2026-10-24T00:19:45Z https://nostr.now majer https://nostr.now/npub1wckh0566nx8y9myxmmt3l0292prhwgtsk8cghh2fdq49h3sknwksjxv88t https://media.mathstodon.xyz/accounts/avatars/109/682/513/879/510/490/original/400dfe2dbe2ff698.jpg https://media.mathstodon.xyz/accounts/avatars/109/682/513/879/510/490/original/400dfe2dbe2ff698.jpg https://nostr.now/nevent1qqs2t73kmf0w4d7fxsv4fgh3cvp4lu0mkxzkca9pytx8nj3gzzwek6szypmz6a7nt2vcushvsm0dw8aag4gywaepwzclpz7af95z5k7xz6d660nkftc In reply to <a href='/nevent1qqsw844d976km9y7qf7n8ks6f5gh0ukr9pkkgh3czxrkmscnuywjf0swwpmmc'>nevent1q…pmmc</a><br/>_________________________<br/><br/>About “they lie on a perfectly straight line is zero”，here is a slightly overengineered proof.<br/><br/>Write the three points as<br/><br/>\( P_i=(X_i,Y_i) \) <br/><br/>The sample space is<br/>\[ D^3\subset\mathbb R^6 \] <br/><br/>where \(D\) is the unit disk.<br/><br/>The three points are collinear if and only if<br/>\[\begin{vmatrix} X_1&amp;Y_1&amp;1\\ X_2&amp;Y_2&amp;1\\ X_3&amp;Y_3&amp;1 \end{vmatrix}=0 \] <br/><br/>In other words,<br/>\[ f(X_1,Y_1,X_2,Y_2,X_3,Y_3)=0 \] <br/><br/>where \(f\) is a polynomial in six variables.<br/><br/>Thus the set of collinear triples is<br/>\(A=f^{-1}(0) \) <br/><br/>an algebraic hypersurface in \(\mathbb R^6\). Its dimension is at most \(5\), and hence it has \(6\)-dimensional Lebesgue measure zero.<br/><br/>On the other hand, the joint distribution of the three independently and uniformly chosen points is absolutely continuous with respect to \(6\)-dimensional Lebesgue measure.<br/><br/>Therefore,<br/><br/>P(A)=0<br/> ​ 2026-10-06T06:56:05Z https://nostr.now/nevent1qqstqjgf2jzf5vyrnp3gn299u0kzk5jddglrfezzccc2dp00ztav0dgzypmz6a7nt2vcushvsm0dw8aag4gywaepwzclpz7af95z5k7xz6d66jx5kd7 In reply to <a href='/nevent1qqsgg8nwqhadd5lz5kl6mwvw67jefjcvkcaeunu5t7gj0fstgapczxq4q2mm4'>nevent1q…2mm4</a><br/>_________________________<br/><br/>Replace 2 with 1662, then 𝑝 is a Wieferich prime to base 1662. Earlier this year, for the first time, a prime 𝑝 was discovered satisfying <br/><br/>\( 1662^{p-1} \equiv 1 \pmod{p^2} \) .<br/><br/>In this case,𝑝=4001605247257.<br/><br/>Previously, no prime below 4 trillion had been found to satisfy this congruence. The discoverer was Richard Fischer.<a href="https://www.fermatquotient.com/FermatQuotienten/FermQ_Sorg.txt">https://www.fermatquotient.com/FermatQuotienten/FermQ_Sorg.txt</a><br/><br/>But there is deeper mathematics here: it is not merely the discovery of a curious prime. From this, one can deduce that there exists a natural number 𝑛 such that the ring of integers of \(\mathbb{Q}(\sqrt[n]{1662})\) is not \( \mathbb{Z}[\sqrt[n]{1662}] \) . The smallest such example is precisely 𝑛=4001605247257.<br/><br/>In the literature (<a href="https://kconrad.math.uconn.edu/blurbs/gradnumthy/integersradical.pdfsee">https://kconrad.math.uconn.edu/blurbs/gradnumthy/integersradical.pdfsee</a> Theorem 5.3), it is stated:<br/><br/>When 𝑎≠±1,𝑛≥2, and \( T^n - a\) is irreducible over 𝑍,the ring of integers of \(\mathbb{Q}(\sqrt[n]{a})\) equals \( \mathbb{Z}[\sqrt[n]{a}] \) if and only if 𝑎 is square-free, and none of the prime divisors of 𝑛 is a Wieferich prime to base 𝑎.<br/><br/>Thus, when 𝑎=1662, the polynomial \( 𝑇^𝑛−1662 \) satisfies:−1662 is divisible by 2 but not by 4,<br/><br/>all middle coefficients are 0, hence divisible by 2,the leading coefficient (=1) is not divisible by 2.<br/><br/>Therefore, by Eisenstein’s criterion with 𝑝=2,\( 𝑇^𝑛−1662 \) is irreducible over 𝑄.<br/><br/>Since 1662 is square-free, for all 𝑛 less than the Wieferich prime to base 1662 (namely <br/>4001605247257), the ring of integers is \( \mathbb{Z}[\sqrt[n]{1662}] \) .<br/><br/>Similarly, for \(\mathbb{Q}(\sqrt[n]{2})\) , when 𝑛=1093, the ring of integers is not \( \mathbb{Z}[\sqrt[n]{2}] \) , because 1093 is the classical Wieferich prime to base 2.<br/> <img src="https://media.mathstodon.xyz/media_attachments/files/116/968/602/918/356/214/original/a54ddeabbf2b0b49.jpg"> <br/> 2026-07-23T09:36:26Z https://nostr.now/nevent1qqsxhsx68ecsx8gznawevcm0kae03fweydqv32f40azrt39eqfufhhqzypmz6a7nt2vcushvsm0dw8aag4gywaepwzclpz7af95z5k7xz6d66prl9sl In reply to <a href='/nevent1qqsqh75hvw6ucpw08e28re6vdh663qdq9fa3nklm4u55eswwcjm5a3gf8p5ld'>nevent1q…p5ld</a><br/>_________________________<br/><br/>Regarding the hospital iPhone failures caused by helium leaks mentioned in several comments，⬇<br/> <img src="https://media.mathstodon.xyz/media_attachments/files/116/822/781/171/549/545/original/519b91f32a00f748.jpg"> <br/> 2026-06-27T15:30:49Z https://nostr.now/nevent1qqs29uf20ae9m80pzy06mxwp697520p6wpwp8qv4jtx4uckmg6gc9aszypmz6a7nt2vcushvsm0dw8aag4gywaepwzclpz7af95z5k7xz6d66kzgr8h In reply to <a href='/nevent1qqs02fnqzhunuquhcmy3ppes7jere3kudlkwgtttuzp40vdxe6nwu4q5620cl'>nevent1q…20cl</a><br/>_________________________<br/><br/>Sorry to go a bit off-topic, but I just came across a newly published paper by Japanese academics on the algebraic analysis of counterpoint. I thought you might find it interesting.<a href="https://arxiv.org/abs/2606.01102">https://arxiv.org/abs/2606.01102</a> 2026-06-02T18:19:12Z https://nostr.now/nevent1qqsfcct7fslsrcmmf9h72ytfjrwgk25qm75hx2f4henczundq6z4n0gzypmz6a7nt2vcushvsm0dw8aag4gywaepwzclpz7af95z5k7xz6d66v056qa In reply to <a href='/nevent1qqsvgt273r5etqs3hwx9gxuha0l9cs8u30j4cwu50l278tntasdjw5c9fur0v'>nevent1q…ur0v</a><br/>_________________________<br/><br/>Before clicking to read, I thought one of the motives was related to the incident a month ago when 200 containers of helium were stranded in the Persian Gulf, disrupting one-third of the global helium supply. 2026-04-23T13:15:01Z https://nostr.now/nevent1qqsps648sccdecvh7s3jp0ukc350z9xwffaud08m4dgsnq7e3gsjrlczypmz6a7nt2vcushvsm0dw8aag4gywaepwzclpz7af95z5k7xz6d66lr3e7s In reply to <a href='/nevent1qqs9sc8j0kur2258k6lu3e0uep4zx7vlt6haf5puzrwfmnw369wd54qu2nqgz'>nevent1q…nqgz</a><br/>_________________________<br/><br/>The day I read that headline, I briefly thought they had discovered something like a naked singularity😅 2025-09-15T04:59:17Z https://nostr.now/nevent1qqs03jly5egjg4qhzakuh52zglgrvwwnayhmmpl5zxda469uck0ydsczypmz6a7nt2vcushvsm0dw8aag4gywaepwzclpz7af95z5k7xz6d6642scqr In reply to <a href='/nevent1qqsqns2dy5yhfyqh2czaag60v3kc00umhut6ymsvcytw7w7ker75dcckazamf'>nevent1q…zamf</a><br/>_________________________<br/><br/>Apparently the American constitution is just too barbaric. The Putin–Medvedev commutativity law at least has a touch of mathematical elegance 2025-08-15T07:07:27Z https://nostr.now/nevent1qqsg99a9cldelqvvql7kk0y5nalhfx2q3zlgtatfgtnn3rku9sw467czypmz6a7nt2vcushvsm0dw8aag4gywaepwzclpz7af95z5k7xz6d66xja6pl In reply to <a href='/nevent1qqs8sntuxlcnzxqtntm3ghc0d03cg2h2s4uqv0t0njgzt4ktgcz5vvs5ndk23'>nevent1q…dk23</a><br/>_________________________<br/><br/>Oh sorry，I found this information from a Chinese source（In the report about the early award-winning film 菊豆 &#39;Ju Dou&#39; by the famous Chinese director 张艺谋Zhang Yimou,it was briefly mentioned that the award is dedicated to Hugo Haas）. However, after seeing your reply, I tried verifying it with English sources, but couldn&#39;t find any. It&#39;s possible that the information reported by Chinese movie news writers might have been based on hearsay also. 2025-02-13T12:00:43Z https://nostr.now/nevent1qqsy0nwywkav2u8reap2ckyrme3s5lukudjw4d3yuasjdhd4p5n4u0gzypmz6a7nt2vcushvsm0dw8aag4gywaepwzclpz7af95z5k7xz6d66m752sx In reply to <a href='/nevent1qqsvszmjwykynf4j26776pkczs3cs85kndtpvprtpw9cwceujud3dgc5m3vs6'>nevent1q…3vs6</a><br/>_________________________<br/><br/>Hugo Haas 2025-02-13T11:27:52Z