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  https://scitechdaily.com/the-hidden-mathematics-of-alzheimers-plaques-could-reveal-new-treatment-strategies/

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  The mathematics of the future will benefit from out of the box thinking, which is sometimes not closely aligned with academia. A bug left my old phone with half a million contacts for years. However, I did not fix it because I discovered that apps trying to read the address book would crash. I had just discovered a wonderful protection because even though I granted granular access to my contacts, if an app attempted to access the entire address book, it simply could not due to architecture and r...

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  Step 7 of "Proof Chain for the 24000/10451 Zero-Density Bound (Friendly Version)" (https://doi.org/10.13140/RG.2.2.11136.60161) is the key to achieve a very small, yet interesting improvement in the 30/13 Zero-Density Bound of Guth-Maynard. And I will never tire of saying it: what matters here is not the numerical improvement itself, but the technique used to achieve it. This approach can be extrapolated to other mathematical problems that reflect onto physics, engineering, m...

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  Can we spot structural anomalies in RSA keys before we even know their vulnerability signature? Our latest research says yes. Today, we are releasing ECGS-RSA, a Python-based research tool that uses the Exact Centered Gram Synthesis (ECGS) to make these public traces visible. By computing algebraic features from public RSA moduli, ECGS evaluates deviations from a reference model. In our testing across a million keys and real Spanish DNIe certificates, it revealed ROCA structures naturally, with...

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  1/4 — Detecting potentially vulnerable RSA keys with ECGS-RSA ECGS-RSA can identify potentially vulnerable RSA public keys by detecting significant structural anomalies. When all three ALERT levels are triggered, the detected anomaly is particularly pronounced and may indicate that the RSA modulus is susceptible to factorization. Such a result constitutes a strong diagnostic signal, although it does not, by itself, establish factorability.

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  A tool that proves it. Detecting ROCA without fingerprint. "ECGS-RSA is a Python-based research tool for analyzing structural anomalies in RSA public keys using the Exact Centered Gram Synthesis (ECGS) statistical model. The software computes algebraic features from public RSA moduli and evaluates their deviation from a predefined reference model. It supports RSA public keys, X.509 certificates, digitally signed PDF documents, TLS endpoints, SSH host keys, and large public-key datasets....

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  @tao , "Some of these errors can be mitigated by running different kernels that have been implemented in different programming languages on different hardware and operating systems.". Better than different kernels, different formalization languages. If the bug is in Lean itself, different kernels may reproduce it. But different formalization languages not. Main problem is this is not always possible. "What mathematicians should know about the Lean Theorem Prover: ques...

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  While I was able to push the zero-density bound down to 24000/10451 (~2.2964), Mark Lewko has already optimized this threshold further down to 39/17 (~2.2941). Congrats Mark! PALOMAR: https://palomar-registry.org/entry?id=PALOMAR-2026-10-11-000010&version=1

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  State of the art of Zero Density. #numbertheory