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We investigate the integer solutions of Diophantine equations related to perfect numbers. These solutions generalize the example, found by Descartes in 1638, of an odd, ``spoof perfect factorization $3^2cdot 7^2cdot 11^2cdot 13^2cdot 22021^1$. More recently, Voight found the spoof perfect factorization $3^4cdot 7^2cdot 11^2cdot 19^2cdot(-127)^1$. No other examples appear in the literature. We compute all nontrivial, odd, primitive spoof perfect factorizations with fewer than seven bases -- there are twenty-one in total. We show that the structure of odd, spoof perfect factorizations is extremely rich, and there are multiple infinite families of them. This implies that certain approaches to the odd perfect number problem that use only the multiplicative nature of the sum-of-divisors function are unworkable. On the other hand, we prove that there are only finitely many nontrivial, odd, primitive spoof perfect factorizations with a fixed number of bases.
Let $f(n)$ and $g(n)$ be the number of unordered and ordered factorizations of $n$ into integers larger than one. Let $F(n)$ and $G(n)$ have the additional restriction that the factors are coprime. We establish the asymptotic bounds for the sums of $
We classify the dual strongly perfect lattices in dimension 16. There are four pairs of such lattices, the famous Barnes-Wall lattice $Lambda _{16}$, the extremal 5-modular lattice $N_{16}$, the odd Barnes-Wall lattice $O_{16}$ and its dual, and one
The concept of a covering system was first introduced by ErdH{o}s in 1950. Since their introduction, a lot of the research regarding covering systems has focused on the existence of covering systems with certain restrictions on the moduli. Arguably,
M. B. Levin used Sobol-Faure low discrepancy sequences with Pascal matrices modulo $2$ to construct, for each integer $b$, a real number $x$ such that the first $N$ terms of the sequence $(b^n x mod 1)_{ngeq 1}$ have discrepancy $O((log N)^2/N)$. Thi
Some new results concerning the equation $sigma(N)=aM, sigma(M)=bN$ are proved. As a corollary, there are only finitely many odd superperfect numbers with a fixed number of distinct prime factors.