Extremal problems for GCDs

高维情况下GCD和LCM的极值问题

Extremal Problems for GCDs and LCMs in Higher Dimensions

本文研究高维整数元组在大GCD和小LCM条件下的极值问题，推导出GCD和LCM问题的最优上界，并证明这些界在ε损失下是最佳的。

受限二项式最大公约数的尖锐 $p$-adic 极值与最小极值行

Sharp $p$-adic Extrema and Least Extremal Rows for Restricted Binomial GCDs

本文证明受限二项式GCD的p-adic极值恰为r_p(m)，并构造达到该值的行，对m=p^a+1精确给出最小极值行p^{3a}+1，结合Kummer定理与显式见证。

限制二项式系数在模-1素数处的最大公因数

Restricted Binomial GCDs at Primes Congruent to -1

本文证明了限制二项式系数最大公因数在素数 p≡-1 mod m 处的完整 p-进赋值公式，利用 Kummer 定理和带符号零和分类，并在 Lean 4 中形式化验证。

关于不相交剩余类的大最大公因数问题

On the problem of large gcd for disjoint residue classes

研究\(k\)个两两不相交剩余类，通过使用边由模的最大公因数染色的完全图等方法，证明了\(\max_{1\leq i<j\leq k}\gcd(m_i,m_j)\)与\(k\)的一个下界关系。

A bound on the multiplicative energy of a sum set and extremal sum-product problems

In recent years some near-optimal estimates have been established for certain sum-product type estimates. This paper gives some first extremal results which provide information about when these bounds may or may not be tight. The main tool is a new result which provides a nontrivial upper bound on the multiplicative energy of a sum set or difference set.

Maximum GCD Among Pairs of Random Integers

Fix $α>0$, and sample $N$ integers uniformly at random from $\{1,2,\ldots ,\lfloor e^{αN}\rfloor \}$. Given $η>0$, the probability that the maximum of the pairwise GCDs lies between $N^{2-η}$ and $N^{2+η}$ converges to 1 as $N\to \infty $. More precise estimates are obtained. This is a Birthday Problem: two of the random integers are likely to share some prime factor of order $N^2/\log [N]$. The proof generalizes to any arithmetical semigroup where a suitable form of the Prime Number Theorem is valid.