154 karma · joined July 17, 2025
Google is great at some things, but this isn't it.
The answer is yes. Assume, for the sake of contradiction, that there exists an \(\epsilon > 0\) such that for every \(k\), there exists a choice of congruence classes \(a_1^{(k)}, \dots, a_k^{(k)}\) for which the set of integers not covered by the first \(k\) congruences has density at least \(\epsilon\).
For each \(k\), let \(F_k\) be the set of all infinite sequences of residues \((a_i)_{i=1}^\infty\) such that the uncovered set from the first \(k\) congruences has density at least \(\epsilon\). Each \(F_k\) is nonempty (by assumption) and closed in the product topology (since it depends only on the first \(k\) coordinates). Moreover, \(F_{k+1} \subseteq F_k\) because adding a congruence can only reduce the uncovered set. By the compactness of the product of finite sets, \(\bigcap_{k \ge 1} F_k\) is nonempty.
Choose an infinite sequence \((a_i) \in \bigcap_{k \ge 1} F_k\). For this sequence, let \(U_k\) be the set of integers not covered by the first \(k\) congruences, and let \(d_k\) be the density of \(U_k\). Then \(d_k \ge \epsilon\) for all \(k\). Since \(U_{k+1} \subseteq U_k\), the sets \(U_k\) are decreasing and periodic, and their intersection \(U = \bigcap_{k \ge 1} U_k\) has density \(d = \lim_{k \to \infty} d_k \ge \epsilon\). However, by hypothesis, for any choice of residues, the uncovered set has density \(0\), a contradiction.
Therefore, for every \(\epsilon > 0\), there exists a \(k\) such that for every choice of congruence classes \(a_i\), the density of integers not covered by the first \(k\) congruences is less than \(\epsilon\).
\boxed{\text{Yes}}
If you just want to make a buck, build a ChatGPT wrapper where people will pay you for the privilege of uploading their deepest secrets and intellectual property to your servers.
If you're ideologically motivated, forget about the profit motive and go FOSS.
By chance I once sat near him and the rest of Farbrausch at a demoparty, but I was too shy to say hi.
Had a Gemini Pro sub for 1 or 2 months but it didn't impress me that much.
When I need real horsepower (usually advanced math stuff) I use DeepSeek, which is both free and unparalleled in my opinion.
I did some experiments with OpenRouter but my total usage is still below $10.
More and more I've been experimenting with local thinking models and while it's quite slow (on my non-Apple, CPU-heavy hardware, at least), for some use cases it's an acceptable trade-off and the results have been satisfactory.
I care a lot about data privacy and I'm just not gonna upload clients' proprietary codebases under NDA to some AI company. I've been considering buying or renting proper hardware for local inference but the TCO (particularly with regards to depreciation) is not intuitive.