Elementary Proof of the Completeness of OEIS A051221 Below 2000
Seiichi Azuma
TL;DR
The paper addresses whether the OEIS sequence A051221, consisting of numbers of the form $10^x - y^2$, is complete up to 2000. It reduces the problem to Pell-type equations by considering odd $x=2u+1$ and expressing solutions via $s+t\sqrt{10}=(a+b\sqrt{10})(19+6\sqrt{10})^K$ with $10b^2-a^2=c$, using the unit $19+6\sqrt{10}$ in $\mathbb{Z}[\sqrt{10}]$ to bound possibilities. A key lemma and modular obstructions on the associated $t_k$ sequences show that $t_k$ cannot equal $\pm 10^u$ for any $u$, reducing the verification to finite checks; a complete computation across all $c\in[0,2000]$ (except a few cases) is carried out, with a refined modulus $p=1601$ used to finish the remaining cases. The authors provide a Google Colab notebook to ensure reproducibility, conclusively establishing that no new values arise for $x\ge 9$ within the interval, thereby confirming the list is complete up to 2000.
Abstract
The OEIS sequence A051221 consists of nonnegative integers of the form 10^x - y^2. The known values are those less than or equal to 2000 with x <= 7, and it is conjectured that no new values in this range appear for x >= 8. In this paper, we give an elementary proof that this is indeed the case by using Pell-type equations.