On searching for solutions of the Diophantine equation

We propose an efficient search algorithm to solve the equation for a fixed value of 0$">. By parametrizing , this algorithm obtains and (if they exist) by solving a quadratic equation derived from divisors of . Thanks to the use of several efficient number-theoretic sieves, the new algorithm is much faster on average than previous straightforward algorithms. We performed a computer search for six values of below 1000 for which no solution had previously been found. We found three new integer solutions for and 931 in the range of .

     

Enumerating solutions to

Let be polynomials with integer coefficients. This paper presents a fast method, using very little temporary storage, to find all small integers satisfying . Numerical results include all small solutions to ; all small solutions to ; and the smallest positive integer that can be written in ways as a sum of two coprime cubes.

     

The distribution of solutions of the congruence

For a cube of size , we obtain a lower bound on so that is nonempty, where is the algebraic subset of defined by

a positive integer and an integer not divisible by . For we obtain that is nonempty if , for we obtain that is nonempty if , and for we obtain that is nonempty if . Using the assumption of the Grand Riemann Hypothesis we obtain is nonempty if .

     

On contractible

We present a very short proof of a well-known result, that for each there exists a contractible -dimensional compactum, non-embeddable into .

     

On the primality of

For each prime , let be the product of the primes less than or equal to . We have greatly extended the range for which the primality of and are known and have found two new primes of the first form ( ) and one of the second (). We supply heuristic estimates on the expected number of such primes and compare these estimates to the number actually found.

     

On the structure of

We show that for with its geometrically induced structure maps is not an Hopf algebroid because neither the augmentation nor the coproduct are multiplicative. As a consequence the algebra structure of is slightly different from what was supposed to be the case. We give formulas for and and show that the inversion of the formal group of is induced by an antimultiplicative involution . Some consequences for multiplicative and antimultiplicative automorphisms of for are also discussed.

     

On the Diophantine equation

Let be an odd prime. In this paper, using some theorems of Adachi and the author, we prove that if and , then the equation , and the equation , have no integral solutions respectively. Here is th Bernoulli number.

     

On searching for solutions of the Diophantine equation

We propose a new search algorithm to solve the equation for a fixed value of . By parametrizing min, this algorithm obtains and (if they exist) by solving a quadratic equation derived from divisors of . By using several efficient number-theoretic sieves, the new algorithm is much faster on average than previous straightforward algorithms. We performed a computer search for 51 values of below 1000 (except ) for which no solution has previously been found. We found eight new integer solutions for and in the range of .

     

On the stability of the

We prove the stability in of the projection onto a family of finite element spaces of conforming piecewise linear functions satisfying certain local mesh conditions. We give explicit formulae to check these conditions for a given finite element mesh in any number of spatial dimensions. In particular, stability of the projection in holds for locally quasiuniform geometrically refined meshes as long as the volume of neighboring elements does not change too drastically.

     

On the diophantine equation

In this note, we find all positive integer solutions of the diophantine equation from the title with a prime power.

     

Efficient computation of the extreme solutions of

We propose a new quadratically convergent algorithm, having a low computational cost per step and good numerical stability properties, which allows the simultaneous approximation of the extreme solutions of the matrix equations and . The algorithm is based on the cyclic reduction method.

     

On and discriminants

We modify the -conjecture for number fields in order to make the support (like the height) well-behaved under field extensions. We show further that the exponent 1$"> of the absolute value of the discriminant cannot be replaced by , and even that an arbitrarily large power of must be present.

     

The smallest solutions to the diophantine equation

In this paper we discuss a method used to find the smallest nontrivial positive integer solutions to . The method, which is an improvement over a simple brute force approach, can be applied to search the solution to similar equations involving sixth, eighth and tenth powers.

     

On the classes of spaces

The two better-known ways of understanding the notion of local unconditional structure allow us to define successive extensions of the well-known class of the spaces of Lindenstrauss and Pelczynski. This paper also studies stability properties of these classes under ultrapowers, biduals and complemented subspaces.

     

The minimal number of solutions to

In 1958, A. Schinzel showed that for each fixed there are at least two solutions to . Using the same method and a computer search, Schinzel and A. Wakulicz extended the bound to all . Here we show that Schinzel's method can be used to further extend the bound when is even, but not when is odd.

     

On integers of the form

In this paper we show that the set of positive odd integers such that has at least three distinct prime factors for all positive integers has positive lower asymptotic density.

     

On the comparison of the spaces

The notion of -variation and the space arise in the study of regularity properties of solutions to perturbed conservation laws. In this article we show that this notion is equivalent to variation in the regular sense, and therefore the space is the same as the space in the sense of Cesari-Tonelli. We also point out some connection between the space and the Favard classes for translation semigroups.

     

On the greatest prime factor of

We prove that for integers b>c>0$">, the greatest prime factor of tends to infinity with . In particular, this settles a conjecture raised by Györy, Sarkozy and Stewart, predicting the same conclusion for the product .

     

On the asphericity of a symplectic

C. H. Taubes asked whether a closed (i.e. compact and without boundary) connected oriented three-dimensional manifold whose product with a circle admits a symplectic structure must fiber over a circle. An affirmative answer to Taubes' question would imply that any such manifold either is diffeomorphic to the product of a two-sphere with a circle or is irreducible and aspherical. In this paper, we prove that this implication holds up to connect sum with a manifold which admits no proper covering spaces with finite index. It is pointed out that Thurston's geometrization conjecture and known results in the theory of three-dimensional manifolds imply that such a manifold is a three-dimensional sphere. Hence, modulo the present conjectural picture of three-dimensional manifolds, we have shown that the stated consequence of an affirmative answer to Taubes' question holds.

     

On peak-interpolation manifolds for

Let be a bounded, weakly convex domain in , , having real-analytic boundary. is the algebra of all functions holomorphic in and continuous up to the boundary. A submanifold is said to be complex-tangential if lies in the maximal complex subspace of for each . We show that for real-analytic submanifolds , if is complex-tangential, then every compact subset of is a peak-interpolation set for .