Additive indecomposability of the primes

No finite modification of the set of primes is a sum A + B of two sets with at least two elements each: Ostmann's conjecture, the inverse Goldbach problem, after OpenAI's 2026 paper. Sieve bounds and collision stability; decorrelation from translated quadratic and higher characters (Bonami's inequality, Weyl sums, Heath-Brown's quadratic large sieve, Kneser's theorem, anchor codes); a supply of nonsparse Fourier transforms from coverage of every large prime; a finite-field tree comparison; a positive statistic and its contradiction.

18 lessons with proofs, examples and exercises with solutions. Written and self-checked by Claude Opus 5.5 (Anthropic); independent review is not complete.

1. The inverse Goldbach problem
2. Square-root bounds and collision stability
3. A biased block and an amplified moment
4. Hypercontractive moments and a small-kernel witness
5. A common centre for the quadratic biases
6. Quadratic kernels and the collision contradiction
7. Scales, cells and a positive statistic for higher characters
8. Templates, transfers and frequency histories
9. Anchor codes and decorrelation of higher characters
10. A tensor budget and a contracting kernel
11. The prime mean and a supply of nonsparse transforms
12. Diagrams, densities and cycles over a finite field
13. Affine actions and the tree comparison
14. A positive statistic and its binary-tree transfers
15. Supports, a progression estimate and the giants
16. Small primes, coincidences and the bulk
17. The signed and norm comparisons
18. The final contradiction

Theorems and assumptions · Reader and editable sources · Course record

Prerequisites

• Cyclotomic fields · Number fields
• Dirichlet characters · Dirichlet L-functions and primes in progressions
• Gauss sums · Dirichlet L-functions and primes in progressions
• The functional equation of Dirichlet L-functions · Dirichlet L-functions and primes in progressions
• Zero-free regions and the exceptional zero · Dirichlet L-functions and primes in progressions
• Counting zeros and the explicit formula for ψ(x, χ) · Dirichlet L-functions and primes in progressions
• Siegel's theorem · Dirichlet L-functions and primes in progressions
• The large sieve inequality · Dirichlet L-functions and primes in progressions
• The least prime in a progression: Linnik's theorem · Dirichlet L-functions and primes in progressions
• Counting primes by elementary means: Chebyshev and Mertens · The Riemann zeta function
• Poisson summation, theta, and the functional equation · The Riemann zeta function
• The prime number theorem · The Riemann zeta function
• The prime number theorem with the classical error term · The Riemann zeta function
• Elementary number theory: congruences, the Chinese remainder theorem, quadratic residues and Jacobi symbols; finite Fourier analysis on Z/qZ and on its unit group; Cauchy-Schwarz and Hölder inequalities.