Additive Number Theory & Series Tests

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Abstractions about representability by sums, arithmetic density, smooth numbers, sum-free sequences, combinatorial identities, and convergence ratios.

6 abstractions in this family — domain-specific abstractions that sit near one another in structural-signature space (k-means over structural-signature embeddings). Each is shown with its short description.

• Coin Problem — Find the largest nonnegative integer not representable as a nonnegative integer combination of given coprime positive denominations—the Frobenius number of their numerical semigroup.
• Pascal's rule — Decompose the family of fixed-size subsets by whether they contain a distinguished element, yielding the binomial-coefficient recurrence that generates Pascal's triangle.
• Ratio Test — Classify absolute convergence or divergence of a series from the eventual magnitude ratio of successive nonzero terms: below one converges, above one diverges, and equality to one is inconclusive.
• Schnirelmann Density — Measure a set of positive integers by the least fraction it occupies in any initial segment, making early gaps permanently visible and enabling quantitative sumset and additive-basis theorems.
• Smooth Number — A positive integer is B-smooth when every prime divisor is at most the declared bound B, making its factorization lie entirely within a small-prime factor base.
• Sum-Free Sequence — A strictly increasing sequence of positive integers in which no term is representable as a sum of a subset of its predecessors, coupling prefix-dependent additive avoidance to sparse-growth and reciprocal-sum questions.