Kyle Pratt

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20242026

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5 papers · 1 filter

math.NT2026

Utilizing Smoothing Techniques to Bound $∣ I^¶ (1 + i t) ∣$

Andrew Christensen, Kyle Pratt

We demonstrate an improved explicit upper bound of $∣ I^¶ (1 + i t) ∣$ for $3 \leq t \leq 1 0^{9}$ using smoothing techniques. Our method sharpens previous bounds relying on the Riemann--Sie…

math.NT2026

Binomial coefficients with divisors avoiding an interval

Hung M. Bui, Slava Naprienko, Kyle Pratt +1

We solve a fifty-year-old conjecture of ErdÅs and Graham concerning whether the binomial coefficient $(k n)$ with $1 \leq k \leq \frac{n}{2}$ must always have a divisor $\…

math.NT2025

Cubes from products of terms in progression with one term missing

Kyle Pratt

Let $5 \leq k \leq 11$ and $0 \leq i \leq k - 1$ be integers. We determine all solutions to the equation \begin{align*} n(n+d)(n+2d)\cdots(n+(i-1)d)(n+(i+1)d) \cdots (n+(k-1)d) = y^3…

math.NT2025

On the Lebesgue-Nagell equation $x^{2} - 2 = y^{p}$

Ethan Katz, Kyle Pratt

We investigate the Lebesgue--Nagell equation \begin{align*} x^2-2=y^p \end{align*} in integers $x, y, p$ with $p \geq 3$ an odd prime. A longstanding folklore conjecture asserts that…

math.NT2024

Nontrivial rational points on ErdÅs-Selfridge curves

Kyle Pratt

We study rational points on the ErdÅs-Selfridge curves \begin{align*} y^\ell = x(x+1)\cdots (x+k-1), \end{align*} where $k, ℓ \geq 2$ are integers. These curves contain "trivial"…

Utilizing Smoothing Techniques to Bound ∣I^¶(1+it)∣

Binomial coefficients with divisors avoiding an interval

Cubes from products of terms in progression with one term missing

On the Lebesgue-Nagell equation x2−2=yp

Nontrivial rational points on ErdÅs-Selfridge curves

Utilizing Smoothing Techniques to Bound $∣ I^¶ (1 + i t) ∣$

On the Lebesgue-Nagell equation $x^{2} - 2 = y^{p}$

Nontrivial rational points on ErdÅs-Selfridge curves