Elliptic Arithmetic & Group Finiteness

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Abstractions about elliptic-curve ranks and divisibility sequences, finitely generated rational points, group periodicity, and arithmetic functions.

5 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.

• Burnside Problem — Ask when a finitely generated group whose elements all have finite order must itself be finite, separating the general, bounded-exponent, and restricted finite-quotient versions whose answers differ.
• Elliptic Divisibility Sequence — An integer divisibility sequence generated by the nonlinear recurrence of elliptic-curve division polynomials, translating multiplication of a rational point into term divisibility, height growth, ranks of apparition, and primitive-divisor structure.
• Mordell–Weil Rank of an Elliptic Curve — Count the independent infinite-order rational points on an elliptic curve over a declared number field by taking the free rank of its finitely generated Mordell–Weil group.
• Mordell–Weil Theorem — For an abelian variety over a number field, the group of rational points is finitely generated.
• Pillai's Arithmetical Function — The multiplicative gcd-sum function P(n)=sum from k=1 to n of gcd(k,n), whose divisor-class decomposition P=id*phi exposes prime-power evaluation, Euler products, and average-order analysis.