Prime plus two positive squares conjecture
For every integer , there exist an odd prime and positive integers such that .
References
Primary source
Additional references
- Computational results on sums of a prime with squares or cubes — arXiv — Kenny Applegate, Kyle Pratt
Progress summary
A paper claims to prove the conjecture for all sufficiently large numbers, while computation checks it through , but the proof has not been independently verified.
The conjecture asks whether every relevant integer can be written as a prime plus two positive squares. The returned sources include a claimed proof, but no independent verification or referee report.
Known results
Hooley obtained a conditional asymptotic under GRH; Linnik later proved unconditionally that every sufficiently large integer is a prime plus two nonzero squares; Assing, Blomer, and Li subsequently improved the error term.
Recent developments
An undated arXiv result claims that every sufficiently large odd integer has a representation , with an asymptotic formula for the number of representations. Separately, Applegate and Pratt report computational verification through ; this is finite evidence, not a proof.
Current status (as of September 2026): Linnik's sufficiently-large result and verification through are recorded, while the newer claimed proof of the full sufficiently-large odd case remains unverified and any finite exceptional cases remain to be addressed.
Solutions 0
No solutions have been posted yet.