The number of solutions of .
The number of solutions to cubic Thue inequalities
The number of solutions to the generalized Pillai equation .
We consider , the number of solutions to the equation in nonnegative integers and integers , for given integers , , , and . When , we show that except for a finite number of cases all of which satisfy for each solution; when , we show that except for three infinite families of exceptional cases. We find several different ways to generate an infinite number of cases giving solutions.
The number of squarefull numbers in an interval
The number of squares and sets
The Number of Steps in a Finite Jacobi Algorithm.
The number of subspaces occurring in the p-adic subspace theorem in diophantine approximation.
The number of topologies on a finite set.
The number of unsieved integers up to x
The number of zeroes of x3 + y3 + cz3 in certain finite fields.
The number of zeros of polynomials in valuation rings of complete discretely valued fields
The number Γ(k) in Waring's problem
The numbers of sums of quadratic residues and of non-residues respectively taken n at a time and congruent to any given integer to an odd prime modulus p.
The Number-System of Algebra [Book]
The number-theoretic content of the Jacobi triple product identity.
The number-theoretic work of Paul Turán
The Oppenheim series expansions and Hausdorff dimensions
The optimality of the Bounded Height Conjecture
In this article we show that the Bounded Height Conjecture is optimal in the sense that, if is an irreducible subvariety with empty deprived set in a power of an elliptic curve, then every open subset of does not have bounded height. The Bounded Height Conjecture is known to hold. We also present some examples and remarks.
The orders of the reductions of a point in the Mordell-Weil group of an elliptic curve
The Ostrogradsky series and related Cantor-like sets