Large Prime Gaps Have Positive Density, Artin's Primitive Root Conjecture and 8 More

Proves unconditionally the all-primary-prime form of Patterson’s first-moment asymptotic: normalized cubic Gauss sums over primary Eisenstein primes of norm at most X, including both conjugates, have an explicit positive main term of order X^5/6/log X. Every fixed nonzero prime-angle Fourier mode has smaller order. (Lean)

Proves Uchida's conjecture: every continuous open homomorphism between Galois groups of possibly infinite solvably closed Galois extensions of number fields comes from a unique equivariant field embedding in the opposite direction. No restriction on the kernel or separate cyclotomic-compatibility assumption is needed.

Resolves the Erdős–Pomerance joint Dickman conjecture: the logarithmic sizes of the largest prime factors of n and n+1 are asymptotically independent in ordinary natural density. In particular, the integers satisfying P⁺(n) < P⁺(n+1) have density 1/2. (Lean)

Proves that the irrationality exponent of π is exactly 2: for every ε > 0 and all sufficiently large denominators q, every rational p/q satisfies |π-p/q| ≥ q^-2-ε. This also proves convergence of the Flint–Hills series ∑_{n ≥ 1}1/(n³ sin² n), with angles in radians. (Lean)

Gives an asymptotic equivalent for the number V(x) of distinct totient values up to x, with a positive bounded phase-dependent factor determined by convergent arithmetic approximations. In particular, V(cx)/V(x) → c for every fixed c > 0, answering Erdős and Hall’s scaling question. (Lean)

Every rational a/b with 1 ≤ a < b is a sum of O(log log b) distinct positive unit fractions. The worst-case minimum number of terms has the same order, resolving Erdős’s conjecture on short Egyptian fractions. (Lean)

For every fixed C > 0, a positive proportion of consecutive prime gaps exceed C log p_n, throughout every sufficiently large initial segment of the primes. The proportion may depend on C. Consequently, the indices where p_n/n increases have positive lower density, answering Erdős and Prachar. (Lean)

Resolves the determinant-one curve case of Litt's integral-density question. For every smooth connected complex algebraic curve and every rank, integral points become Zariski dense in every component of its SLr character variety over the full ring of integers of one number field. Prescribed quasi-unipotent boundary conjugacy classes are allowed, including nonsemisimple classes. (Lean)

Proves the infinitude assertion in Artin's primitive root conjecture for every integer a that is neither −1 nor a square. For each such base, at least c_a x/(log x)² primes in every sufficiently large interval (x,2x) have primitive root a, with c_a > 0.

Proves the modularity conjecture for elliptic curves over imaginary quadratic fields: every elliptic curve over every imaginary quadratic field is modular, with matching local parameters at every place.