Units generating the ring of integers of complex cubic fields. Robert Tichy and Volker Ziegler. Project Area(s): Algorithmische Diophantische Probleme

Transcription

1 FoSP Algorithmen & mathematische Modellierung FoSP Forschungsschwerpunkt Algorithmen und mathematische Modellierung Units generating the ring of integers of complex cubic fields Robert Tichy and Volker Ziegler Project Area(s): Algorithmische Diophantische Probleme Institut für Analysis und Computational Number Theory (Math A) Report , September 2007

2 UNITS GENERATING THE RING OF INTEGERS OF COMPLEX CUBIC FIELDS ROBERT F. TICHY AND VOLKER ZIEGLER Abstract. All purely cubic fields such that their maximal order is generated by its units are determined. 1. Introduction In 1954 Zelinsky [15] showed that, if V is a vector space over a division ring D, then every linear transformation can be written as the sum of two automorphisms unless dim V = 1 and D is the field of two elements. Later many authors investigated similar problems for various classes of rings. This gives raise to the following definition (see Goldsmith, Pabst and Scott [6]). Definition 1. Let R be a ring (with identity). An element r is called k-good if r = e e k, with e 1,..., e k R. If every element of R is k-good we call also the ring k-good. The unit sum number u(r) is defined as min{k : R is k good}. If the minimum does not exist but the units generate R additively we set u(r) = ω. If the units do not generate R we set u(r) =. For some historic information on this topic and several examples we refer to recent papers of Ashrafi and Vámos [1], and Vámos [14]. Endomorphism rings have been studied in great detail and also some other classes of rings were investigated from this point of view. Which rings of integers are k-good has been investigated by Ashrafi and Vámos [1]. In particular, they proved that the ring of integers of quadratic fields, complex cubic fields and cyclotomic fields Q(ζ 2 N ), with N 1, are not k-good for any integer k. Jarden and Narkiewicz [9] proved that every finitely generated integral domain of characteristic zero has unit sum number ω or. In other words, they proved that no ring of integers has finite unit sum number. However, the question which rings 1991 Mathematics Subject Classification. 11D25, 11R16. Key words and phrases. units, complex cubic fields, cubic diophantine equations. The second author gratefully acknowledges support from the Austrian Science Fund (FWF) under project Nr. P18079-N12. 1

3 2 R.F. TICHY AND V. ZIEGLER of integers are generated by their units remains. In case of quadratic fields Belcher [2] and Ashrafi and Vámos [1] answered independently this question. Similar questions arose in 1964 when Jacobson [8] asked which number fields K have the property that all algebraic integers of K can be written as the sum of distinct units. Let us denote by U the set of number fields that have this property. Jacobson [8] proved that the number fields Q( 2) and Q( 5) are members of U. Some years later Śliwa [12] proved that these two fields are the only quadratic fields with this property. Moreover Śliwa showed that there is no field of the form Q( d) that lies in U. Criteria for which a number field lies in U were given by Belcher [2, ]. In particular Belcher [] proved that K U if 2 is the sum of two distinct units and the ring of integers of K is generated by its units. By an application of this criterium Belcher [] characterized all cubic number fields with negative discriminant that lie in U. The aim of this paper is to consider which rings of integers of complex cubic fields, in particular purely cubic fields, are generated by their units. Theorem 1. Let X BX C be an irreducible polynomial having a complex root, and let α be any root of the polynomial, possibly not complex. Let O = Z[α]. Then O is generated by its units if and only if there exists a solution (X, Y ) to the Diophantine equation X + BXY 2 CY = ±1, such that there is a unit of Z[α] of the form Z + Xα + Y α 2 with Z an integer. This theorem together with results of Delaunay [5] and Nagell [10] yields: Corollary 1. Let d be a cube-free integer and K = Q( d) the corresponding purely cubic field. Then the order Z[ d] is generated by its units, i.e. there exist ɛ 1, ɛ 2 Z[ d] such that ɛ 1 and ɛ 2 generate Z[ d], if and only if d = a ± 1 with a Z. As our main result we will establish the following theorem. Theorem 2. Let d be a cube-free integer and let O d be the maximal order of Q( d). The ring O d is generated by its units if and only if d is square-free, d ±1 mod 9 and d = a ± 1 for some integer a or d = 28.

4 UNITS GENERATING THE RING OF INTEGERS Since in general Q( d) has no integral power basis the proof of Theorem 2 is far from being straight forward. 2. The quadratic case revised The aim of this section is to present the basic ideas for the proofs of our results. For this purpose we start with the quadratic case and give a simple proof of the result due to Ashrafi and Vámos [1], Theorems 7 and 8. Proposition 1. Let d Z be square free, then O = Z[ d] is generated by its units, if and only if d = a 2 ± 1 for a Z. Before we prove Proposition 1. We want to state and prove following helpful lemma. Lemma 1. If ɛ is a unit of some number field K with deg K = d and some powers of ɛ generate the additive group of integers (or some order of K) then then also 1, ɛ,..., ɛ d 1. Proof. It is enough to show that the Z-module generated by 1, ɛ,..., ɛ d 1 contains ɛ k for all k Z. This is easy to see since ɛ is an algebraic integer, and we have ɛ d = a 0 + a 1 ɛ + a d 1 ɛ d 1 with a i Z, with i = 0, 1,..., d 1. Now by induction we see that every positive power of ɛ is a linear combination of 1, ɛ,... ɛ d 1 with integral coefficients. Similarly we can express ɛ 1, ɛ 2,... as integral linear combinations of 1, ɛ,..., ɛ d 1. Proof of Proposition 1. Assume ɛ 1, ɛ 2 O generate O. Then also 1 and ɛ generate O, where ɛ is the fundamental unit of O. Therefore we may assume without loss of generality that 1 and ɛ generate O. Let ɛ = x + y d, then the statement that 1 and ɛ generate O is equivalent to (1, 0) and (x, y) generate the lattice Z 2, hence y = ±1. Since ɛ is a unit we have x 2 dy 2 = ±1 and therefore x 2 d = ±1 or d = 1 + x 2. This shows one direction. Now let us assume d = a 2 ±1. Every unit ɛ = x+y d O fulfills the equation x 2 dy 2 = x 2 (a 2 ± 1)y 2 = ±1 with mixed signs. Obviously one solution is x = a and y = 1. Since (1, 0) and (a, 1) generate Z 2 also 1 and ɛ = a + d generate O. Proposition 2. Let d Z be square-free. Then the ring of integers of Q( d) is generated by its units if and only if the following holds: d = a 2 ± 1, d 1 mod 4, d = a 2 ± 4, d 1 mod 4.

5 4 R.F. TICHY AND V. ZIEGLER Proof. Use the same method as above where d is replaced by 1+ d 2 in the case of d 1 mod 4. Note that the ring of integers is generated by 1 and d if d 1 mod 4 and by 1 and 1+ d 2, otherwise.. The general cubic case This section is devoted to the proof of Theorem 1 and Corollary 1. Proof of Theorem 1. Since we assume that Q(α) has a complex embedding into C the complex numbers the unit structure of Z[α] is very simple. By Dirichlet s unit theorem we know Z[α] = ζ, ɛ, where ζ is some root of unity and ɛ is the fundamental unit. Since Q(α) is of degree, the root of unity ζ can only have degree 1 or. Since φ(n) = has no solution, ζ is of degree 1, hence ζ = 1. With ɛ 1, ɛ 2, ɛ also ±ɛ 1, ±ɛ 2, ±ɛ generate Z[α]. Thus we may assume ɛ 1 = ɛ k 1, ɛ 2 = ɛ k 2 and ɛ = ɛ k with k 1, k 2, k Z. Therefore we may assume by Lemma 1 that 1, ɛ, ɛ 2 generate Z[α]. Let us write ɛ = a + bα + cα 2 then a short computation shows that b:= ã:= {}}{{}}{{}}{ ɛ 2 = a 2 + 2bcC +( 2ab + 2bcB + c 2 C)α + ( b 2 + 2ac + c 2 B)α 2 =ã + bα + cα 2. Therefore the vectors (1, 0, 0), (a, b, c) and (ã, b, c) generate the lattice Z, i.e. det M = ±1 with M = a b c. ã b c A short computation shows c:= det M = b c bc = b bc 2 B c C = ±1, and ɛ has the desired form. The other direction is quite easy. Assume ɛ = a + bα + cα 2 has the properties described in Theorem 1 then the the vectors (1, 0, 0), (a, b, c) and (ã, b, c) generate Z, where ɛ 2 = ã + bα + cα 2. Hence 1, ɛ and ɛ 2 generate Z[α]. Next we prove Corollary 1. We apply Theorem 1 with B = 0, C = d and put α = d. Hence O = Z[α] is generated by its units if and only if there is a unit ɛ O of the form ɛ = a + bα + cα 2, with a, b, c Z such that b dc = ±1. By a theorem of Delaunay [5] we know that the equation X dy = ±1 has at most one solution beside except the trivial solution X = ±1 and Y = 0. Moreover, Delaunay showed

6 UNITS GENERATING THE RING OF INTEGERS 5 that for a solution (X, Y ) to X dy = 1 the quantity X + dy is a fundamental unit. Assuming, b dc = ±1 we have by the proof of Theorem 1 that the fundamental unit satisfies ɛ = ±(a + b d + c d). On the other hand by Delaunay [5] ɛ = ±( b + c d), where ( b, c) is the non-trivial solution to X dy = 1. If (b, c) is a non-trivial solution then we get a contradiction, therefore b = ±1 and c = 0. Hence we see that ɛ = a ± d. This yields a ± d = ±1 or equivalently d = a ± 1 for some integer a. 4. Purely cubic fields of the first kind The next two sections are devoted to the proof of Theorem 2. At the beginning of the proof of Theorem 2, we remind the well known fact (e.g. see [4, section 6.4.]) that if d = ab 2 with a, b Z square-free and coprime, then O d is generated by 1, ab 2 and a 2 b if d ±1 mod 9 and by 1 (1+a ab 2 +b a 2 b), ab 2 and a 2 b otherwise. In the case where d is square-free and d ±1 mod 9, Corollary 1 yields Theorem 2. Let us consider now the case d ±1 mod 9 and b 1. In view of Lemma 1, we assume that there exists a unit ɛ = X + Y ab 2 + Z a 2 b such that {1, ɛ, ɛ 2 } generates O d. Since (1) ɛ 2 = X 2 + 2abY Z + (az 2 + 2XY ) ab 2 + (by 2 + 2XZ) a 2 b, we have to investigate the equation det M = ±1, where M = X Y Z. X 2 + 2abY Z az 2 + 2XY by 2 + 2XZ Therefore (Y, Z) has to be a solution to the Diophantine equation (2) by az = ±1. It is obvious that with {1, ɛ, ɛ 2 } also {1, ɛ 1, ɛ 2 } generates the algebraic integers (see Lemma 1). Since ɛ 1 = (X 2 aby Z) + (az 2 XY ) ab 2 + (by 2 XZ) a 2 b also (az 2 XY, by 2 XZ) is a solution to (2). By a theorem of Delaunay [5] and Nagell [10] we know that (2) has at most one solution with Y 0. Suppose (Y, Z) is such a solution, we have az 2 XY = ±Y and by 2 XZ = ±Z. Note that the signs for Y and Z must be the same. Eliminating X from these equations yields by az = 0 which is a contradiction. Note that Y Z 0, since b 1 and a 1.

7 6 R.F. TICHY AND V. ZIEGLER 5. Purely cubic fields of the second kind Now the situation is more complicated. Since ab 2 ±1 mod 9 we have a 1 mod and a ±b mod 9. Let a eb mod 9 with e {±1}. Then with 1(1 + a ab 2 + b a 2 b), ab 2 and a 2 b also 1, ab 2 and 1(1 + ab 2 + e a 2 b) is an integral basis. Therefore we write hence ɛ = X + Ỹ ab 2 + Z a 2 b = ξ + η a 2 b + ζ 1 + ab 2 + e a 2 b, X = ξ + ζ/, Ỹ = η + ζ/, Z = eζ/. Moreover, let X = X, Y = Ỹ and Z = Z. We can express ɛ 2 in the new basis and obtain ({}} ){ ɛ 2 = ξ 2 ebη 2 2 e(2ab b) 1 e(ab b) + ζ + 2ηζ + 9 η:= ξ:= ({}}{ ebη 2 + ζ 2 a eb + 2ξη + 2ηζ 1 eb ) 9 ζ:= ab2 + {( }}{ ebη 2 + ζ eb ) 1 + 2ξζ + 2ebηζ (1 + ab 2 + e a 2 b). Therefore we have to investigate the equation det M = ±1, with M := ξ η ζ ξ η ζ. This yields the equation which is equivalent to eb(η + ζ) aζ = ±9, () by az = e 1 9, where e 1 {±1}. With {1, ɛ, ɛ 2 } also {1, ɛ 1, ɛ 2 } generates O d. Therefore with (Y, Z) also ( az2 XY, by 2 XZ ) Z Z is a solution to the Diophantine equation (4) by az = ±9.

8 UNITS GENERATING THE RING OF INTEGERS 7 Let us assume ( az2 XY, by 2 XZ ) fulfills (4) with e 19 on the right side, where e 1 {±1}. As above we see that the two solutions ( az 2 XY ±, by ) 2 XZ and ± (Y, Z) are distinct since otherwise az 2 XY = ±Y, by 2 XZ = ±Z. These two equations imply ±9 = by az = 0, ±6Y Z depending on the signs. However each of these cases is impossible since X, Y, Z Z. Note that Z and Y is impossible, since otherwise both Y and Z are divisible by and this implies 27 9, a contradiction. On the other hand a famous result due to Siegel [11] tells us that there is at most one solution to if with ax n by n c ab n/2 1 λ n c 2n 2, λ n = 4 n p n p 1/(p 1) In our case this yields ab > However, by this estimate too much cases remain to be checked individually. So we have to refine this method. Now we take into account that ɛ is a unit. Therefore we find (5) X + ab 2 Y + a 2 bz abxy Z = e 2 27, with e 2 {±1}. Let us assume a 10. Since by az = ±9 and Z 0 we see that Y and Z have the same sign. Without loss of generality we may assume that Y, Z > 0. Moreover we may assume that ɛ < 1. Since Y ab 2 + Z a 2 b ab( a + b) > ab > n. > ɛ = X + Y ab 2 + Z a 2 b, we have X < 0. Let us compute the asymptotics of X and Y in terms of Z and of X and Z in terms of Y. Since we need exact error terms we use the so called L-notation (cf. [7]). This notations allows us to keep track of how large the constants of the usual O-terms get. The L-notation is defined as follows: For two functions g(t 1,..., t k ) and h( t 1,..., t k ) and positive numbers u 1,..., u k we write g(t 1,..., t k ) =

9 8 R.F. TICHY AND V. ZIEGLER L u1,...,u k (h( t 1,..., t k )) if g(t 1,..., t k ) h( t 1,..., t k ) for all t 1,..., t k with absolute value at least u 1,..., u k respectively. Note that all the following computations have been performed with Mathematica r First we compute Y in terms of Z: Y = az + e 1 9 b = Z a/b + e 1 a/b 9 a/b + O(1/Z 6 ) az 2 a 2 Z 5 For further computations we need an L-term instead of an O-term. Let Computations show Y + =Z a/b + e 1 a/b a/b + 11 az 2 a 2 Z, 5 Y =Z a/b + e 1 a/b a/b 11 az 2 a 2 Z. 5 (b(y + ) az e 1 9)(b(Y ) az e 1 9) = ( ζ e 1 ζ 14520ζ e 1 ζ ζ e 1 ζ ζ 8) /(a 10 Z 0 ), where ζ = az. This quantity is positive if ζ > 28.66, in particular if a 29 and Z 1. This shows (6) Y (a, b, Z) = Z a/b + e ( ) 1 a/b a/b + L az 2 29,1,1 11. a 2 Z 5 Similarly we obtain (7) Z(a, b, Y ) = Y b/a + e ( ) 1 b/a b/a + L by 2 29,1,1 11. b 2 Y 5 Now let us compute X. Remember that (8) (9) p 1 := by az e 1 9 = 0, p 2 := X + ab 2 Y + a 2 bz abxy Z e 2 27 = 0. We compute the Groebner basis of the ideal generated by p 1 and p 2 with respect to the lexicographic term order such that X Z Y.

10 UNITS GENERATING THE RING OF INTEGERS 9 The first component of the Groebner basis is p :=729a b e a 2 b 2 e abe e a 2 b 2 X 1458abe 1 e 2 X X + 27abe 1 X 6 81e 2 X 6 + X a 4 b Z 2916a b 2 e 1 e 2 Z + 474a 2 bz 15a b 2 e 1 X Z 24a 2 be 2 X Z + 6a 2 bx 6 Z + 108a 5 b e 1 Z 6 24a 4 b 2 e 2 Z 6 15a 4 b 2 X Z 6 + 8a 6 b Z 9. Since p is a polynomial of degree in terms of X, it has either 1 or real roots. Because p comes from a Groebner basis with lexicographic order the solutions of p for some fixed Z are the same as those of p 2 with (Y, Z) a fixed solution to p 1, with the same Z. Since the constant term is positive (remember Y, Z 1 and a 10) either all roots of p 2 are negative or only one is negative. The fact that the coefficient of X 2 of p 2 is zero shows that not all three roots can be negative. Therefore we deduce that there is exactly one negative root of p for positive Z. If we compute the asymptotics of the solutions to p in terms of Z we find that one asymptotic has the form 2Z a 2 b + abe 1 a 4/ b 2/ Z abe 1(abe 1 1) + O(1/Z 6 ). 2 a 10/ b 5/ Z 5 Indeed this is the desired approximation to X. Let us compute p (X +, Z)p (X, Z) = Z96 a 68 b 6 + Z 90 a 60 b 0, where the rest of the numerator is a polynomial of lower degree (in each variable) and X + = 2Z a 2 b + abe 1 + 9ab(ab + 1) + 26, a 4/ b 2/ Z2 a 10/ b 5/ Z 5 X = 2Z a 2 b + abe 1 + 9ab(ab + 1) 26. a 4/ b 2/ Z2 a 10/ b 5/ Z 5 Since the numerator is positive for a 41, b 1 and Z 1, we deduce (10) X(a, b, Z) = 2Z a 2 b + abe 1 a 4/ b 2/ Z 2 + L 41,1,1 Similarly we obtain (11) X(a, b, Y ) = 2Y ab abe 1 a 2/ b 4/ Y 2 + L 51,1,4 ( 2 ( 2 ) 6 + 9ab(ab + 1). a 10/ b 5/ Z 5 ) 6 + 9ab(ab + 1). a 5/ b 10/ Y 5 Because of the form of the L-terms we assume from now on a 51, b 1, Y 4 and Z 1.

11 10 R.F. TICHY AND V. ZIEGLER If we substitute (6) and (10) in az2 XY, and (7) and (11) in by 2 XZ we obtain (12) Y := az2 XY = az 2 + e 1 Z 1 abz + az e 1 4 a 2 bz + 4 and (1) L 51,1,1 ( 40 az a b 2 Z a 2 bz a 2 Z a 4 b 2 Z a bz 7 Z := by 2 XZ = by 2 e 1 Y 1 aby + by 4 + e 1 ab 2 Y 4 + L 51,1,4 ( 40 by a 2 b Y ab 2 Y b 2 Y a 2 b 4 Y ab Y 7 respectively. Note that Z = by 2 + R 1, where R 1 is small if Y, a, b are large. Remember that we assume Y 4, a 51 and b 1. In the case of (1) we see that R 1 < Since Z is an integer also R 1 has to be an integer, hence R 1 = 0 and Z = by 2. Similar, if we assume Z 4, a 51 and b 1 we obtain Y = az 2 + R 2, with R 2 < Hence R 2 = 0 and Y = az 2. If Z = 2 then Y = az 2 + e 1 /2 + R. From (12) we compute R < 0.01 if a 51 and b 1. But this implies that Y is not an integer and we have a contradiction. In the case of Z = 1 we find Y = a + e 1 + R 4, with R 4 < 0.55 if a 51 and b 1, hence Y = a + e 1. Since (Y, Z ) is a solution to (4) and Y = a + e 1 and Z = by 2 we obtain (14) b(a+e 1 ) a(by 2 ) 9e 1 = a b+9a 2 be 1 +27ab+27be 1 9e 1 ab Y 6 = 0 and therefore b 9. Since ab 2 ±1 mod 9 we find b = 1. Now (14) has the following form: a + 9a 2 e a + 27e 1 9e 1 ay 6 = 0. This is a 18 or a 6. Since we assume a 51 we have a contradiction. Now, if we assume Y 4, a 51 and b 1, then we have Y = az 2 and Z = by 2. Moreover, we obtain ba Z 6 ab Y 6 = ±9, hence ba 9, which is again a contradiction to a Small a We still have to consider the case a 50 or Y. In this section we want to exclude the case a 50. Since ab 2 ±1 mod 9 we have a 1 mod and b ±a mod 9. Since we assume a and b square-free, ), ),

12 UNITS GENERATING THE RING OF INTEGERS 11 gcd(a, b) = 1 and a > b 1, there are only finitely many possibilities left for the pair (a, b). For all possible pairs (a, b) we will solve the Diophantine equation by az = ±9 with Z > 0. If an equation has more than two solutions, the quantity d = ab 2 is a possible candidate to fulfill Theorem 2. In particular we prove the following lemma. Lemma 2. Let 0 < b < a Z, a 50, a and b square-free and gcd(a, b) = 1, with ab 2 ±1 mod 9, then (a, b) P with P ={(46, 7), (46, 5), (46, 19), (46, 17), (46, 1), (4, 8), (4, 4), (4, 29), (4, 11), (4, 7), (4, 2), (7, 5), (7, 26), (7, 17), (7, 10), (7, 1), (4, 29), (4, 11), (4, 7), (1, 2), (1, 22), (1, 14), (1, 1), (1, 5), (22, 1), (22, 5), (19, 17), (19, 10), (19, 1), (1, 5), (10, 1), (7, 2)}. Moreover all solutions (Y, Z) Z Z to by az = ±9 with Z > 0 and (a, b) P are listed in table 1. Table 1. Solutions (Y, Z) to by az = ±9, with Z > 0. a b Y Z Y Z Proof. The first part of the lemma is clear. The second part of the lemma is due to a computation in PARI [1]. In particular we solved all Thue equations of the form Y ab 2 Z = (by ) ab 2 Z = 9b 2, with (a, b) P and only considered solutions (Y, Z) such that b Y. Indeed all solutions have this property. The computation took only a few seconds on a common work station. Lemma 2 tells us that the only candidate is d = = 28. From (5) we obtain e 2 = 1 and X = 1. Hence ξ = η = 0 and ζ = 1.

13 12 R.F. TICHY AND V. ZIEGLER Therefore ɛ = 1 ( ) and ɛ 2 = Since (1, θ 1 := 28, θ2 := 1 ( )) is a Z-basis of O 28, we have ɛ = θ 2 and ɛ 2 = θ 1. Moreover we have M:= {}}{ θ 1 = 1 ɛ. 1 0 θ 2 ɛ 2 Since det M = 1 also 1, ɛ, ɛ 2 is a Z-basis of O 28 and therefore O 28 is generated by its units. 7. The case Y = 1 We are left to check the case Y. Since Y we have to consider the cases Y = 1 and Y = 2. Because of the section above we may assume a 51. First we consider the case Y = 1. From () we get b az = ±9 or Z = b 9. Since a > b and a 51 we deduce Z = 1 and a = b 9. a If we substitute this in (5) we obtain 81b + 27b 2 + 2b 27e 2 27bX b 2 X + X = 0. If we put X = ξ + b and η = b we obtain 27e 2 + ξ + 81η 27ξ η + ξ 2 η = 0, hence ξ. If we put ξ = ξ we get the Diophantine equation (15) e 2 + ξ + η ξη + ξ 2 η = 0. If we solve (15) for η we obtain η = ξ e 2 ξ 2 ξ + = ξ 6 ξ e 2 ξ ( ) 2 8 = ξ L 5, ξ ( ) 1 + O ξ i.e if ξ 9 then η = ξ. But η = ξ yields 6ξ = 9 + e 2. Since ξ Z, this is a contradiction. So we compute for each ξ with 8 ξ 8 the quantity η. In the case of e 2 = 1 we find the solutions (ξ, η) = (1, 0), (2, 7), (4, 9) and in the case of e 2 = 1 we find (ξ, η) = (, 0), (1, 2), (, 9). Note that η = b > 0. None of these solutions yields a proper b.

14 UNITS GENERATING THE RING OF INTEGERS 1 8. The case Y = 2 Now we discuss the case Y = 2, this is 8b az = ±9 or Z = (8b 9)/a. Since 8b 9 is odd also Z must be odd. Since a 51 we also have Y Z > 0, hence Z = 1. Therefore a = 8b + 9e 1 with e 1 = ±1. If we put Y = 2, Z = 1 and a = 8b + 9e 1 into (5) we get 128b 27e b 2 e b 48b 2 X 54be 1 X + X = 0. If we use the transformation indicated by X = ξ + 4b and b = η, we get 27e 2 + ξ + 81η 54e 1 ξ η + 12ξ 2 η = 0. Note that ξ, hence we put ξ = ξ and obtain (16) e 2 + ξ + η 6e 1 ξη + 4ξ 2 η = 0. We solve (16) for η and obtain ξ e 2 η = 4ξ 2 6e 1 ξ + = ξ 4 e 1 8 8ξ + 8e 2 9e 1 2ξ 2 = ξ 4 e ( ) L 6 = 2ξ + e ( L 6 2ξ 8 2ξ ). ( ) 1 + O ξ We see that η cannot be an integer if ξ 6. So we compute for each ξ with 6 ξ 6 the quantity η. We find that the only integral solutions are (ξ, η) = (, 0), (6, 1) if e 1 = e 2 = 1, (ξ, η) = (, 0), (, 2) if e 1 = e 2 = 1, (ξ, η) = (, 0), (, 2) if e 1 = e 2 = 1, (ξ, η) = (, 0), ( 6, 1) if e 1 = e 2 = 1. So we are reduced to b = 2 and e 1 = 1 or b = 1 and e 1 = 1. Hence a = 7 or a = 1. Thus the only proper pair is (a, b) = (7, 2), which has been found above. Acknowledgement We are grateful to W. Narkiewicz for drawing our attention to this kind of problems.

15 14 R.F. TICHY AND V. ZIEGLER References [1] N. Ashrafi and P. Vámos. On the unit sum number of some rings. The Quarterly Journal of Mathematics, 56(1):1 12, [2] P. Belcher. Integers expressible as sums of distinct units. Bull. Lond. Math. Soc., 6:66 68, [] P. Belcher. A test for integers being sums of distinct units applied to cubic fields. J. Lond. Math. Soc., II. Ser., 12: , [4] H. Cohen. A course in computational algebraic number theory, volume 18 of Graduate Texts in Mathematics. Springer, Berlin, 199. [5] B. Delaunay. Vollständige Lösung der unbestimmten Gleichung X q + Y = 1 in ganzen Zahlen. Mathematische Zeitschrift, 28:1 9, [6] B. Goldsmith, S. Pabst, and A. Scott. Unit sum numbers of rings and modules. Q. J. Math., Oxf. II. Ser., 49(195):1 44, [7] C. Heuberger, A. Pethő, and R. F. Tichy. Thomas family of Thue equations over imaginary quadratic fields. J. Symbolic Comput., 4(5):47 449, [8] B. Jacobson. Sums of distinct divisors and sums of distinct units. Proc. Am. Math. Soc., 15:179 18, [9] M. Jarden and N. W ladys law. On sums of units. page 10, [10] T. Nagell. Solution complète de quelques équations cubiques à deux indéterminées. J. Math. Pures et Appl., pages , [11] C. L. Siegel. Die Gleichung ax n by n = c. Math. Ann., 114:57 68, 197. [12] J. Śliwa. Sums of distinct units. Bull. Acad. Pol. Sci., 22:11 1, [1] The PARI Group, Bordeaux. PARI/GP, version 2.1.5, available from [14] P. Vámos. 2-good rings. The Quarterly Journal of Mathematics, 56():417 40, [15] D. Zelinsky. Every linear transformation is a sum of nonsingular ones. Proc. Am. Math. Soc., 5:627 60, R.F. Tichy Institute of Computational Number Theory and Analysis, Graz University of Technology Steyrergasse 1, A-8010 Graz, Austria address: tichy@tugraz.at V. Ziegler Institute of Mathematics, University of Natural Resources and Applied Life Sciences, Vienna Gregor-Mendelstr. 1, A-1180 Vienna, Austria address: ziegler@finanz.math.tugraz.at