Computational Mathematics & Problem Solving

The arithmetic of the integers: division with remainder, greatest common divisors, and Bézout’s identity &m

Fundamental theorem of arithmetic, the sieve of Eratosthenes and its linear/segmented cousins, multiplicative functions,

Working in \(\Z/n\Z\): congruences, inverses, Fermat and Euler, the Chinese Remainder Theorem, and computing \(a^b\bmod \lt /div\gt \lt div class="lc-meta"\gt \lt span\gt Intermediate\lt /span\gt · \lt span\gt 60 min\lt /span\gt \lt /div\gt \lt /a\gt \lt a class="lesson-card" data-module-id="p19-exact-arithmetic" href="exact-arithmetic.html"\gt \lt div class="lc-top"\gt \lt span class="lc-num"\gt 19.4\lt /span\gt \lt span class="lesson-badge full"\gt Full Lesson\lt /span\gt \lt /div\gt \lt div class="lc-title"\gt Exact Arithmetic: Big Integers, Rationals, and Floating-Point Traps\lt /div\gt \lt div class="lc-sub"\gt When \(10^{16}+1\) equals \(10^{16}\), and what to do about it: IEEE-754, catastrophic cancellation, exact rationals, an

Bijections, the twelvefold basics, Pascal’s recurrence, binomials mod a prime, and inclusion–exclusion as th

From Fibonacci to Catalan: solving linear recurrences in closed form, and turning a whole sequence into a single analyti

Optimal substructure, overlapping subproblems, top-down vs bottom-up - and, as a payoff, combinatorial game theory

Subsets as integers: enumerating \(2^n\) states, popcount and lowbit tricks, subset-sum DP, and the \(O(2^n n^2)\) trave

Modelling a problem as a graph is usually the whole solution; after that, four classical algorithms cover most of what y

How to explore an exponential space without visiting all of it - plus the computational geometry predicates that l

p(n), Euler’s product, the pentagonal number theorem, and the family of counting structures (Stirling, Bell, Catal

The best rational approximations to a real number, the periodic expansion of a quadratic surd, and how that periodicity

Any linear recurrence is a matrix product; square-and-multiply then evaluates term \(10^{18}\) in \(O(k^3\log n)\) opera

The two questions to ask before writing any code: will it finish, and will it be right? Plus the disciplined optimizatio