Shisheng Li

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math.NT2026

Coprime pairs with equal sum of divisors, assuming smooth shifted primes

Shisheng Li, Zhong Zhuang

Let $h (x)$ be the number of pairs $1 \leq a < b < x$ with $g cd (a, b) = 1$ and $σ (a) = σ (b)$ . Erdős Problem #824 asks whether $h (x) > x^{2 - o (1)}$ . By an argument of Pollack and Pomerance, this…

math.NT2026

Unit fractions with semiprime denominators: an elementary proof of Erdős Problem #306

Shisheng Li

We give an elementary proof that every positive rational number $a / b$ with $b$ squarefree is a finite sum of distinct unit fractions $1/ n$ , where each $n$ is a product of two disti…

math.NT2026

Every natural number is a sum of distinct semiprime unit fractions

Shisheng Li

We prove that every natural number is a finite sum of distinct unit fractions whose denominators are semiprimes (products of two distinct primes). This is the $ω = 2$ integer case of…