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I started reading Silverman and Tate’s introductory book on elliptic curves. In the introductory chapter they mention that for the Bachet equation $x^2 - y^3 = c$, there are infinitely many rational s …

So one of the approaches to proving the equality in the question is via the following three steps: First differentiate both sides of the equation to see that they agree up to a constant. This reduce …

I ran into a "well-known identity" on page 345 of Shepp and Lloyd's On ordered cycle lengths in a random permutation: $$\int_x^{\infty} \frac{\exp(-y)}y dy = \int_0^x \frac{1-\exp(-y)}y dy - \log x - …

What about the AKS primality test? Years ago Avi Wigderson gave a talk at the Newton Institute of Mathematical Sciences at Cambridge and he related the result to his general method of converting a har …

I am looking for any remotely related reference for the following problem, for which I have not the least clue what techniques would be useful. Consider a discrete probability space $\Omega = \{x, y, …

This has been solved in a joint paper with a colleague that is forthcoming. The key is to consider an interpolating probability measure $\rho$ of $\mu$ and $\nu$ defined by $\rho_i = \mu_i \nu_i / \su …

Markov chains on symmetric groups converging to various distributions other than uniform provide a fertile ground for the marriage between modern combinatorics such as Macdonald polynomials and hard a …

For $k :=|\lambda| \ge r$, the statement that all $s_\lambda(x_1, \ldots, x_r)$ vanish is equivalent to all the elementary polynomials $e_j(x_1, \ldots, x_r) := \sum_{i_1 < \ldots < i_j} x_{i_1} \ldot …