On the diophantine equation x(x + 1)(x + 2)…(x + (m − 1)) =g(y)

Let g(y) ? Q[Y] be an irreducible polynomial of degree n ≥ 3. We prove that there are only finitely many rational numbers x, y with bounded denominator and an integer m ≥ 3 satisfying the equation x(x + 1) (x + 2)…(x + (m − 1) ) = g(y). We also obtain certain finiteness results when g(y) is not an irreducible polynomial.     

The central configurations of four masses x, −x, y, −y

The configuration of a homothetic motion in the N-body problem is called a central configuration. In this paper, we prove that there are exactly three planar non-collinear central configurations for masses x, −x, y, −y with x≠y (a parallelogram and two trapezoids) and two planar non-collinear central configurations for masses x, −x, x, −x (two diamonds). Except the case studied here, the only known case where the four-body central configurations with non-vanishing masses can be listed is the case with equal masses (A. Albouy, 1995-1996), which requires the use of a symbolic computation program. Thanks to a lemma used in the proof of our result, we also show that a co-circular four-body central configuration has non-vanishing total mass or vanishing multiplier.     

On the Diophantine Equation x+q=y

In this paper it has been proved that if q is an odd prime, q?7 (mod 8), n is an odd integer ?5, n is not a multiple of 3 and (h,n)=1, where h is the class number of the filed Q(√−q), then the diophantine equation x²+q^2k+1=yⁿ has exactly two families of solutions (q,n,k,x,y).     

On the diophantine equation y2 − D = 2k

The equation of the title is studied for 1 ≤ D ≤ 100. It is shown that for such values of D the above equation is really interesting only if D = 17, 41, 73, 89, 97. Then, for these values of D, (i) necessary conditions are given for the solvability of the diophantine equations y² = 2x⁴ + D and y² = 8x⁴ + D, and (ii) y² ? D = 2^k is solved.     

On elliptic curves y=x−nx with rank zero

In this paper we determine all elliptic curves E_n:y²=x³−n²x with the smallest 2-Selmer groups S_n=Sel₂(E_n(Q))={1} and S_n′=Sel₂(E_n′(Q))={±1,±n}(E_n′:y²=x³+4n²x) based on the 2-descent method. The values of n for such curves E_n are described in terms of graph-theory language. It is well known that the rank of the group E_n(Q) for such curves E_n is zero, the order of its Tate-Shafarevich group is odd, and such integers n are non-congruent numbers.     

On the number of solutions of Goormaghtigh equation for given x and y

In this note, we extend a result obtained by Bugeaud and Shorey in Pacific J. Math.207 (2002) 61-75. In fact, we show that the Goormaghtigh equation

     

The differential equation x = f ⊥ x

A theory is presented for absolutely continuous solutions of the general scalar first order autonomous o.d.e. Necessary and sufficient conditions are given for local and global existence and uniqueness, and further topics include existence-uniqueness duality, structure of the solution set and weak solutions.     

Regions of absolute stability of explicit Runge-Kutta-Nyström methods for y″ = f(x, y, y′)

We examine regions of absolute stability of s-stage explicit Runge-Kutta-Nyström (R-K-N) methods of order s for s = 2, 3, 4 for y″ = f(x, y, y′) by applying these methods to the test equation: y″ + 2αy′ + β²y = O, α, β ? 0, α + β > 0. We also examine the regions of absolute stability of Runge-Kutta (R-K) methods for first order differential equations of respective orders. Interestingly, it turns out that regions of absolute stability of R-K methods and R-K-N methods of respective orders for which the asymptotic relative error does not deteriorate are identical. Our present investigations are in continuation of the recent results of Chawla and Sharma [1].     

Solutions of the equation zez = a(z + b)

The theory of complex variables is used to develop exact closed-form solutions for the, in general, complex zeros of the exponential polynomial F(z) = z exp z ? a(z + b), a complex and b real. The established zeros are related to canonical solutions of suitably posed Riemann problems and are expressed ultimately in terms of elementary quadratures.