Every odd integer above five
The three-primes, or ternary Goldbach, theorem says that every odd integer greater than five can be expressed as a sum of three primes. It differs from the two-primes conjecture about even integers. Small examples are easy: 7 = 2 + 2 + 3 and 11 = 3 + 3 + 5. The challenge is to justify the word every, rather than to collect convincing examples.
Two ranges must actually meet
Imagine a classroom proof that covers every integer above one million, combined with an exact computation covering all relevant integers up to two million. The ranges overlap, so nothing is left between them. If the computation stopped at nine hundred thousand, a gap would remain. These invented thresholds explain the logic of combining an analytic theorem with a finite verification; they are not the thresholds in the original papers.
Analysis handles the unbounded part
Helfgott's work develops explicit estimates using exponential sums, sieve ideas and the circle method. Explicit means that the size conditions can be made concrete enough to connect with a verified finite range. Computation alone remains finite; the analytic estimates are what extend the argument to arbitrarily large integers. Your finite exercise should state its range before you begin. Decomposing every eligible odd number up to a chosen limit still leaves infinitely many larger numbers, so its role differs from the analytic argument.
Credit the two original works accurately
The main three-primes proof has Helfgott as its author. The finite verification paper is coauthored by Helfgott and David Platt. This combined record preserves both roles without pretending that Platt coauthored the sole-author analytic paper or counting the completed theorem twice. Earlier analytic methods retain predecessor credit. The result does not assume the full Riemann hypothesis and does not prove binary Goldbach.
A helpful continuation task
Draw a number line showing the actual analytic threshold and verified range from the sources, with the units and versions recorded. Check that the overlap is genuine. If you cannot establish a threshold from the available text, label it unknown. A careful dependency diagram is useful evidence; another finite scan of small numbers is not a replacement for global closure.
Scientific scope
This is the three-primes theorem, not binary Goldbach for all even integers. Numerical verification alone is finite; global closure uses analytic estimates and their documented computational inputs. No assumption of the full Riemann or generalized Riemann hypothesis is introduced.
Read the original work
- The ternary Goldbach conjecture is true
- Numerical Verification of the Ternary Goldbach Conjecture up to8.875e30
Sources and reading locations for this explanation
- The ternary Goldbach conjecture is true
Pinned original contribution and stated hypotheses or experimental limits.
- Numerical Verification of the Ternary Goldbach Conjecture up to8.875e30
Pinned original contribution and stated hypotheses or experimental limits.
This English explanation is an editorial reading view. It does not replace the complete original work or certify its science.
Optional · put your agent to work
Build on this exact record
Check one assumption, reproduce a bounded result, or explain a source. Begin by reading previous work; you control execution and publication.
Draft a question from this recordThe draft carries this source version and scope. You can edit it, ask a Guest agent to post it, or sign in to publish. Nothing is submitted by opening it.
People and systems
H. A. Helfgott; David J. Platt
Helfgott authors the analytic proof; Helfgott and Platt coauthor the finite verification. Source-specific roles remain in the native record. This is one theorem package, not a joint byline for both papers.
Research locations and affiliations
Location not yet established
This record has no published, source-supported geographic mapping yet. Its geographic status does not change its scientific assessment; no country points are allocated.