Breakthrough Tracker record
The Friedlander–Iwaniec Gaussian-primes conjecture
For each n congruent to 0 or 4 modulo 6, the authors obtain an asymptotic count of primes of the form p² + nq² with p and q prime. The case n=4 settles the Friedlander–Iwaniec Gaussian-primes conjecture.
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- Stable ID
math-friedlander-iwaniec-gaussian-primes-2024- Revision
math-friedlander-iwaniec-gaussian-primes-2024.v1- Field
- Mathematics · Analytic number theory
- Evidence
- Tier 1 · Peer reviewed: Yes
- Record state
- Current · Accepted by Acta Mathematica; final publication pending
- Last checked
AI role
No generative-AI role was disclosed.
Record details
- Problem or result
- Friedlander–Iwaniec Gaussian-primes conjecture
- Authors
- Ben Green and Mehtaab Sawhney
- Institutions
- University of Oxford and Columbia University at announcement
- Result date
- Preprint submitted October 5, 2024; accepted July 22, 2025
Why it matters
The proof overcomes a long-standing parity barrier for simultaneous prime-variable constraints in sieve theory.
Limits
The theorem applies to specified congruence classes of n. It is not a complete classification of all quadratic forms that take prime values. Version three corrected a partial-summation error in Section 8.3 and a minor inaccuracy in Section 8.7. Acta Mathematica lists the paper as accepted and awaiting publication; no final issue or DOI was located.
Sources
- Primary: Acta Mathematica, accepted papers awaiting publication
- Primary: Ben Green and Mehtaab Sawhney, Primes of the form p²+nq²
- Independent: Quanta, Mathematicians Uncover a New Way to Count Prime Numbers
- Independent: Mehtaab Sawhney publication list
Correction and revision history
- 2026-07-22 — Added with the n modulo 6 restriction and the special n=4 conjecture case stated explicitly.
- 2026-07-22 — Corrected the review status after the journal's accepted-paper list and the author's publication list showed that the paper was accepted by Acta Mathematica on July 22, 2025; final publication is still pending. Also disclosed the partial-summation error and minor inaccuracy corrected in arXiv version three.
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