Lucas' square pyramid problem revisited
Markoff numbers and ambiguous classes
The Markoff conjecture states that given a positive integer , there is at most one triple of positive integers with that satisfies the equation . The conjecture is known to be true when is a prime power or two times a prime power. We present an elementary proof of this result. We also show that if in the class group of forms of discriminant , every ambiguous form in the principal genus corresponds to a divisor of , then the conjecture is true. As a result, we obtain criteria in terms of...
Matrix theoretic interpretation of the classical Markoff theory. (L'interprétation matricielle de la théorie de Markoff classique.)
Mock Heegner Points and Congruent Numbers.
Mordell-Weil ranks of families of elliptic curves associated to Pythagorean triples
We study the family of elliptic curves y² = x(x-a²)(x-b²) parametrized by Pythagorean triples (a,b,c). We prove that for a generic triple the lower bound of the rank of the Mordell-Weil group over ℚ is 1, and for some explicitly given infinite family the rank is 2. To each family we attach an elliptic surface fibered over the projective line. We show that the lower bounds for the rank are optimal, in the sense that for each generic fiber of such an elliptic surface its corresponding Mordell-Weil...
More cubic surfaces violating the Hasse principle
We generalize L. J. Mordell’s construction of cubic surfaces for which the Hasse principle fails.
Několik poznámek k problému W. Mnicha
Neuer Beweis eines Satzes von Gerono.
New upper bounds for taxicab and cabtaxi numbers.
Non-congruent numbers, odd graphs and the Birch-Swinnerton-Dyer conjecture
Note on a Diophantine Equation.
Note on Arithmetical Progressions with Equal Products of Five Terms.
Note on J. H. E. Cohn's paper "The Diophantine equation xⁿ = Dy² + 1"
Note sur quelques équations indéterminées
Number of solutions of cubic Thue inequalities with positive discriminant
Let F(X,Y) be an irreducible binary cubic form with integer coefficients and positive discriminant D. Let k be a positive integer satisfying . We give improved upper bounds for the number of primitive solutions of the Thue inequality .
O jednej sústave kongruencií. Poznámka k predchádzajúcemu článku J. Sedláčka.
O trojúhelnících racionálních
On a characterization of Shimura's elliptic curve over ℚ(√37)
On a class of Diophantine equations.
On a Diophantine Equation.