A prime x for which x² + x + 1 is square-full
2026-09-29
Research publication
A prime x for which x² + x + 1 is square-full
Preprint
- Original date
- Authors
- Yair Lavi
- Number theory
- Primes
- Square-full numbers
About this work
In Question 9 of On odd perfect numbers with exactly one even exponent greater than 2, Ochem and Zelinsky ask whether there exists a prime \(x\) such that \({x^2+x+1}\) is square-full. We answer the question in the affirmative. The number
\[ x = 47116128896261596331524418282573 \]
is prime, and \({x^2+x+1=7^3a^2}\) with
\[ a = 2544031832644333879196802002161. \]
The note also gives a short, self-contained proof that \(x\) is prime, based on Pocklington’s criterion. Of course, any modern primality-proving algorithm confirms this just as well.
Supplementary material
A Python script checks every computation in the note, including the primality certificate. It needs only Python 3 and runs in under a second.