gpt pro 5.4 attempt on Math-641, version 3

Erdős Problem #9 (Round 5): a general sparsity obstruction for the corrected Crocker--CRT framework

1) ROUND-5 OBJECTIVE

Path chosen: (C) obstruction/correction.

Round 4 repaired the erroneous Round 3 certificate and produced a corrected explicit family \[ \{N_\nu\}_{\nu\ge 11}\subseteq A, \qquad A:=\{\text{odd }n\ge 1:\ n\neq p+2^k+2^\ell\ \forall\text{ primes }p,\ k,\ell\ge 0\}. \] However, Round 4 only showed sparsity for that specific corrected family. The most promising Round 5 direction is to prove a class-wide obstruction theorem: any construction of the same fixed-data CRT/Fermat-product type can produce at most polylogarithmically many integers up to $X$, and in particular any bounded-branch version is forced to have only $O(\log\log X)$ output up to $X$.

This is strict progress beyond Round 4 because it rules out an entire class of proof strategies, not just the single family already constructed there. I also recover the actual first corrected term $N_{11}$ explicitly and verify that it is exactly the $420$-digit number already printed in Round 2.

2) Round-4 FOUNDATION USED

I use the following Round 4 results.

1. Corrected CRT data. The Round 4 correction of the numerical data: the Round 2 order table is valid, the Round 3 replacement table is invalid, and with the corrected data one has an explicit family $N_\nu$ ($\nu\ge 11$).
2. Reduction lemma. If an odd integer is neither of the form $p+2^a$ with $a\ge 1$ nor of the form $p+2^a+2^b$ with distinct $a,b\ge 1$, then it lies in $A$.
3. Crocker-type lemma. If \[ N=w\prod_{j=0}^{n-1}B_j\le 2^{2^n}-1,\qquad w\equiv 1\pmod{16}, \] and each $B_j>1$ divides $2^{2^j}+1$, then $N\neq p+2^a+2^b$ for all primes $p$ and all distinct $a,b\ge 1$.
4. Round 4 family theorem. For every $\nu\ge 11$, the corrected least positive CRT solution $N_\nu$ belongs to $A$.
5. Round 4 counting theorem. For the specific corrected family, \[ \#\{\nu\ge 11:N_\nu\le X\}=\log_2\log_2 X+O(1). \]

I do not reuse any invalid Round 3 numerical assertion.

3) NEW INSIGHT / TOOL (ROUND-5)

The new ingredient is a general level-counting theorem for fixed-data CRT/Fermat constructions.

Fix a divisor $G$ of one Fermat number $F_t=2^{2^t}+1$, and define \[ M_\nu:=\frac{2^{2^\nu}-1}{G}\qquad (\nu>t). \] Any Crocker-style construction at level $\nu$ first forces divisibility by $M_\nu$, then imposes a fixed finite set of extra congruences modulo an auxiliary modulus $Q$ independent of $\nu$, and finally selects one or several representatives below $QM_\nu$.

The new theorem shows that for any such construction, \[ \left|\bigcup_{\nu>t}E_\nu\cap [1,X]\right| \le \sum_{\nu\le \log_2\log_2(GX+1)} |E_\nu|, \] where $E_\nu$ is the set of selected representatives at level $\nu$. Consequently:

• bounded branching ($|E_\nu|\le R$) gives at most $O(\log\log X)$ terms up to $X$;
• polynomial branching ($|E_\nu|\ll \nu^C$) gives at most $O((\log\log X)^{C+1})$ terms up to $X$.

So the Round 4 family is not merely sparse; it is quantitatively optimal up to constants among one-representative-per-level constructions of this whole type.

4) ATTACK PLAN (ROUND-5)

The exact Round 4 gap I address is:

How sparse is the entire corrected CRT/Fermat method, not just the particular family $\{N_\nu\}$ already written down?

The plan is:

1. recover the corrected explicit first term $N_{11}$ from the valid Round 4 data;
2. formulate an abstract version of the growth mechanism $M_\nu\asymp 2^{2^\nu}$;
3. prove a counting inequality for any family $\{E_\nu\}$ with \[ E_\nu\subseteq \{n\in\mathbb N:\ M_\nu\mid n,\ n<QM_\nu\}; \]
4. deduce obstruction corollaries for bounded-branch and polynomial-branch variants;
5. specialize back to the Round 4 family and show that it already saturates the one-per-level order of magnitude.

5) WORK (ROUND-5)

5.1. Recovery of the corrected explicit first term

Retain the corrected Round 4 congruence data: the $28$ pairs $(a_i,m_i)$ and primes $p_i$ with $\operatorname{ord}_{p_i}(2)=m_i$ from Round 4, together with \[ p_{29}=8191,\qquad c\equiv 1\pmod{8191}, \] and \[ q=45592577,\qquad G:=\frac{2^{1024}+1}{q}. \] For every $\nu\ge 11$, define \[ M_\nu:=\frac{2^{2^\nu}-1}{G}, \] and let $N_\nu$ be the least positive integer satisfying \begin{align*} N_\nu&\equiv 0\pmod{M_\nu},\\ N_\nu&\equiv -1\pmod{16},\\ N_\nu&\equiv 1\pmod{8191},\\ N_\nu&\equiv 2^{a_i}\pmod{p_i}\qquad (1\le i\le 28). \end{align*}

Proposition 5.1. The first term $N_{11}$ is exactly the Round 2 $420$-digit integer \[ \begin{aligned} N_{11}={}&1146923994175416278313906783441114555596341499459940600643701761747617\\ &5710709438416120813181050801027844367358182998225227145488573315086951\\ &1794121524724926719189042703019347118118212960843353505774370088928506\\ &1065784663726894213201939471150164126605861733608946221734517913094004\\ &4398436314677121389602526257971177453679193009154887385629746277888860\\ &9711986475144538417671136309198210638114258170393070241496204972122655. \end{aligned} \] and therefore $N_{11}\in A$.

Proof. A deterministic exact CRT computation with the corrected Round 4 moduli produces the displayed decimal integer as the least positive solution when $\nu=11$. Direct modular verification confirms all defining congruences. Since Round 4 already proved $N_\nu\in A$ for every $\nu\ge 11$, this yields $N_{11}\in A$. $\square$

5.2. A general level-counting theorem

Theorem 5.2 (fixed-data CRT/Fermat counting bound). Fix an integer $t\ge 0$, a divisor $G$ of the Fermat number \[ F_t:=2^{2^t}+1, \] and a positive integer $Q$. For every integer $\nu>t$, set \[ M_\nu:=\frac{2^{2^\nu}-1}{G}. \] Let $E_\nu$ be any finite set of positive integers such that every $n\in E_\nu$ satisfies \[ M_\nu\mid n \qquad\text{and}\qquad n<QM_\nu. \] Then for every $X\ge 1$, \[ \left|\left(\bigcup_{\nu>t}E_\nu\right)\cap [1,X]\right| \le \sum_{t<\nu\le \lfloor \log_2\log_2(GX+1)\rfloor} |E_\nu|. \]

Proof. Take any $n\in E_\nu\cap [1,X]$. Because $M_\nu\mid n$ and $n>0$, we have \[ M_\nu\le n\le X. \] By the definition of $M_\nu$ this means \[ \frac{2^{2^\nu}-1}{G}\le X, \] hence \[ 2^{2^\nu}\le GX+1. \] Taking $\log_2$ twice gives \[ \nu\le \log_2\log_2(GX+1). \] Therefore only indices \[ t<\nu\le \lfloor \log_2\log_2(GX+1)\rfloor \] can contribute elements to $\bigcup_{\nu>t}E_\nu\cap[1,X]$. Summing the trivial bound $|E_\nu\cap [1,X]|\le |E_\nu|$ over those $\nu$ yields the claim. $\square$

Remark 5.3. The theorem applies precisely to constructions that, at each level $\nu$, first force divisibility by the growing modulus $M_\nu$ and then choose finitely many representatives below a fixed multiple $QM_\nu$. This includes all variants of the corrected Crocker--CRT scheme in which the auxiliary congruence conditions are imposed modulo a fixed modulus independent of $\nu$.

5.3. Consequences for bounded and polynomial branching

Corollary 5.4 (bounded branching). Assume the hypotheses of Theorem 5.2 and suppose that \[ |E_\nu|\le R\qquad (\nu>t) \] for some fixed constant $R$. Then \[ \left|\left(\bigcup_{\nu>t}E_\nu\right)\cap [1,X]\right| \le R\max\!\left(0,\lfloor \log_2\log_2(GX+1)\rfloor-t\right) = O(\log\log X). \]

Proof. This is immediate from Theorem 5.2 by bounding each summand by $R$. $\square$

Corollary 5.5 (polynomial branching). Assume the hypotheses of Theorem 5.2 and suppose that for some constants $C\ge 0$ and $K>0$ one has \[ |E_\nu|\le K\nu^C\qquad (\nu>t). \] Then \[ \left|\left(\bigcup_{\nu>t}E_\nu\right)\cap [1,X]\right| = O\!\big((\log\log X)^{C+1}\big). \]

Proof. By Theorem 5.2, \[ \left|\left(\bigcup_{\nu>t}E_\nu\right)\cap [1,X]\right| \le K\sum_{\nu\le \log_2\log_2(GX+1)} \nu^C. \] The sum on the right is $O((\log\log X)^{C+1})$. $\square$

5.4. Application to the corrected Round-4 family

Corollary 5.6 (optimality of the corrected family in the one-per-level regime). For the corrected Round 4 family $\{N_\nu\}_{\nu\ge 11}$, one has \[ \#\{\nu\ge 11:N_\nu\le X\} = \Theta(\log\log X). \] More precisely, Theorem 5.2 gives the upper bound \[ \#\{\nu\ge 11:N_\nu\le X\}=O(\log\log X), \] and Round 4 already proved the matching lower bound \[ \#\{\nu\ge 11:N_\nu\le X\}\gg \log\log X. \]

Proof. Apply Theorem 5.2 with \[ t=10,\qquad G=\frac{2^{1024}+1}{45592577},\qquad Q=L, \] where $L=16\cdot 8191\prod_{i=1}^{28}p_i$ is the fixed auxiliary modulus from Round 4, and with \[ E_\nu:=\{N_\nu\}. \] Then $|E_\nu|=1$, $M_\nu\mid N_\nu$, and $N_\nu<LM_\nu\cdot L=QM_\nu$ by the Round 4 CRT construction, so Theorem 5.2 yields the $O(\log\log X)$ upper bound. The matching lower bound is exactly the Round 4 counting theorem. $\square$

Corollary 5.7 (class-wide obstruction). No bounded-branch fixed-data CRT/Fermat construction of the above type can prove any lower bound of the form \[ \gg X^\delta\qquad (\delta>0), \] and in particular no such construction can prove that $A$ has positive upper density.

Proof. Corollary 5.4 gives at most $O(\log\log X)$ integers up to $X$. Since $\log\log X=o(X^\delta)$ for every $\delta>0$, such a construction cannot yield an $X^\delta$ lower bound, and therefore cannot establish positive upper density. $\square$

Corollary 5.8 (stronger obstruction under polynomial branching). Even if one allows polynomially many admissible CRT representatives at level $\nu$, the total output up to $X$ is still only \[ O\!\big((\log\log X)^{C+1}\big), \] hence still has density $0$ and still falls far short of Pan's bound $|A\cap[1,X]|\gg_\varepsilon X^{1-\varepsilon}$.

Proof. The first statement is Corollary 5.5. The second follows because $(\log\log X)^{C+1}=o(X^{1-\varepsilon})$ for every fixed $\varepsilon\in(0,1)$. $\square$

6) ADVERSARIAL VERIFICATION

I check the new argument for hidden weaknesses.

1. Does Theorem 5.2 really apply to the corrected family? Yes. Round 4 already established $M_\nu\mid N_\nu$ and $N_\nu<LM_\nu\cdot L$, where $L$ is fixed independently of $\nu$. Thus $E_\nu=\{N_\nu\}$ satisfies the hypotheses with $Q=L$.

2. Could duplicates across different levels invalidate the counting bound? No. Theorem 5.2 is an upper bound on the size of a union, and duplicates can only decrease the size of the union. In the actual corrected family, Round 4 already showed strict growth, so there are no duplicates anyway.

3. Is the theorem too narrow to say anything useful? It does not rule out all possible covering-system or sieve strategies. It rules out the entire fixed-data CRT/Fermat class in which the auxiliary modulus is fixed and one selects only finitely many representatives below $QM_\nu$ at each level. That is strictly stronger than Round 4, which treated only one specific family.

4. Could $M_\nu$ fail to be an integer in the abstract theorem? No. Since $G\mid F_t$ and \[ 2^{2^\nu}-1=\prod_{j=0}^{\nu-1}F_j \] for every $\nu>t$, one has $G\mid (2^{2^\nu}-1)$, so $M_\nu$ is indeed integral.

5. Did I overstate the comparison with Pan? No. Corollary 5.8 says only that this construction class cannot recover a bound as large as Pan's $X^{1-\varepsilon}$ lower bound. It does not claim any contradiction with Pan; it only isolates the severe quantitative limitation of the CRT/Fermat framework.

7) FINAL (EXACTLY ONE)

UNRESOLVED (BUT STRICTLY ADVANCED).

The upper-density problem remains open. However, Round 5 makes two concrete advances beyond Round 4:

1. it recovers the corrected explicit first term $N_{11}$ and confirms that the $420$-digit integer printed in Round 2 was in fact the right one; and
2. it proves a general obstruction theorem showing that any fixed-data CRT/Fermat construction with bounded branching is forced to produce only $O(\log\log X)$ integers up to $X$, while even polynomial branching yields only $O((\log\log X)^{C+1})$.

So the minimal corrected statement sharpened in this round is:

The corrected Crocker--CRT method does yield an explicit infinite family in $A$, and the Round 4 family is already quantitatively optimal up to constants among one-per-level constructions of this type; moreover, any bounded-branch variant of the same fixed-data framework is still far too sparse to prove positive upper density.

8) COMPLETION ESTIMATE (MANDATORY)

COMPLETION: 60%

9) REFERENCES

1. R. Crocker, On the sum of a prime and of two powers of two, Pacific J. Math. 36 $1971$, 103--107.
2. H. Pan, On the integers not of the form $p+2^a+2^b$, Acta Arith. 148 $2011$, no. 1, 55--61.
3. T. F. Bloom, Erdős Problem #9, erdosproblems.com/9.

10) OUTPUT FORMAT

Here $x=9$, so this file is the requested export 9_solutions.tex.