arXiv subjects

Peter J. Campbell

Publications and source records attributed to Peter J. Campbell.

3 recordsLinked to original sources

arXiv2026

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 every integer $n>24$ that is not divisible by $997\#$, where $997\#=\prod_{p\leq 997}p$. In particular, any counterexample must be divisible by every prime up to $997$, and hence exceeds $10^{415}$. The proof uses an elementary congruence argument together with finite computation.

math.NT↗

arXiv2026

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 $Ω(ab)\leq33$, where $Ω(n)$ 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 Friedlander--Iwaniec $Λ^-Λ^2$ lower-bound sieve to a sequence derived from the products $n(N-n)$. The main new ingredient is pre-sieving at the prime $3$, which eliminates the extremal small-prime case in the dimension condition while keeping the resulting remainder terms under explicit control. We complete the proof using analytic estimates for large $N$, finite verification over an intermediate range, and explicit prime-gap data for small $N$.

math.NT↗

arXiv2026

On the Existence of Integers with at Most 3 Prime Factors Between Every Pair of Consecutive Squares

We prove an explicit analogue of Legendre's conjecture for almost primes. Namely, for every integer $n \geq 1$, the interval $(n^2,(n+1)^2)$ contains an integer having at most $3$ prime factors, counted with multiplicity. This improves the previous best result of Dudek and Johnston, who showed that every such interval contains an integer with at most $4$ prime factors. The proof is divided into two ranges. For $n^2 \leq 10^{31}$, we use prior computational results on primes in short intervals between consecutive squares, together with explicit bounds on maximal prime gaps. For $n^2 > 10^{31}$, we give a sieve-theoretic argument with explicit constants, adapting Richert's logarithmic weights to intervals between consecutive squares and employing an explicit linear sieve of Bordignon, Johnston, and Starichkova.

math.NT↗