A positive $D(n)$-$m$-tuple is a set $A=\{a_1,\ldots,a_m\}$ of distinct positive integers such that $a_i a_j+n$ is a square for every $i\ne j$. In 2005, Dujella and Luca obtained an absolute bound for the cardinality of a $D(p)$- or $D(-p)$-tuple of positive integers, uniformly in the prime $p$. We extend the underlying gap argument to prime powers. An explicit toric elimination certificate, proved by elementary linear algebra, replaces the unsaturated homogeneous elimination step and is valid modulo every prime power. Explicit degree and height bounds for this eliminant yield a uniform gap principle. Combined with the general bound for bounded $|n|$, this shows that every positive reduced $D(\pm p^r)$-tuple (i.e. tuple with elements not divisible by $p$) has less than $2^{121}$ elements, independently of $p$ and $r$. Thus, every positive $D(\pm p)$-tuple has at most $2^{121}$ elements, positive $D(\pm p^2)$-tuples have less than $2^{122}$ elements, and the maximal cardinality of a positive $D(\pm p^r)$-tuple is $O(r)$ uniformly in $p$.
A $m$-tuple with the property $D(n)$ is a tuple of $m$ positive integers $(a_1, a_2, \dots, a_m)$ such that $a_i a_j + n$ is an square, for $1 \le i<j \le m$. The $k$th triangular number is $T_k = \frac{k(k+1)}{2}$ for nonnegative integers $k$. We consider $D(1)$ tuples consisting only of triangular numbers. We prove t...
Let $b_1,\dots,b_d\ge2$ be fixed integers, and let $k$ be their multiplicative rank. We study representations $n=a+b_1^{u_1}\cdots b_d^{u_d}$, where $a$ belongs to a set $\mathcal{A}$ of positive integers and $u_1,\dots,u_d$ are positive integers, counting distinct tuples of exponents separately. Under density and corr...
For a positive integer $\alpha$, let $p_\alpha(n)$ denote the number of $\alpha$-colored partitions of $n$. The Rademacher-type expansion of Iskander, Jain, and Talvola is valid for every real $\alpha>0$, but when $\alpha>24$ it involves several polar terms and a two-parameter family of exponential sums $A_k^{(\alpha)}...
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...
Sylvester's denumerant $d(t; \boldsymbol{A})$ counts the number of nonnegative integer solutions to $\sum_{i=1}^{N} a_i x_i = t$, where $\boldsymbol{A} = (a_1, \dots, a_N)$ is a sequence of positive integers with $\gcd(\boldsymbol{A}) = 1$. In 2025, Xin and Zhang gave a polynomial time algorithm in $N$ for computing $d...
Let $p_1,\dots,p_r\geq 5$ be distinct primes, let $P=p_1\cdots p_r$, and put $M=6P$. For a positive integer $n$, let $g_P(2n)$ denote the number of unordered representations $2n=h+k$, with $1\leq h\leq k$, such that $\gcd(h,M)=\gcd(k,M)=1$. Using the canonical remainder operator $\delta_q(x)=x-q\lfloor x/q\rfloor$, we...
Andres M. Salazar· 0 citations
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