939 81672176 99147074111209484113149952 x 28 +4044 72570762 17292533523320942836580352 x 29 − 8406 00427917 75646091063152898350252032 x 30 +4072 48409875 87942318458896159738953728 x 31 +3408 88013041 11660197683822288919592960 x 32 − 3487 30255389 17765121024203000119296000 x 33 , P (38) 2 = 2 02496000 − 8467 1104000 x + 1 59614046 59200 x 2 − 18 17819283 938560 x 3 +1410 42261097 575040 x 4 − 79 45786419 559994432 x 5 + 33 69704828 90735391136 x 6 − 1094 81027065 58839518064 x 7 + 27 21012517 99491505044720 x 8 − 4990 11094718 2458236154960 x 9 + 57 74716517 2968723279034760 x 10 +4251 37569084 1730042108460 x 11 − 19 66410241 2813111595220034000 x 12 +5865 39601535 060491103255831780 x 13 − 11 64828083 2868820874871506994648 x 14 +1851 68754164 459407231412940918408 x 15 − 2524 02714973 9792644483439537740736 x 16 +3061 22641602 02676427790224166656736 x 17 − 3367 56368430 758251349398256374549440 x 18 +3374 92890500 4383352843307682288939648 x 19 − 3059 48295776 94461795851875047251759104 x 20 +2475 50135999 641906395555078053550042624 x 21 − 1761 69486079 1556801623390940580862476288 x 22 +1088 05853001 65439414813579169355207311360 x 23 − 5765 14638318 86251900194559835893548711936 x 24 +2592 70510361 927197193957311877476593434624 x 25 − 9779 78052427 489585499761822245900827754496 x 26 +3043 30551555 5663644471442318841392857612288 x 27 − 7593 09162998 9468917294503828609326603304960 x 28 +1431 77209029 33442365662690637880059263713280 x 29 − 1733 74181721 94871339769688830180251691646976 x 30 +3546 87980940 4692840748046019057281136590848 x 31 High or der F u ch sian ODE 21 +3290 42237333 04447370184725984806679848419328 x 32 − 6107 02550957 17193234874579385327453575577600 x 33 +2728 71600115 87832026533318214423160423448576 x 34 +4753 81885163 82446352727572349627507901726720 x 35 − 5644 55746860 08125172119780480189438504206336 x 36 − 4318 90470369 2797702055795738669690860339200 x 37 +2361 50085518 19589708322104774634383815475200 x 38 , P (43) 1 = − 25088000 0 0 + 131387 2896000 x − 3 13495056 179200 x 2 +4540 25813150 51520 x 3 − 45 02030899 704432640 x 4 + 32 66962412 78915100672 x 5 − 1807 67648587 22283537408 x 6 + 78 26861273 10817603163904 x 7 − 2691 36541994 85748976447296 x 8 + 73 78954260 74343351817982240 x 9 − 1594 19065139 87915790530627104 x 10 + 25 87733318 15879690900773968400 x 11 − 2607 96230636 0230233373492782176 x 12 − 59 52736815 704779243433578988240 x 13 +9869 94067078 072761785220512495568 x 14 − 2714 68448845 53280870530528088810192 x 15 +5178 53080131 584647940304813906843912 x 16 − 8012 48102106 3135055260920360860291792 x 17 +1067 70798207 835443855237151398845884336 x 18 − 1266 45062311 5899739824560105189293118336 x 19 +1363 31889142 02542686825030295715897195712 x 20 − 1343 27854309 114583892390549684219213327360 x 21 +1210 06401578 9594600623288132568732268617728 x 22 − 9885 95092561 3173310943030286291198265745664 x 23 +7240 92583849 98425181940033322470418064579584 x 24 − 4697 43224562 744760675515167582702668512534528 x 25 +2668 37743533 9727605940145954082991900173762560 x 26 − 1313 03692478 54115699188867418934857934469332992 x 27 +5535 10596069 88527054614355506855140926785847296 x 28 − 1972 23974876 465329508996857363261585516738379776 x 29 +5824 89670248 346892198679343375535443002292961280 x 30 − 1378 04130057 1967278991550115889940047326999478272 x 31 +2424 92058179 4299143009574014342980105306252509184 x 32 − 2497 99592378 5565959357957923036374193256014020608 x 33 − 9152 21239768 968177447587513000776938544384966656 x 34 +8834 50927749 1743877951520843172281230968074797056 x 35 − 1473 94332339 07061551935598195219565259254451404800 x 36 +5672 24785018 1350880926829983674652540556997033984 x 37 +1686 56765653 74917674763783610716646680263347142656 x 38 − 2305 78763846 47717319687179507181219615528382889984 x 39 High or der F u ch sian ODE 22 − 1650 49741660 3706423024253737913692788809736912896 x 40 +1795 03285786 10802277327697770535083294270546771968 x 41 − 3724 97000124 3182786619005376755715492471782768640 x 42 − 5981 41334140 0069058756778898532874294556360704000 x 43 , P (47) 0 / 16 = 588 41859686 400 x − 1 23282432 000 − 1309 95528665 70240 x 2 +1817 26952372 0161280 x 3 − 17 68800126 91621796864 x 4 +1288 04418934 60632329216 x 5 − 72 99108513 92566766105088 x 6 +3301 41413923 29879832166784 x 7 − 12 10942171 584302533599014752 x 8 +3630 88444740 92544015578885632 x 9 − 88 91290880 58672919373638221264 x 10 +1750 83812715 90013090109310169040 x 11 − 2634 94886656 617518206756373588932 x 12 +2493 59171550 4400008185935972185648 x 13 +5772 65277735 7046820837948335210000 x 14 − 8801 12646375 062548999453842320020740 x 15 +2349 34943167 13860255651067234081149257 x 16 − 4380 58417861 614862884693737035489286345 x 17 +6643 94376786 3126335261566491851505292189 x 18 − 8675 66999012 68061114746560625886582904365 x 19 +1006 18662523 4680761751520210312145980149549 x 20 − 1057 35222229 31154420271493264607253396390520 x 21 +1018 94357518 227690884588911318694326634020120 x 22 − 9044 89874897 014837177389619617321458811380360 x 23 +7377 02519715 7259422622822297421365236335307120 x 24 − 5485 77503795 33672661182684179932897350723993600 x 25 +3680 77157510 764846299472690339090755869412496960 x 26 − 2203 47357683 6831766402446311571835988588370992640 x 27 +1163 89483681 94240385082022186232592838207372154880 x 28 − 5364 03165616 68843524196008022033100643589470191616 x 29 +2130 35315257 870225008043406258186703365521254907904 x 30 − 7175 84853510 007605413068883853037631249102250442752 x 31 +2000 77912690 0084461641442809746125018650394950107136 x 32 − 4414 46329778 6097513664235192893813927566161255333888 x 33 +6904 89143578 7610921130882736916279097844736823656448 x 34 − 4551 72405068 4467601081502988404586537388373763424256 x 35 − 1157 18370659 95769727688883612933393577503693010370560 x 36 +4360 40713149 66497936511910817544815142484611319201792 x 37 − 6404 66954752 93378492360343847354456353947960484036608 x 38 +1722 95208999 52062417015850756255391466062797246300160 x 39 +9891 33050273 17465024954824787190137389180923821424640 x 40 High or der F u ch sian ODE 23 − 1456 93357979 556257119098588624331861246512084740472832 x 41 − 1743 76320003 7842518500493452602647084799741447372800 x 42 +1599 68299464 829816606333313819738117801053481636724736 x 43 − 6845 31337104 64189864730237770717937227743579749744640 x 44 − 8497 45253909 92986946108353023934616304288806232653824 x 45 +4140 70974406 32033071894561954752886956699467613470720 x 46 +2869 86098546 75644415679733396189051258415886630912000 x 47 . 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