PROBLEM SECTION

AMTHOR'S SOLUTION OF ARCHIMEDES' CATTLE PROBLEM*

RUdiger Loos

V is i t i ng Professor, Universi ty of Cal i fo rn ia , Berkeley On leave from the Universi t~t Kaiserslautern

I . INTRODUCTION

Archimedes' problem for his colleagues in Alexandria, as stated in th is Bu l le t in [ l ] , amounts to solving the system ( I ) - (9 ) for posi t ive integers W, X, Y, Z and w, x, y, z.

(l) (2) (3) (4) (5) (6) (7)

W = ( l / 2 + I / 3 ) X + Y ,

X = ( I / 4 + I / 5 ) Z + Y ,

Z = ( I /6+ I /7 )W + Y ,

w = ( I /3 + I /4)(X +x) ,

x = ( ] / 4 + 1/5)(Z + z) ,

z : (115 + I / 6 ) ( Y +y ) ,

y = ( I / 6 + I / 7 ) ( W + w )

and

(8) (9)

W+X = a square

Y + Z = a t r iangu lar number .

In 1880 the problem was completely solved by A (Amthor [2 ] ) . He broke the problem into manageable parts. We checked his extensive hand calculat ions with MACSYMA, and except for the long f loa t ing point test , SAC-2; the computer calculat ions ve r i f y a l l hand calculated results as stated by Amthor. This sett les the question among competitive hand calculat ions giv ing d i f fe ren t results in his favor. We give a resume of his solut ion and state the algorithms involved.

2. THE REDUCTION OF THE PROBLEM

For any posi t ive integer n the l inear Diophan- t ine system ( I ) - (7 ) has the solutions

(10)

W = 2.3.7-53-4657n ,

X = 2-32-89.4657n ,

y = 34.11.4657n

Z = 22.5.79.4657n ,

w = 23.3-5.7-23.373n ,

x = 2-32.17.15991n ,

y = 32.13.46489n

z = 22-3-5.7.11.761n .

This research has been supported in part by National Science Foundation Grant MCS74-13278-AOI.

In order to sat is fy (8) set W+X = q2 = Q.n with Q = 22-3.11-29.4657. Let Qf be the squarefree part and Qs be the square par t of 0 : Qs'Qf. Since q2 is a square, n m~s~ be o f t h e form Q f . t ~ and consequent ly q~ = QsN~t z. In ( I0 ) n becomes

( I I ) n = 3.11,29-4657t 2

which sat is f ies now ( I ) - ( 8 ) . In order to f i n a l l y sat is fy (9) set ~+Z = nT = v(v+ l ) /2 for posi t ive v. With n = Qft one gets

v(v+ l ) /2 : nT = Qft2T

and af ter mu l t ip l i ca t ion by (8)

(12) u 2 -D t 2 = l ,

with T - 7-353.4657, ~ = 2 v + l , D = 8QfT = 23.3.7.11.29.353.4657 ~.

Equation (12) is the Pell equation, which always has the t r i v i a l solut ion t = ~, u = ± l , which is excluded here. That u has to be odd in our case is no res t r i c t i on since D is even and also Dt 2. Before we fol low Author's solut ion we would l i ke to consider the Pell equation from a constructive point of view related to well known algorithms and problems for algebraic systems.

3. THE PELL EQUATION

T r i v i a l l y , one may consider (12) as an instance of the extended Euclidean algorithm for l and -D. Since u 2 and t 2 have the same prime fac- tors as u and t , a l l solutions must be re l a t i ve l y prime.

Further, a solut ion of (12) sa t is f ies u 2 ~ l mod D which says that l is a quadratic residue of D. Since such u always ex is t , u 2 = l +sD , reduc- ing the Rroblem to the question whether s can be a square t ~.

To set t le th is question le t us rewrite (12), t = 0 being excluded, as

For s u f f i c i e n t l y large t th is can be viewed as an approximation of the "radical" r u b y the rat ional number ( u / t ) , i .e . as an approximate solut ion of

x 2-D = O. Lagrange isolates a real root by an interval of length l with integer endpoints. A simple search gives a 0 = Ix ] , and b 0 = a n + l , thereby ref ining the star t interval [ 0 , ~ to [ao,bo). For further refinement he transforms the interval [ao,bo) into [0,~) again, sending a 0 to and b 0 to O, with the transformation

l x = x 0 = a 0 + ~ I

and searches fo r a I = LXlJ now. The a t t r a c t i v e po in t f o r an a lgebra ic system is the observat ion tha t a l l ref inements can be done by using a 0 instead o f v~, working there fo re on ly w i t h i n t e - gers. Let us consider a n o n - t r i v i a l example, say D = 31 w i th a 0 = Lv~] = 5 and b 0 = 6. The ca l cu la - t i o n proceeds as f o l l ows :

v ~ i - : 5 + ( v ~ T - 5) : 5 + - - ~ T + 5

v~T+5 - 1 + v ~ i - - I _ 1 ÷ _ _ _ 5 _ _ ,

6 6 v~i-+ l

v~T+l v~T-4 3 - - - I + - I + - - 5 5 v~i-+

v~T+4 - 3 + v ~ - - 5 _ 3 ÷ _ _ _ 2 - ,

3 3 v~T+5

v~T+5 v'3T- 5 3 - T - 5 + 2 - 5 ÷ - - ,

v~T+5

v~T+5 / 3 T - 4 5 - - - 3 + - 3 + - -

4

3 3 v~i-+ 4 '

3 ~ + 4 - I + v'~"-1 - 1 ÷----6--- 5 5 v~T+I '

v~'T + l ,73T- 5 I - - - I + - - - 1 ÷ ~

6 6 v~i-+ 5

6

~ i -+5 v~T+5 = 10+(3~-5) = l O + - -

The sequence repeats i t s e l f from here i n f i n i t e l y . The calculation can be described by

(14) x 0 = v~ = a 0 +-]- Xn '

v~+P x _ n_ a + 1 n Qn n Xn+ l '

where a n = ~(l~. +Pn)/Qn] ~ L(ao+Pn)/QnJ = Lxn] and PO = O, QO . Table l summarizes the periods and in the periods symmetric sequences {a i } , {Qi } and {Pi } of the example.

Table 1

{a i } , {Qi} and {Pi} of

i 0 l 2 3 4 5 6 7 8

a. 5 1 l 3 5 3 1 l lO 1

Qi 1 6 5 3 2 3 5 6 1

Pi 0 5 l 4 5 5 4 l 5

From (14) one finds by back substitution

(15)

where

÷ AnX n An_ l

X 0 - BnXn+Bn_ l '

(16) A-l = l ' A O=a O, A n=An_la n+An_ 2 n > 2 .

B_l=O, B O=l , B n=Bn_la n+Bn_ 2 ,

The connection of Lagrange's real square root iso- lat ion algorithm, sometimes called continued frac- t ion expansion, with the Pell equation becomes sim- ple now. Substitution of (14) into (15) gives . xo = ,/~ = (An(v~+Pn)+An-IQn)/IBn-l(V~+Pn)+BnQn) which sp l i t s immediately into an i r ra t ional part An_ l = Bn_iPn+Bn_2Qn and a rational part DBn-I = An_iPn+An_2Qn . The difference of the i r ra t iona l part mult ipl ied by An_ l and the rational part mul- t i p l i ed by Bn_ l y ields

(17) AR-I - DBn2-1 = (An-IBn-2-AR-2Bn-I)QR

The expression in parenthesis can be simpl i f ied i f one eliminates a n from (16): AnBn_ l -An_IB n = - (An-l.Bn-2-An- 2Bn-l ) . . . . (-l)n(AOB_l -A-l BO) =

n+l ( - l ) . Therefore (17) is s impl i f ied to

(18) A 2 -DB~ 1 = (-l)nOn n-I - • "

From Table l we conclude 00 = 08 = Ql6 . . . . . l , hence

A 2 2 = l m = O,l ,2 . . . . 8.m-I - DBo.m_l

and one computes from (16) for m = l for example A 7 = 1520 and B 7 = 273. Our observations do not hold accidentally. Let k denote the period length, in the example 8, then

u = I , t = , k even (19) Amk-' Bmk-l m = 0, I ,2 . . . .

u = A2m~-l'K t = B2m~-'K l ' k odd

give a l l solutions of the Pell equation. In gen- eral , every continued fract ion of the square root of a rational number D > l has the f i n i t e periodic form

/D= [ao;al,a 2 . . . . . a2,al,2a O]

and has the symmetries

= a n = l , k - l an k-l ' " " ' Qn = Ok-n ' n = 0 . . . . . k ,

= Pk-n Pn , n = 0 . . . . . k-l .

Moreover, no two adjacent Pi or Qi are equal, except Pi = Pi+l for an even period k = 2i , and Qi = Qi+l for an odd period k = 2i + I . The symme- t ry cuts the computation cost by a factor of 2.

We col lect our considerations in the following two algorithms given in extended ALDES, hopefully explaining i t s e l f .

L ÷ CF2(D)

[Continued fraction expansion of a square root. D > I is an integer. L =[aO;al,...,ak_l,2aO], a l i s t of integers, k is the period length, which is 0 for a perfect square D.]

a,D,P,Q,r E Z. [The def ini t ion of the algebra allows for polymorphic operators.]

( 1 ) [ I n i t i a l i ze ] r÷ L÷

FLOOR(D**( I /2 ) ) ; P÷r; Q~I; (2*r); L ' ÷ ( ) . [two symmetric

halves]

(2)[Expand] repeat

unti l

Q'÷Q; Q÷(D-P**2)/Q; a÷(r+P)/Q; P'÷P; Pc-a'Q-P; a on L; a on L' Q=Q, vp=p, .

(3)[Combine symmetric parts] i f P=P' then [period odd] a from L'; L ÷ r on CONC(INV(L'),L); return []

t(CF2,D) < v~.

PELL(D;A,B)

[Pell equation. D is a positive integer, not a perfect square. A, B are the smallest positive integral solutions of A2-DB 2 = I . ]

a,A,Ao,B,B 0 E Z.

(1)[L= (ao;al,a 2 . . . . . a2,al,2ao)] L÷CF2(D).

(2)[Recurrence A.=anAn l+An_2, Bn: anBn_l+Bn_2 until Yi

A O~l; A from L; Bn ÷ l ; B÷ l ; while RED(L)~() ~o

{a from L; M÷A; A÷a*A+Ao; ~÷M; M÷B; B÷a*B+Bo; ÷M}; return D

This algorithm seems to be a classical instance where exact integer arithmetic is mandatory. The output tends to be large and i t does not make much sense to solve a Diophantine problem by floating point numbers.

3. AMTHOR'S SOLUTION

The given algorithm solves the problem com- pletely-- in principle. Remembera al l that was le f t to compute was n = 3.11-29.4657t ~ with t such t h a t

u 2-23"3-7"II-29-353.46572t2 = l

Amthor sets D' = 2-3.7.11.29.353, A = u, B = 2-4657t and solves the f i r s t

A 2- D'B 2 = 1 .

The continued fraction expansion unt i l the middle element (~) is D~= [2174;1,2,1,5,2,25,3,1,I,I,I, l 1,15,1,2,16,1,2,1,I ,8,6,1,21,I, I ,3,1,I , I ,2,2,6,1, 1,5,1,17,1,I,47,3,1,I,(~) . . . . . 4348] and has a period of 91. Amthor computes by hand

A = I09931986732829734979866232821433543901088049

B = 50549485234315033074477819735540408986340

(20)

as smallest positive solutions. All other positive solutions An, B n can be computed, alternatively to (19), by

A n+Bn/D= (A+Bv~) n

Since a solution of the original Pell equation exists i t remains to find an n such that 2.46571B n. He uses the recurrence relations

(21) Am+ n = + B B D AmAn m n

Bm+ n = AmB n + AnB m

which show that a l l B n w i l l be even i f B is. Hence i t suffices to compute n such that B n z 0 mod M with M = 4657. Instead of computing B2,B 3 . . . . modM he proves the interesting theorem that only such B o z 0 mod M, M a prime, for which pI(M+l) i f (D/M) = - l or pl(M-l) i f (D/M) = +l holds. Now, (D/M) : - l , M+l = 4658 = 2.17.137 and the only Bo's to be checked have an index p = 2, 17, 34, 137, 274, 2329 or 4658. I t turns out that B2329 z Omod4657. Therefore, i f I r denotes the irrat ional part,

(22) t = Ir(A + Bye) 232g ,

A and B taken from (20). Instead of using this ingenious method we applied (21) i te ra t ive ly :

TEST.

A,B,D,M EZ; u ,u ' , u " , t , t ' EZ(M).

(1 ) [ I n i t i a l i ze search for B n z Omod M] BEGIN2; PELL(2*3*7*ll*29*353;A,B); M~4657; u ÷ l ; u ÷ A mod M; t ÷ B mod M; d ÷ D mod M; u '÷u; t ' ÷ t .

(2)[Apply recurrence (21)] while t ' #O do

{u÷ u+l ; u"÷u' ; u ' ÷ ( u * u ' + t ' * t * d ) mod M; t ' ÷ (u* t '+ u"*t) mod M};

WRITE("A =",A,"B =",B,"n =",n) []

and veri f ied A, B in (20) and n = 2329. To use (1) and (2) in the above algorithm without mod M ar i th- metic is as wasteful as using the PELL algorithm direct ly. Amthor uses logarithms

loglot = log(A+ Bye) 2329 = 2329.44.3421541

= I03272.8769

and gives log t 2 = 2~6530.5388 and W = 1598.10206541

The last four lineswere veri f ied only by R. Fateman's impressive Big-Floating-Point Arith- metic [3] in MACSYMA. Since the precision can be programmed, (22) can be computed to any desired accuracy. However, " i t does not make sense to

l enerate a number as large as a phone book" D.H. Lehmer).

6

4. HISTORICAL REMARKS

Fermat knew that Pe l l ' s equation has i n f i n i t e - ly many solut ions. As usual, he did not publish his proof. The theorem on which the continued f rac t ion expansion algorithm is based is known as the Euler-Lagrange Theorem. The time saving te r - mination c r i t e r i on is due to Muir. A comprehen- sive treatment is given in O. Perron, "Die Lehre von den Kettenbr~chen," Leipzig, 1913. The Pell equation is t reated, unfortunately, in a non- construct ive manner, for example in W. Adams and L. Goldstein, Introduct ion to the Theory o.f_ Numbers, Englewood C l i f f s , N.J., 1976, containing many programming exercises. The solut ions of the Pell equation are units in the real quadratic extension f i e l d ~(v'D). I t is therefore reasonable to require D to be squarefree, the case used by Amthor. This context is treated in depth by H. Hasse, "Vorlesungen Uber Zahlentheorie," Ber l in , 1950_~where we took the example for the expansion of /31. Algorithms to solve l inear Diophantine equations can be found in Knuth's The Art of Computer Programming, Ch. 4, or in the Col lect ion o_f_Algorithms of the ACM. There is no analysis known to me. Before Amthor, Gauss is believed to have solved Archimedes' problem completely. For the phi lo logica l questions--which arose par t ly because of the incredib ly large numbers--see L. Dickson, History of the Theory of Numbers, Vol. I I , New York, Chelsea, 1952. I was not aware of Amthor's paper when I submitted the problem to the Bu l le t in .

ACKNOWLEDGMENTS

I thank R. Fateman for his MACSYMA and E. and M. Lauer for algorithms in ALDES in Madison.

assistance with programming the

REFERENCES

[ I ] Problem Section, "The Catt le Problem of Archimedes," SIGSAM Bu l le t in of the ACM, New York, Vol. 9, Aug. 1975.

[2] "Das Prolema bovinum des Archimedes," B. Krumbiegel und A. Amthor, Histor isch- l i t e ra r i sche Abtei lung, Ze i t s ch r i f t fur Physik und Mathematik, XXV, 1880, 153-171.

[3] R. Fateman, "The MACSYMA 'B ig-F loat ing-Point ' Ar i thmetic System," Proceedings of 1976 ACM Symposium on Symbolic and Algebraic Computa- t ion, Aug. 10-12, 1976, Yorktown Heights, N.Y., ed. R. Jenks, 209-213.

ABSTRACTS (Cont. from page S)

necessary to abtain a canonical form for functions obtained by integration and exponentiation from the set of rational functions. These aspects include a new algorithm for symbolic integration of functions involving logarithms and exponentials which avoids factorization of polynomials in those cases where algebraic extension of the constant field is not re- quired, avoids partial fraction decompositions, and only s].oves linear systems with a small number of unknowns. We also found a theorem which states, roughly speaking, that if the integrals which can be represented as logarithms are presented as such, the only algebraic dependence that a new logarithm can satisfy is given by the law of exponents or the law of logarithms.

An Avera6e-Case Analysis of the Bucketsort Algorithm for Sparse Polynomial Multiplication, by Frank Teer Informatika Rapport Nr. 14, JuG 1~6, VriJe Univer- siteit, The Netherlands. An analysis is given of the average number of arith- metic operations in multiplying two sparse polynomi- als having m and n terms respectively with a new algorithm, called the bucketsort algorithm, recently developed by the author. It is shown that the aver- age number of arithmetic operations in multiplying two polynomials is O(mn) in contrast to the O(mn" logn) of other algorithms.

Sparse Complex Polynomials and Polynomial Reducibi- lity, by D. A. Plaisted, Dept. of Comp. Science, University of Illinois at Urbana-Champaign, Urbana, Illinois 61801 We show that certain problems involving sparse poly- nomials with integer coefficients are at lease As hard as any problem in NP. These problems include determining the degree of the least common multiple of a set of such polynomials, and related problems. The proofs make use of a homomorphism from Boolean expressions over the predicate symbols (P_,...,P) onto divisors of the polynomial xN-1, where N isnthe product of the first n primes. Various combinato- rial and number theoretic applications are also pre- sented. (To appear).

Some Polynomial and Inte6er Divisibility Problems a~ NP-hard, by D. A. Plaisted, Dept. of Comp. Science, Univ. of Illinois at Urbana-Champaign, Urbana, Ill. 61801. In an earlier paper (see previous abstract, Ed.) the author showed that certain problems involving sparse polynomials and integers are NP-hard. In this paper we show that many related problems are also NP-hard. In addition, we exhibit some new NP-comple- te problems in which the non-determinism is hidden. That is the problems are not explicitly stated in temns of one of a number of possibilities being true. Furthermore, most of these problems are in the areas of number theory or the theory of functions of a comp- lex variable. Thus there is a rich mathematical theory that can be brought to bear. These results therefore introduce a class of NP-hard and NP-complete problems different from those known previously. (Proceedings of the IEE SWAT Foundations of Computer Science Conference, Houston, 1976)

Aspects of Symbolic Integration and Simplification of Exponentia& and Primitive Functions, by M. Roth- stein, University of Wisconsin Ph.D. Thesis (Dec, 76) 119 p. (Available from Xerox Univ. Microfilm, Ann Arbor, Michigan) In this thesis we cover sQme aspects of the theory