Author

Peter J. Campbell

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Preprint Sep 2026

Prime Plus a Non-Square-Free Integer: An Elementary Approach

Lee and O'Clarey recently conjectured that every integer $n>24$ can be written as the sum of a prime and a positive integer that is not square-free. They proved this for odd $n$, and for all $n>24$ assuming the generalised Riemann hypothesis for Dirichlet $L$-functions. We prove their conjecture unconditionally for eve...

Peter J. Campbell · 0 citations

Preprint Aug 2026

A Sharper Explicit Result for the Sum of Two Almost Primes

We prove that for every integer $N\geq2$, there exist positive integers $a$ and $b$ such that $N=a+b$ and $\Omega(ab)\leq33$, where $\Omega(n)$ denotes the number of prime factors of $n$, counted with multiplicity. This improves the previous bound of $40$ obtained by Dudek and Dunn. The proof applies the explicit Fried...

Peter J. Campbell · 0 citations