Shisheng

Coding / Math / AI enthusiast.

Recent work

• 2026-09-28 A negative answer to Erdős Problem #786
  arXiv:2609.37471. Shows that an admissible set (distinct products a₁···a_r = b₁···b_s force r = s) in {1, …, N} has ∑ 1/a ≤ ½ log N + (log log N + 2)² and size < (1−η)N for an absolute η > 0, answering both of Erdős's questions negatively; elementary proofs, formally verified in Lean 4 against the Formal Conjectures statements.
• 2026-09-26 Unit fractions with semiprime denominators: an elementary proof of Erdős Problem #306
  arXiv:2609.32140. An elementary proof that every positive rational a/b with b squarefree is a finite sum of distinct unit fractions whose denominators are products of two distinct primes; the only inputs about primes are Chebyshev-type bounds, and the proof is formalised in Lean 4 (apart from a cited inequality of Ramanujan).
• 2026-09-19 Local representatives and shortest completions for next-to-shortest paths in directed graphs
  arXiv:2609.22940. Gives an O(n³(m + n log n)) algorithm for the next-to-shortest s–t path problem on positively weighted digraphs (O(n³m) unweighted), improving the O(n⁴m³ log n) bound of Chen, Wein and Zhang; the key is that only O(n³) “local representative” prefixes need to be kept, via an uncrossing lemma.
• 2026-08-18 Asymptotics of the Tchoukaillon array and a conjecture of Beluhov
  arXiv:2608.17517. Proves N. Beluhov's conjectured asymptotic T_i,j = (πi+2j+2)²/(4π) + O((i+j+1)^4/3) for the general entry of the Tchoukaillon array arising from one-row Mancala solitaire.
• 2026-06-13 Every natural number is a sum of distinct semiprime unit fractions
  arXiv:2606.15159. Proves every natural number is a finite sum of distinct unit fractions with semiprime denominators — the ω = 2 (and hardest) case of a problem of Erdős and Graham.
• 2026-05-11 SB(3, n) has no Hamiltonian cycle when n is even
  arXiv:2605.09489. Resolves Knuth's TAOCP Pre-Fascicle 8A, Exercise 7.2.2.4–224: the digraph SB(3, n) has no Hamiltonian cycle for even n, via a sign-of-permutation obstruction. Included in TAOCP Pre-Fascicle 8A.
• 2026-05-07 Nonexistence of whirling-knight tours at half coil count for n ≡ 4, 6 (mod 8)
  arXiv:2605.04603. Settles a conjecture of Beluhov: no whirling knight's tours at coil count c = n/2 for n ≡ 4, 6 (mod 8), via Farkas dual certificates for a cycle-cover LP. Included in TAOCP Pre-Fascicle 8A.
• 2026-05-01 A note on the parameter ℓ in Buchbinder–Feldman's deterministic submodular matroid algorithm
  arXiv:2604.27362. Elementary tightenings shrink the hidden constant in Buchbinder–Feldman's Õ_ε(nr) deterministic (1−1/e−ε)-approximation; Lean 4 formalization at github.com/daizisheng/bf24-note, listed on Moran Feldman's publications page.