Abstract. We find the primitive integer solutions to x 2 + y 3 = z 7. A nonabelian descent argument involving the simple group of order 168 reduces the problem to the determination of the set of rational points on a finite set of twists of the Klein quartic curve X. To restrict the set of relevant twists, we exploit the isomorphism between X and the modular curve X(7), and use modularity of elliptic curves and level lowering. This leaves 10 genus-3 curves, whose rational points are found by a combination of methods. 1.

We propose a new algorithm to find worst cases for correct rounding of an analytic function. We first reduce this problem to the real small value problem — i.e. for polynomials with real coefficients. Then we show that this second problem can be solved efficiently, by extending Coppersmith’s work on the integer small value problem — for polynomials with integer coefficients — using lattice reduction [4, 5, 6]. For floating-point numbers with a mantissa less than, and a polynomial approximation of ¡ degree, our al-gorithm finds all worst cases ¢ at distance a machine number �� � § ¥�©������� � in time ¡��¤ �. For, this improves �� � �� � � on the complexity from Lefèvre’s algorithm

Abstract—We propose a new algorithm to find worst cases for the correct rounding of a mathematical function of one variable. We first reduce this problem to the real small value problem—i.e., for polynomials with real coefficients. Then, we show that this second problem can be solved efficiently by extending Coppersmith’s work on the integer small value problem—for polynomials with integer coefficients—using lattice reduction. For floating-point numbers with a mantissa less than N and a polynomial approximation of degree d, our algorithm finds all worst cases at distance less than N d2 2dþ1 from a machine number in time OðN dþ1 2dþ1þ " Þ. For d 2, a detailed study improves on the OðN 2=3þ " Þ complexity from Lefèvre’s algorithm to OðN 4=7þ " Þ. For larger d, our algorithm can be used to check that there exist no worst cases at distance less than N k in time OðN 1=2þ " Þ. Index Terms—Computer arithmetic, multiple precision arithmetic, special function approximations. æ 1

Cut the unit circle S¹ = R/Z at the points = x mod 1, and let J 1 , . . . , JN denote the complementary intervals, or gaps, that remain. We show that, in contrast to the case of random points (whose gaps are exponentially distributed), the lengths |/N are governed by an explicit piecewise real-analytic distribution F (t) dt with phase transitions at t = 1/2 and t = 2. The gap distribution is related to the probability p(t) that a random unimodular lattice translate # meets a fixed triangle S t of area t; in fact p ## (t) = (t). The proof uses ergodic theory on the universal elliptic curve E = (SL 2 (R) # R )/(SL 2 (Z) # Z and Ratner's theorem on unipotent invariant measures.

Abstract. We provide algorithms to count and enumerate representatives of the (right) ideal classes of an Eichler order in a quaternion algebra defined over a number field. We analyze the run time of these algorithms and consider several related problems, including the computation of two-sided ideal classes, isomorphism classes of orders, connecting ideals for orders, and ideal principalization. We conclude by giving the complete list of definite Eichler orders with class number at most 2. Key words. quaternion algebras, maximal orders, ideal classes, number theory AMS subject classifications. 11R52 Since the very first calculations of Gauss for imaginary quadratic fields, the problem of computing the class group of a number field F has seen broad interest. Due to the evident close association between the class number and regulator (embodied in the Dirichlet class number formula), one often computes the class group and unit group in tandem as follows. Problem (ClassUnitGroup(ZF)). Given the ring of integers ZF of a number field F, compute the class group Cl ZF and unit group Z ∗ F.

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Abstract. A perfect power is a positive integer of the form ax where a ≥ 1 and x ≥ 2 are rational integers. Subbayya Sivasankaranarayana Pillai wrote several papers on these numbers. In 1936 and again in 1945 he suggested that for any given k ≥ 1, the number of positive integer solutions (a, b, x, y), with x ≥ 2 and y ≥ 2, to the Diophantine equation ax − by = k is finite. This conjecture amounts to saying that the distance between two consecutive elements in the sequence of perfect powers tends to infinity. After a short introduction to Pillai’s work on Diophantine questions, we quote some later developments and we discuss related open problems.

Abstract. Let k and n be positive integers. Define R(n, k) to be the minimum positive value of |ei s1 + e2 s2 + ···+ ek sk − t |, where s1,s2, ·· ·,sk are positive integers no larger than n, t is an integer and ei ∈{1, 0, −1} for all 1 ≤ i ≤ k. It is important in computational geometry to determine a good lower and upper bound of R(n, k). In this paper we show that this problem is closely related to the shortest vector problem in certain integral lattices and present an algorithm to find lower bounds based on lattice reduction algorithms. Although we can only prove an exponential time upper bound for the algorithm, it is efficient for large k when an exhaustive search for the minimum value is clearly infeasible. It produces lower bounds much better than the root separation technique does. Based on numerical data, we formulate a conjecture on the length of the shortest nonzero vector in the lattice, whose validation implies that our algorithm runs in polynomial time and the problem of comparing two sums of square roots of small integers can be solved in polynomial time. As a side result, we obtain constructive upper bounds for R(n, k) whennis much smaller than 22k. 1.

If k ≥ 2 is a positive integer the number of representations of a positive integer N as either x k 1 + x k 2 = N or x k 1 − x k 2 = N, with integers x1 and x2, is finite. Moreover it is easily shown to be Oε(N ε), for any ε> 0. It is known that if k = 2 or 3 then the number of representations is unbounded as N varies, but it

Abstract. We report on our search for solutions of the Diophantine equation x 3 + y 3 + z 3 = n for n<1000 and |x|, |y|, |z | < 10 14. 1.