Diophantine equations with products of consecutive members of binary recurrences

We prove a finiteness result for the number of solutions of a Diophantine equation of the form \(u_n u_{n+1}\cdots u_{n+k}\pm 1 =\pm u_m^2\), where \(\{ u_n\}_{n\ge 1}\) is a binary recurrent sequence whose characteristic equation has roots which are real quadratic units.     

On the diophantine equation x 2 + p 2k = y n

Let 2 ≤ p < 100 be a rational prime and consider equation (3) in the title in integer unknowns x, y, n, k with x > 0, y > 1, n ≥ 3 prime, k ≥ 0 and gcd(x, y) = 1. Under the above conditions we give all solutions of the title equation (see the Theorem). We note that if in (3) gcd(x, y) = 1, our Theorem is an extension of several earlier results [15], [27], [2], [3], [5], [23]. Received: 25 April 2008     

On the largest prime factor of numerators of Bernoulli numbers

We prove that for almost all $n$, the numerator of the Bernoulli number $B_{2n}$ is divisible by a large prime.     

Finiteness results for F-Diophantine sets

Monatshefte für Mathematik - Diophantine sets, i.e. sets of positive integers A with the property that the product of any two distinct elements of A increased by 1 is a perfect square, have a...     

A one-way function based on norm form equations

In this paper we present a new one-way function with collision resistance. The security of this function is based on the difficulty of solving a norm form equation. We prove that this function is collision resistant, so it can be used as a one-way hash function. We show that this construction probably provides a family of one-way functions. This revised version was published online in August 2006 with corrections to the Cover Date.     

Computational experiences on the distances of polynomials to irreducible polynomials

In this paper we deal with a problem of Turán concerning the `distance' of polynomials to irreducible polynomials. Using computational methods we prove that for any monic polynomial of degree there exists a monic polynomial with deg() = deg() such that is irreducible over and the `distance' of and is .

     

An application of index forms in cryptography

We investigate the possibility of using index forms as basic ingredients of cryptographically important functions. We suggest the use of a hash function based on index forms and we prove some important properties of the suggested function. The research was supported in part by the Hungarian Academy of Sciences, by grants T048791 and K67580 of the Hungarian NFSR, by the National Office for Research and Technology and by the grant JP-26/2006.     

On the number of solutions of norm form equations

In 1985 Evertse and Gyory [5] gave explicit upper bounds for the number of solutions of norm form equations of the form (1.1) under the hypotheses that (i) x m &nequiv; 0, &alpha; 1 = 1, &alpha; 2 , ... ,&alpha; m-1 are Q-linearly independent and has degree at least 3 overQ( &alpha; 1 ..., &alpha; m-1 ), or that (ii) the degree of i is at least 3 over Q(&alpha; 1 , ..., &alpha; i-1 ) for i = 2, m. Later Gyry [9], Evertse [3] and Evertse and Gy}ory [6] derived general upper bounds for arbitrary norm form equations which include the case (ii), but not the case (i). In the present paper we considerably improve the bounds of [5], and we give a further improvement which is valid for all but at most finitely many possible values of the constant term b of the equation. Our bound obtained under the assumption (ii) is better for almost all b than the general bounds of [9], [3] and [6].