https://jasper-aa64.github.io/categories/number-theory/Recent content in Number Theory on JasperHugoen-usTue, 28 Apr 2026 00:00:00 +0800https://jasper-aa64.github.io/posts/number-theory-0-common-formulae/Tue, 28 Apr 2026 00:00:00 +0800https://jasper-aa64.github.io/posts/number-theory-0-common-formulae/Legendre’s formula, divisor and multiple convolutions in O(n log n), offline-difference batch processing for H(n)=Σf(d)g(⌊n/d⌋).https://jasper-aa64.github.io/posts/number-theory-1-elementary/Tue, 28 Apr 2026 00:00:00 +0800https://jasper-aa64.github.io/posts/number-theory-1-elementary/GCD/LCM properties, Bézout/Frobenius, Euler’s theorem with power-tower reduction and φ-chain technique, congruence system merging via exGCD, CRT, and n! prime factorisation via Legendre’s formula.https://jasper-aa64.github.io/posts/number-theory-1-1-divisibility-blocking/Tue, 28 Apr 2026 00:00:00 +0800https://jasper-aa64.github.io/posts/number-theory-1-1-divisibility-blocking/Divisibility blocking for O(sqrt n) floor-sum evaluation: lemma, single-variable loop, O(1) interval sum requirements, two-variable min-trick, multi-variable pseudocode, and ceiling-division blocking.https://jasper-aa64.github.io/posts/number-theory-2-ex-mobius-practice/Tue, 28 Apr 2026 00:00:00 +0800https://jasper-aa64.github.io/posts/number-theory-2-ex-mobius-practice/Practice problems for Möbius inversion: LCMSUM, two-variable lcm sum (O(n^3/4)), FSF’s game with offline difference, divisor-count sum d(ij), Omega Numbers with Dirichlet suffix sum.https://jasper-aa64.github.io/posts/number-theory-2-ex-mobius-techniques/Tue, 28 Apr 2026 00:00:00 +0800https://jasper-aa64.github.io/posts/number-theory-2-ex-mobius-techniques/Seven core techniques for Möbius inversion problems: loop shortening, gcd indicator expansion, coprimality reduction, Dirichlet convolution via T=dk, divisor symmetry, diagonal symmetry, and value-domain cnt[] tricks.https://jasper-aa64.github.io/posts/number-theory-2-multiplicative-functions/Tue, 28 Apr 2026 00:00:00 +0800https://jasper-aa64.github.io/posts/number-theory-2-multiplicative-functions/Multiplicative functions, single-point and linear-sieve evaluation; Möbius function, both forms of Möbius inversion with full derivations and examples; Dirichlet convolution, key identities, Du Jiao sieve; generalized Möbius inversion on posets.https://jasper-aa64.github.io/posts/number-theory-3-sieve-mobius-inclusion-exclusion/Tue, 28 Apr 2026 00:00:00 +0800https://jasper-aa64.github.io/posts/number-theory-3-sieve-mobius-inclusion-exclusion/Eratosthenes sieve for μ; g/f/h framework; Conclusions 1 and 2 with derivations and tables; GCD table intuition; CF 803F Coprime Subsequences with bottom-up and Möbius approaches.https://jasper-aa64.github.io/posts/number-theory-3-1-sieve-problems/Tue, 28 Apr 2026 00:00:00 +0800https://jasper-aa64.github.io/posts/number-theory-3-1-sieve-problems/Greedy optimality for divisibility-chain coin systems, DP over the maximum denomination, and a sieve-style transition by adding one larger coin.https://jasper-aa64.github.io/posts/number-theory-4-min25-du-jiao-sieve/Tue, 28 Apr 2026 00:00:00 +0800https://jasper-aa64.github.io/posts/number-theory-4-min25-du-jiao-sieve/Min_25 sieve: theory, two complete examples, Lagrange interpolation for power sums. Du Jiao sieve: template, P3768, GCD sum, LCM sum.https://jasper-aa64.github.io/posts/number-theory-4-primality-factorisation-residues/Tue, 28 Apr 2026 00:00:00 +0800https://jasper-aa64.github.io/posts/number-theory-4-primality-factorisation-residues/Miller-Rabin primality test, Pollard Rho factorisation, Tonelli-Shanks for quadratic residues, cubic residues via F_p[t]/(t^3-a), quartic residues by double square root, and general k-th residues via primitive root + BSGS, plus N-th residues mod composite.