GPT 6 Astra Ultra attempt on Math-5673
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Erdős Problem 471
Statement.
Starting with a finite set $Q_0$ of primes, repeatedly adjoin every prime expressible as the sum of three distinct primes already present. Can the sets $Q_i$ have unbounded size?External input and distinctness.
The quantitative form of Vinogradov's three-primes theorem gives, for every sufficiently large odd integer $N$, at least $cN^2/(\log N)^3$ ordered representations $N=p_1+p_2+p_3$, where $c>0$ is absolute. This standard analytic theorem is the external dependency. For a primary modern statement of the classical weighted formula, see Frei, Koymans and Sofos, Section 1.2, https://arxiv.org/abs/1804.00573. Its singular series for odd $N$ is bounded below by $2\prod_{p\geq3}(1-(p-1)^{-2})>0$. Dividing the weighted lower bound by $(\log N)^3$, an upper bound for each representation's weight, gives the unweighted bound used here.Representations having two equal entries number at most $3N$: choose one of the three pairs of equal positions and their common prime; the third entry is then determined. This overcounts triples with all entries equal, which is harmless. Since $N^2/(\log N)^3$ grows faster than $N$, every sufficiently large odd integer is therefore a sum of three distinct primes. Every summand is smaller than $N$.
Choose the seed and check the iteration.
Choose an integer $N_0$ past the threshold just obtained, and let $Q_0$ contain every prime at most $N_0$. We prove by induction through the ordered primes that each prime eventually belongs to some $Q_i$. The assertion holds for the seed. If $p>N_0$, write $p=a+b+c$ with three distinct smaller primes. By induction, they belong to stages $Q_{i_a},Q_{i_b},Q_{i_c}$. Since the stages are nested, all three belong to $Q_{\max(i_a,i_b,i_c)}$, and the next stage contains $p$. This completes the induction.Every stage is finite, because only finitely many triples can be formed from a finite set. Their union contains all primes and is infinite, so their cardinalities are unbounded. This is exactly the requested existence conclusion.