![](\"https:\/\/www.johndcook.com\/chakerian1.png\")
<\/p>\nG. D. Chakerian [1] discovered the following generalization of the Pythagorean theorem.<\/p>\n
Take any triangle and attach squares to each side. From each triangle vertex, extend a line perpendicular to the opposite side and extend the line through the square, as shown in the figure below. Then the areas of the rectangles with the same labels are equal.<\/p>\n
<\/p>\n
It’s not clear at first why this is a generalization of the Pythagorean theorem, but if we make our triangle closer to a right triangle it’s easier to see the connection.<\/p>\n
![](\"https:\/\/www.johndcook.com\/chakerian2.png\")
<\/p>\n
As the angle at B becomes closer to a right angle, the area of the orange triangles labeled II goes to zero. In the limit we’re left a theorem saying that the areas of the blue and green squares add up to the area of the square attached to the hypotenuse.<\/p>\n
The paper [1] caught my eye because it gives a connection between a geometry problem and a circuit problem, much like this post<\/a>. However, I find the theorem itself more interesting than the connection to circuits.<\/p>\n The first diagram is my reproduction of the first figure from the paper, with color added. I then moved the vertex A to make the connection to the Pythagorean theorem easier to see.<\/p>\n [1] G. D. Chakerian and M. S. Klamkin. A Generalization of the Pythagorean Theorem Seen as a Problem of Equivalent Resistances. Mathematics Magazine, Vol. 59, No. 3 (Jun., 1986), pp. 149\u2013153.<\/p>\n The first line of the paper is “The following result (observed several years ago by the first author) may be viewed as a generalization of the Pythagorean Theorem.” The paper also gives two other sources of the theorem. Presumably Chakerian independently discovered a theorem that was already known.<\/p>\n","protected":false},"excerpt":{"rendered":" G. D. Chakerian [1] discovered the following generalization of the Pythagorean theorem. Take any triangle and attach squares to each side. From each triangle vertex, extend a line perpendicular to the opposite side and extend the line through the square, as shown in the figure below. Then the areas of the rectangles with the same […]<\/p>\n","protected":false},"author":1,"featured_media":0,"comment_status":"open","ping_status":"closed","sticky":false,"template":"","format":"standard","meta":{"_acf_changed":false,"footnotes":""},"categories":[9],"tags":[224],"class_list":["post-248041","post","type-post","status-publish","format-standard","hentry","category-math","tag-geometry"],"acf":[],"aioseo_notices":[],"aioseo_head":"\n\t\t\n\t\n\t\n\t\n\t\n\t\n\t\t\n\t\t\n\t\t\n\t\t\n\t\t\n\t\t\n\t\t\n\t\t\n\t\t\n\t\t\n\t\t\n\t\t\n\t\t\n\t\t\n\n","aioseo_head_json":{"title":"A generalization of the Pythagorean theorem","description":"A theorem for general triangles that reduces to the Pythagorean theorem when the triangle is a right triangle.","canonical_url":"https:\/\/www.johndcook.com\/blog\/2026\/10\/14\/pythagorean-variation\/","robots":"max-image-preview:large","keywords":"geometry","webmasterTools":{"miscellaneous":""},"schema":null,"og:locale":"en_US","og:site_name":"John D. Cook | Applied Mathematics Consulting","og:type":"article","og:title":"A generalization of the Pythagorean theorem","og:description":"A theorem for general triangles that reduces to the Pythagorean theorem when the triangle is a right triangle.","og:url":"https:\/\/www.johndcook.com\/blog\/2026\/10\/14\/pythagorean-variation\/","article:published_time":"2026-10-14T11:11:11+00:00","article:modified_time":"2026-10-14T11:11:11+00:00","twitter:card":"summary","twitter:title":"A generalization of the Pythagorean theorem","twitter:description":"A theorem for general triangles that reduces to the Pythagorean theorem when the triangle is a right triangle.","twitter:image":"https:\/\/www.johndcook.com\/blog\/wp-content\/uploads\/2022\/05\/twittercard.png"},"aioseo_meta_data":{"post_id":"248041","title":null,"description":"A theorem for general triangles that reduces to the Pythagorean theorem when the triangle is a right triangle.","keywords":null,"keyphrases":{"focus":{"keyphrase":"","score":0,"analysis":{"keyphraseInTitle":{"score":0,"maxScore":9,"error":1}}},"additional":[]},"primary_term":null,"canonical_url":null,"og_title":null,"og_description":null,"og_object_type":"default","og_image_type":"default","og_image_url":null,"og_image_width":null,"og_image_height":null,"og_image_custom_url":null,"og_image_custom_fields":null,"og_video":"","og_custom_url":null,"og_article_section":null,"og_article_tags":null,"twitter_use_og":false,"twitter_card":"default","twitter_image_type":"default","twitter_image_url":null,"twitter_image_custom_url":null,"twitter_image_custom_fields":null,"twitter_title":null,"twitter_description":null,"schema":{"blockGraphs":[],"customGraphs":[],"default":{"data":{"Article":[],"Course":[],"Dataset":[],"FAQPage":[],"Movie":[],"Person":[],"Product":[],"ProductReview":[],"Car":[],"Recipe":[],"Service":[],"SoftwareApplication":[],"WebPage":[]},"graphName":"Article","isEnabled":true},"graphs":[]},"schema_type":"default","schema_type_options":null,"pillar_content":false,"robots_default":true,"robots_noindex":false,"robots_noarchive":false,"robots_nosnippet":false,"robots_nofollow":false,"robots_noimageindex":false,"robots_noodp":false,"robots_notranslate":false,"robots_max_snippet":"-1","robots_max_videopreview":"-1","robots_max_imagepreview":"large","priority":null,"frequency":"default","location":null,"local_seo":null,"breadcrumb_settings":null,"limit_modified_date":false,"created":"2026-10-14 10:45:43","updated":"2026-10-14 13:48:59","ai":{"faqs":[],"keyPoints":[],"schemas":[],"titles":[],"descriptions":[],"socialPosts":{"email":{"subject":"","preview":"","content":""},"linkedin":[],"twitter":[],"facebook":[],"instagram":[]}},"seo_analyzer_scan_date":null,"focus_keyword":null,"additional_keywords":null,"truseo_locale":null},"aioseo_breadcrumb":" What’s a connection between Edgar Allan Poe and Friedrich Wilhelm Bessel?<\/p>\n Poe’s story The Pit and the Pendulum describes a pendulum with a semicircular blade being lowered on a cable. This is an interesting variation on the usual harmonic oscillator because the length L<\/em> of the pendulum is not constant but rather an increasing function of time L<\/em>(t<\/em>).<\/p>\n Using the small angle<\/a> approximation sin(\u03b8) \u2248 \u03b8, the equation of motion is<\/p>\n L<\/em> \u03b8\u2033 +\u00a0L<\/em>\u2032 \u03b8\u2032 + g<\/em> \u03b8 = 0.<\/p>\n If we assume the length is increasing at a constant rate, then with a change of variables described in [1], this differential equation becomes Bessel’s equation and its solutions are Bessel functions.<\/p>\n The authors of [1] state that their primary interest is whether a pendulum can have the motion that Poe describes. But as they indicate in their opening sentence,<\/p>\n The study of the motion of a steadily descending pendulum has practical applications (e.g., to the oscillations of a load being lowered by a crane).<\/p>\n A more recent application would be describing the motion of a package being delivered by a Zipline drone.<\/p>\n [1] Robert L. Borrelli, Courtney S. Coleman, and Dana D. Hobson. Poe’s Pendulum. Mathematics Magazine, Vol. 58, No. 2 (Mar., 1985), pp. 78\u201383<\/p>\n","protected":false},"excerpt":{"rendered":" What’s a connection between Edgar Allan Poe and Friedrich Wilhelm Bessel? Poe’s story The Pit and the Pendulum describes a pendulum with a semicircular blade being lowered on a cable. This is an interesting variation on the usual harmonic oscillator because the length L of the pendulum is not constant but rather an increasing function […]<\/p>\n","protected":false},"author":1,"featured_media":0,"comment_status":"open","ping_status":"closed","sticky":false,"template":"","format":"standard","meta":{"_acf_changed":false,"footnotes":""},"categories":[9],"tags":[47,129],"class_list":["post-248036","post","type-post","status-publish","format-standard","hentry","category-math","tag-differential-equations","tag-special-functions"],"acf":[],"aioseo_notices":[],"aioseo_head":"\n\t\t\n\t\n\t\n\t\n\t\n\t\n\t\t\n\t\t\n\t\t\n\t\t\n\t\t\n\t\t\n\t\t\n\t\t\n\t\t\n\t\t\n\t\t\n\t\t\n\t\t\n\t\t\n\n","aioseo_head_json":{"title":"Poe, Bessel, and Drones","description":"A connection between Edgar Allan Poe, Friedrich Wilhelm Bessel, and drone delivery.","canonical_url":"https:\/\/www.johndcook.com\/blog\/2026\/10\/14\/poe-bessel-and-drones\/","robots":"max-image-preview:large","keywords":"differential equations,special functions","webmasterTools":{"miscellaneous":""},"schema":null,"og:locale":"en_US","og:site_name":"John D. Cook | Applied Mathematics Consulting","og:type":"article","og:title":"Poe, Bessel, and Drones","og:description":"A connection between Edgar Allan Poe, Friedrich Wilhelm Bessel, and drone delivery.","og:url":"https:\/\/www.johndcook.com\/blog\/2026\/10\/14\/poe-bessel-and-drones\/","article:published_time":"2026-10-14T10:00:22+00:00","article:modified_time":"2026-10-14T10:03:15+00:00","twitter:card":"summary","twitter:title":"Poe, Bessel, and Drones","twitter:description":"A connection between Edgar Allan Poe, Friedrich Wilhelm Bessel, and drone delivery.","twitter:image":"https:\/\/www.johndcook.com\/blog\/wp-content\/uploads\/2022\/05\/twittercard.png"},"aioseo_meta_data":{"post_id":"248036","title":null,"description":"A connection between Edgar Allan Poe, Friedrich Wilhelm Bessel, and drone delivery.","keywords":null,"keyphrases":{"focus":{"keyphrase":"","score":0,"analysis":{"keyphraseInTitle":{"score":0,"maxScore":9,"error":1}}},"additional":[]},"primary_term":null,"canonical_url":null,"og_title":null,"og_description":null,"og_object_type":"default","og_image_type":"default","og_image_url":null,"og_image_width":null,"og_image_height":null,"og_image_custom_url":null,"og_image_custom_fields":null,"og_video":"","og_custom_url":null,"og_article_section":null,"og_article_tags":null,"twitter_use_og":false,"twitter_card":"default","twitter_image_type":"default","twitter_image_url":null,"twitter_image_custom_url":null,"twitter_image_custom_fields":null,"twitter_title":null,"twitter_description":null,"schema":{"blockGraphs":[],"customGraphs":[],"default":{"data":{"Article":[],"Course":[],"Dataset":[],"FAQPage":[],"Movie":[],"Person":[],"Product":[],"ProductReview":[],"Car":[],"Recipe":[],"Service":[],"SoftwareApplication":[],"WebPage":[]},"graphName":"Article","isEnabled":true},"graphs":[]},"schema_type":"default","schema_type_options":null,"pillar_content":false,"robots_default":true,"robots_noindex":false,"robots_noarchive":false,"robots_nosnippet":false,"robots_nofollow":false,"robots_noimageindex":false,"robots_noodp":false,"robots_notranslate":false,"robots_max_snippet":"-1","robots_max_videopreview":"-1","robots_max_imagepreview":"large","priority":null,"frequency":"default","location":null,"local_seo":null,"breadcrumb_settings":null,"limit_modified_date":false,"created":"2026-10-14 02:19:22","updated":"2026-10-14 11:01:11","ai":{"faqs":[],"keyPoints":[],"schemas":[],"titles":[],"descriptions":[],"socialPosts":{"email":{"subject":"","preview":"","content":""},"linkedin":[],"twitter":[],"facebook":[],"instagram":[]}},"seo_analyzer_scan_date":null,"focus_keyword":null,"additional_keywords":null,"truseo_locale":null},"aioseo_breadcrumb":" Pascal’s triangle is the most famous triangular arrangement of numbers. I’ve written about other numerical triangles, such as Bell’s triangle<\/a> and the trinomial triangle<\/a>. There’s another well-known triangle that somehow I never heard of until recently: the harmonic triangle, first explored by Leibniz around 1700.<\/p>\n To create the harmonic triangle, start with the harmonic series<\/p>\n 1, 1\/2, 1\/3, 1\/4, 1\/5, \u2026<\/p>\n This is the first column of the triangle. Next take the differences of the terms:<\/p>\n (1 \u2212 1\/2), (1\/2 \u2212 1\/3), (1\/3 \u2212 1\/4), \u2026 = 1\/2, 1\/6, 1\/12, \u2026<\/p>\n This, shifted down one space, is the second column of the triangle. Taking differences again gives the third column, and so forth. Column c<\/em> of the triangle consists of the c<\/em>th differences of the harmonic series.<\/p>\n Each entry is the difference of the entries to its west and northwest. If we let L<\/em>(r<\/em>, c<\/em>) be the element in the r<\/em>th row and c<\/em>th column, then<\/p>\n Every numerator is 1, though it’s not immediately obvious why this should be the case. And every number is a value of the beta function, the function that gives the normalizing constant for the beta distribution [1].<\/p>\n [1] Ayoub B. Ayoub. The Harmonic Triangle and the Beta Function. Mathematics Magazine, Vol. 60, No. 4 (Oct., 1987), pp. 223-225<\/p>\n","protected":false},"excerpt":{"rendered":" Pascal’s triangle is the most famous triangular arrangement of numbers. I’ve written about other numerical triangles, such as Bell’s triangle and the trinomial triangle. There’s another well-known triangle that somehow I never heard of until recently: the harmonic triangle, first explored by Leibniz around 1700. To create the harmonic triangle, start with the harmonic series […]<\/p>\n","protected":false},"author":1,"featured_media":0,"comment_status":"open","ping_status":"closed","sticky":false,"template":"","format":"standard","meta":{"_acf_changed":false,"footnotes":""},"categories":[9],"tags":[197],"class_list":["post-248028","post","type-post","status-publish","format-standard","hentry","category-math","tag-combinatorics"],"acf":[],"aioseo_notices":[],"aioseo_head":"\n\t\t\n\t\n\t\n\t\n\t\n\t\n\t\t\n\t\t\n\t\t\n\t\t\n\t\t\n\t\t\n\t\t\n\t\t\n\t\t\n\t\t\n\t\t\n\t\t\n\t\t\n\t\t\n\n","aioseo_head_json":{"title":"Harmonic triangle","description":"The harmonic triangle is analogous to Pascal's triangle. First explored by Liebniz around 1700. 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If you find general topology more familiar than modal logic, this correspondence could be helpful. If you think the motivation for general topology is evident but the motivation for modal logic is suspect, the correspondence between the two fields might relieve some of your suspicion of the latter.<\/p>\n Topological models, however, are quite limited. You can’t model provability logic<\/a>, for example, by the usual approach to constructing topological models, though there is an alternative construction.<\/p>\n Categorical models are less limited. You can model a wide variety of logics using categories via the Curry-Howard-Lambek correspondence. This may be cold comfort if you find category theory and logic equally mysterious. Even so, there’s something intriguing about a correspondence between two things you find mysterious.<\/p>\n Categorical semantics is deep and broad, but I’ll keep this post brief. The correspondence between logic [1] and categories [2] is summarized in the table below.<\/p>\n It’s interesting how model logic fits into this picture. Axiom K, required by normal modal logics, says<\/p>\n In categorial semantics, this says this says the box operator \u25a1 is a functor, mapping objects representing propositions to objects representing propositions, and mapping morphisms representing proofs to mapping morphisms representing proofs.<\/p>\n [1] More precisely, intuitionistic propositional calculus.<\/p>\n [2] More precisely, bicartesian closed categories.<\/p>\n","protected":false},"excerpt":{"rendered":" A few days ago I wrote about topological models of modal logic. If you find general topology more familiar than modal logic, this correspondence could be helpful. If you think the motivation for general topology is evident but the motivation for modal logic is suspect, the correspondence between the two fields might relieve some of […]<\/p>\n","protected":false},"author":1,"featured_media":0,"comment_status":"open","ping_status":"closed","sticky":false,"template":"","format":"standard","meta":{"_acf_changed":false,"footnotes":""},"categories":[9],"tags":[144,150],"class_list":["post-248025","post","type-post","status-publish","format-standard","hentry","category-math","tag-category-theory","tag-logic"],"acf":[],"aioseo_notices":[],"aioseo_head":"\n\t\t\n\t\n\t\n\t\n\t\n\t\n\t\t\n\t\t\n\t\t\n\t\t\n\t\t\n\t\t\n\t\t\n\t\t\n\t\t\n\t\t\n\t\t\n\t\t\n\t\t\n\t\t\n\n","aioseo_head_json":{"title":"Categorical models of logics","description":"A very brief introduction to modeling varieties of logic by categories.","canonical_url":"https:\/\/www.johndcook.com\/blog\/2026\/10\/12\/categorical-models-of-logics\/","robots":"max-image-preview:large","keywords":"category theory,logic","webmasterTools":{"miscellaneous":""},"schema":null,"og:locale":"en_US","og:site_name":"John D. 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Recently these two worlds collided: I became aware of work that uses modal logic to reason about data privacy.<\/p>\n The story begins with Helen Nissenbaum’s paper\u00a0Privacy as Contextual Integrity<\/em> [1]. Rather than simply classifying data as public or private, Nissenbaum looks at norms around the context in which data exists and moves.<\/p>\n \u2026 the benchmark of privacy is contextual integrity; that in any given situation, a complaint that privacy has been violated is sound in the event that one or the other types of the informational norms has been transgressed.<\/p><\/blockquote>\n A couple years later Nissenbaum coauthored a paper [2] with three Stanford computer scientists using modal logic, specifically linear temporal logic<\/strong> (LTL<\/strong>), to reason about contextual integrity. The paper uses four modal operators: the usual box and diamond, plus past tense versions with a minus sign across the middle.<\/p>\n These four operators can be read as “henceforth”, “eventually”, “historically”, and “once.”<\/p>\n Henceforth means something will always be true into the future; historically means something was always true in the past.<\/p>\n Eventually means something will happen at some time in the future; once means something occurred at some time in the past.<\/p>\n You could imagine using these operators to formalize statements such as data sharing in some context is permissible (henceforth) if the data subject has (once) signed a consent form.<\/p>\n Understanding simple stand-alone policies does not require the machinery of formal logic, but analyzing large collections of interacting policies may. It may be, for example, that a collection of policies cannot all be satisfied, though this is not obvious from looking at the individual policies. Formalizing policies in the language modal logic makes their analysis amenable to well established algorithms.<\/p>\n [1] H. Nissenbaum. Privacy as contextual integrity. Washington Law Review, 79(1):119\u2013158, 2004.<\/p>\n [2] A. Barth, A. Datta, J. C. Mitchell, and H. Nissenbaum, “Privacy and contextual integrity: Framework and applications,” in Proc. 2006 IEEE Symp. Security and Privacy (S&P’06), Berkeley\/Oakland, CA, USA, 2006, pp. 184\u2013198, doi: 10.1109\/SP.2006.32.<\/p>\n","protected":false},"excerpt":{"rendered":" If you’ve followed this blog for a while, you may know that I do a lot of work with data privacy and that I have an interest in modal logic. Recently these two worlds collided: I became aware of work that uses modal logic to reason about data privacy. 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Data privacy meets modal logic.","canonical_url":"https:\/\/www.johndcook.com\/blog\/2026\/10\/08\/privacy-policies-and-modal-logic\/","robots":"max-image-preview:large","keywords":"logic,privacy","webmasterTools":{"miscellaneous":""},"schema":null,"og:locale":"en_US","og:site_name":"John D. Cook | Applied Mathematics Consulting","og:type":"article","og:title":"Privacy policies and modal logic","og:description":"Using linear temporal logic to reason about privacy policies. Data privacy meets modal logic.","og:url":"https:\/\/www.johndcook.com\/blog\/2026\/10\/08\/privacy-policies-and-modal-logic\/","article:published_time":"2026-10-08T13:43:02+00:00","article:modified_time":"2026-10-08T13:43:02+00:00","twitter:card":"summary","twitter:title":"Privacy policies and modal logic","twitter:description":"Using linear temporal logic to reason about privacy policies. Data privacy meets modal logic.","twitter:image":"https:\/\/www.johndcook.com\/blog\/wp-content\/uploads\/2022\/05\/twittercard.png"},"aioseo_meta_data":{"post_id":"248022","title":null,"description":"Using linear temporal logic to reason about privacy policies. Data privacy meets modal logic.","keywords":null,"keyphrases":{"focus":{"keyphrase":"","score":0,"analysis":{"keyphraseInTitle":{"score":0,"maxScore":9,"error":1}}},"additional":[]},"primary_term":null,"canonical_url":null,"og_title":null,"og_description":null,"og_object_type":"default","og_image_type":"default","og_image_url":null,"og_image_width":null,"og_image_height":null,"og_image_custom_url":null,"og_image_custom_fields":null,"og_video":"","og_custom_url":null,"og_article_section":null,"og_article_tags":null,"twitter_use_og":false,"twitter_card":"default","twitter_image_type":"default","twitter_image_url":null,"twitter_image_custom_url":null,"twitter_image_custom_fields":null,"twitter_title":null,"twitter_description":null,"schema":{"blockGraphs":[],"customGraphs":[],"default":{"data":{"Article":[],"Course":[],"Dataset":[],"FAQPage":[],"Movie":[],"Person":[],"Product":[],"ProductReview":[],"Car":[],"Recipe":[],"Service":[],"SoftwareApplication":[],"WebPage":[]},"graphName":"Article","isEnabled":true},"graphs":[]},"schema_type":"default","schema_type_options":null,"pillar_content":false,"robots_default":true,"robots_noindex":false,"robots_noarchive":false,"robots_nosnippet":false,"robots_nofollow":false,"robots_noimageindex":false,"robots_noodp":false,"robots_notranslate":false,"robots_max_snippet":"-1","robots_max_videopreview":"-1","robots_max_imagepreview":"large","priority":null,"frequency":"default","location":null,"local_seo":null,"breadcrumb_settings":null,"limit_modified_date":false,"created":"2026-10-07 23:48:38","updated":"2026-10-08 20:40:41","ai":{"faqs":[],"keyPoints":[],"schemas":[],"titles":[],"descriptions":[],"socialPosts":{"email":{"subject":"","preview":"","content":""},"linkedin":[],"twitter":[],"facebook":[],"instagram":[]}},"seo_analyzer_scan_date":null,"focus_keyword":null,"additional_keywords":null,"truseo_locale":null},"aioseo_breadcrumb":" The Riemann Hypothesis (RH) is the conjecture that all the zeros of the Riemann zeta function \u03b6(s<\/em>) in the critical strip, i.e. the region of the complex plane with real part between 0 and 1, have real part equal to \u00bd.<\/p>\n The Quasi Riemann Hypothesis (QRH) says that there exists a constant \u03b8 < 1 such that no zeros of \u03b6(s<\/em>) have real part greater than \u03b8. OpenAI<\/a> has published a paper claiming QRH with \u03b8 = 7\/8.<\/p>\n The RH is so important to number theory that even partial results can have big consequences. This post will focus on one consequence: the error term in the Prime Number Theorem.<\/p>\n The Prime Number Theorem says that \u03c0(x<\/em>), the number of primes less than x<\/em>, is asymptotically equal to Li(x<\/em>). We’d like to know more specifically at what rate \u03c0(x<\/em>) approaches Li(x<\/em>).<\/p>\n The best known result before the QRH announcement was<\/p>\n If the QRH holds for some \u03b8, such as OpenAI’s assertion that \u03b8 = 7\/8,<\/p>\n Incidentally, you may have seen the Prime Number Theorem stated with\u00a0x<\/em>\/log(x<\/em>) rather than Li(x<\/em>). These two functions are asymptotically equal, so they give the same theorem, if you’re not interested in quantifying the rate of convergence. The function Li(x<\/em>) gives better error bounds.<\/p>\n","protected":false},"excerpt":{"rendered":" The Riemann Hypothesis (RH) is the conjecture that all the zeros of the Riemann zeta function \u03b6(s) in the critical strip, i.e. the region of the complex plane with real part between 0 and 1, have real part equal to \u00bd. The Quasi Riemann Hypothesis (QRH) says that there exists a constant \u03b8 < 1 […]<\/p>\n","protected":false},"author":1,"featured_media":0,"comment_status":"open","ping_status":"closed","sticky":false,"template":"","format":"standard","meta":{"_acf_changed":false,"footnotes":""},"categories":[9],"tags":[94],"class_list":["post-248019","post","type-post","status-publish","format-standard","hentry","category-math","tag-number-theory"],"acf":[],"aioseo_notices":[],"aioseo_head":"\n\t\t\n\t\n\t\n\t\n\t\n\t\n\t\t\n\t\t\n\t\t\n\t\t\n\t\t\n\t\t\n\t\t\n\t\t\n\t\t\n\t\t\n\t\t\n\t\t\n\t\t\n\t\t\n\n","aioseo_head_json":{"title":"Consequences of progress toward the Riemann Hypothesis","description":"Consequences of the recent announcement of a proof of the Quasi Riemann Hypothesis","canonical_url":"https:\/\/www.johndcook.com\/blog\/2026\/10\/07\/consequences-of-qrh\/","robots":"max-image-preview:large","keywords":"number theory","webmasterTools":{"miscellaneous":""},"schema":null,"og:locale":"en_US","og:site_name":"John D. Cook | Applied Mathematics Consulting","og:type":"article","og:title":"Consequences of progress toward the Riemann Hypothesis","og:description":"Consequences of the recent announcement of a proof of the Quasi Riemann Hypothesis","og:url":"https:\/\/www.johndcook.com\/blog\/2026\/10\/07\/consequences-of-qrh\/","article:published_time":"2026-10-07T22:29:28+00:00","article:modified_time":"2026-10-07T22:29:28+00:00","twitter:card":"summary","twitter:title":"Consequences of progress toward the Riemann Hypothesis","twitter:description":"Consequences of the recent announcement of a proof of the Quasi Riemann Hypothesis","twitter:image":"https:\/\/www.johndcook.com\/blog\/wp-content\/uploads\/2022\/05\/twittercard.png"},"aioseo_meta_data":{"post_id":"248019","title":null,"description":"Consequences of the recent announcement of a proof of the Quasi Riemann Hypothesis","keywords":null,"keyphrases":{"focus":{"keyphrase":"","score":0,"analysis":{"keyphraseInTitle":{"score":0,"maxScore":9,"error":1}}},"additional":[]},"primary_term":null,"canonical_url":null,"og_title":null,"og_description":null,"og_object_type":"default","og_image_type":"default","og_image_url":null,"og_image_width":null,"og_image_height":null,"og_image_custom_url":null,"og_image_custom_fields":null,"og_video":"","og_custom_url":null,"og_article_section":null,"og_article_tags":null,"twitter_use_og":false,"twitter_card":"default","twitter_image_type":"default","twitter_image_url":null,"twitter_image_custom_url":null,"twitter_image_custom_fields":null,"twitter_title":null,"twitter_description":null,"schema":{"blockGraphs":[],"customGraphs":[],"default":{"data":{"Article":[],"Course":[],"Dataset":[],"FAQPage":[],"Movie":[],"Person":[],"Product":[],"ProductReview":[],"Car":[],"Recipe":[],"Service":[],"SoftwareApplication":[],"WebPage":[]},"graphName":"Article","isEnabled":true},"graphs":[]},"schema_type":"default","schema_type_options":null,"pillar_content":false,"robots_default":true,"robots_noindex":false,"robots_noarchive":false,"robots_nosnippet":false,"robots_nofollow":false,"robots_noimageindex":false,"robots_noodp":false,"robots_notranslate":false,"robots_max_snippet":"-1","robots_max_videopreview":"-1","robots_max_imagepreview":"large","priority":null,"frequency":"default","location":null,"local_seo":null,"breadcrumb_settings":null,"limit_modified_date":false,"created":"2026-10-07 21:54:44","updated":"2026-10-07 23:07:00","ai":{"faqs":[],"keyPoints":[],"schemas":[],"titles":[],"descriptions":[],"socialPosts":{"email":{"subject":"","preview":"","content":""},"linkedin":[],"twitter":[],"facebook":[],"instagram":[]}},"seo_analyzer_scan_date":null,"focus_keyword":null,"additional_keywords":null,"truseo_locale":null},"aioseo_breadcrumb":" The Fast Fourier Transform (FFT) algorithm can compute the discrete Fourier transform of a sequence of length\u00a0n<\/em> in time<\/p>\n O<\/em>(n<\/em> log\u00a0n<\/em>).<\/p>\n OpenAI<\/a> recently posted a paper saying there is an algorithm that could compute the discrete Fourier transform in<\/p>\n O<\/em>(n<\/em> (log\u00a0n<\/em>)1 \u2212 \u03b5<\/sup>)<\/p>\n time for \u03b5 = 10\u221213<\/sup>.<\/p>\n This result is amazing. It seemed that O<\/em>(n<\/em> log\u00a0n<\/em>) was as good as you could do, which it provably is for sorting algorithms.<\/p>\n The result is also of absolutely no practical value, for now. But since the theorem shows that our assumptions were wrong, however slightly, about what is possible, maybe we’re in for further surprises. Maybe the \u03b5 crack will grow. It wouldn’t be the first time.<\/p>\n The Fast Fourier Transform (FFT) algorithm can compute the discrete Fourier transform of a sequence of length\u00a0n in time O(n log\u00a0n). OpenAI recently posted a paper saying there is an algorithm that could compute the discrete Fourier transform in O(n (log\u00a0n)1 \u2212 \u03b5) time for \u03b5 = 10\u221213. This result is amazing. It seemed that […]<\/p>\n","protected":false},"author":1,"featured_media":0,"comment_status":"open","ping_status":"closed","sticky":false,"template":"","format":"standard","meta":{"_acf_changed":false,"footnotes":""},"categories":[9],"tags":[178],"class_list":["post-248015","post","type-post","status-publish","format-standard","hentry","category-math","tag-fourier-analysis"],"acf":[],"aioseo_notices":[],"aioseo_head":"\n\t\t\n\t\n\t\n\t\n\t\n\t\n\t\t\n\t\t\n\t\t\n\t\t\n\t\t\n\t\t\n\t\t\n\t\t\n\t\t\n\t\t\n\t\t\n\t\t\n\t\t\n\t\t\n\n","aioseo_head_json":{"title":"Faster Fourier Transform","description":"The fast Fourier transform is no longer the fastest way to compute the discrete Fourier transform, in some VERY theoretical sense.","canonical_url":"https:\/\/www.johndcook.com\/blog\/2026\/10\/07\/faster-fourier-transform\/","robots":"max-image-preview:large","keywords":"fourier analysis","webmasterTools":{"miscellaneous":""},"schema":null,"og:locale":"en_US","og:site_name":"John D. Cook | Applied Mathematics Consulting","og:type":"article","og:title":"Faster Fourier Transform","og:description":"The fast Fourier transform is no longer the fastest way to compute the discrete Fourier transform, in some VERY theoretical sense.","og:url":"https:\/\/www.johndcook.com\/blog\/2026\/10\/07\/faster-fourier-transform\/","article:published_time":"2026-10-07T21:41:47+00:00","article:modified_time":"2026-10-07T23:35:18+00:00","twitter:card":"summary","twitter:title":"Faster Fourier Transform","twitter:description":"The fast Fourier transform is no longer the fastest way to compute the discrete Fourier transform, in some VERY theoretical sense.","twitter:image":"https:\/\/www.johndcook.com\/blog\/wp-content\/uploads\/2022\/05\/twittercard.png"},"aioseo_meta_data":{"post_id":"248015","title":null,"description":"The fast Fourier transform is no longer the fastest way to compute the discrete Fourier transform, in some VERY theoretical sense.","keywords":null,"keyphrases":{"focus":{"keyphrase":"","score":0,"analysis":{"keyphraseInTitle":{"score":0,"maxScore":9,"error":1}}},"additional":[]},"primary_term":null,"canonical_url":null,"og_title":null,"og_description":null,"og_object_type":"default","og_image_type":"default","og_image_url":null,"og_image_width":null,"og_image_height":null,"og_image_custom_url":null,"og_image_custom_fields":null,"og_video":"","og_custom_url":null,"og_article_section":null,"og_article_tags":null,"twitter_use_og":false,"twitter_card":"default","twitter_image_type":"default","twitter_image_url":null,"twitter_image_custom_url":null,"twitter_image_custom_fields":null,"twitter_title":null,"twitter_description":null,"schema":{"blockGraphs":[],"customGraphs":[],"default":{"data":{"Article":[],"Course":[],"Dataset":[],"FAQPage":[],"Movie":[],"Person":[],"Product":[],"ProductReview":[],"Car":[],"Recipe":[],"Service":[],"SoftwareApplication":[],"WebPage":[]},"graphName":"Article","isEnabled":true},"graphs":[]},"schema_type":"default","schema_type_options":null,"pillar_content":false,"robots_default":true,"robots_noindex":false,"robots_noarchive":false,"robots_nosnippet":false,"robots_nofollow":false,"robots_noimageindex":false,"robots_noodp":false,"robots_notranslate":false,"robots_max_snippet":"-1","robots_max_videopreview":"-1","robots_max_imagepreview":"large","priority":null,"frequency":"default","location":null,"local_seo":null,"breadcrumb_settings":null,"limit_modified_date":false,"created":"2026-10-07 21:27:22","updated":"2026-10-08 00:37:01","ai":{"faqs":[],"keyPoints":[],"schemas":[],"titles":[],"descriptions":[],"socialPosts":{"email":{"subject":"","preview":"","content":""},"linkedin":[],"twitter":[],"facebook":[],"instagram":[]}},"seo_analyzer_scan_date":null,"focus_keyword":null,"additional_keywords":null,"truseo_locale":null},"aioseo_breadcrumb":"Related posts<\/h2>\n
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\n\t\t\tHome<\/a>\n\t\t<\/span>\u00bb<\/span>\n\t\t\tMath<\/a>\n\t\t<\/span>\u00bb<\/span>\n\t\t\tA generalization of the Pythagorean theorem\n\t\t<\/span><\/div>","aioseo_breadcrumb_json":[{"label":"Home","link":"https:\/\/www.johndcook.com\/blog\/"},{"label":"Math","link":"https:\/\/www.johndcook.com\/blog\/category\/math\/"},{"label":"A generalization of the Pythagorean theorem","link":"https:\/\/www.johndcook.com\/blog\/2026\/10\/14\/pythagorean-variation\/"}],"_links":{"self":[{"href":"https:\/\/www.johndcook.com\/blog\/wp-json\/wp\/v2\/posts\/248041","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/www.johndcook.com\/blog\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/www.johndcook.com\/blog\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/www.johndcook.com\/blog\/wp-json\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"https:\/\/www.johndcook.com\/blog\/wp-json\/wp\/v2\/comments?post=248041"}],"version-history":[{"count":1,"href":"https:\/\/www.johndcook.com\/blog\/wp-json\/wp\/v2\/posts\/248041\/revisions"}],"predecessor-version":[{"id":248043,"href":"https:\/\/www.johndcook.com\/blog\/wp-json\/wp\/v2\/posts\/248041\/revisions\/248043"}],"wp:attachment":[{"href":"https:\/\/www.johndcook.com\/blog\/wp-json\/wp\/v2\/media?parent=248041"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.johndcook.com\/blog\/wp-json\/wp\/v2\/categories?post=248041"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.johndcook.com\/blog\/wp-json\/wp\/v2\/tags?post=248041"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}},{"id":248036,"date":"2026-10-14T05:00:22","date_gmt":"2026-10-14T10:00:22","guid":{"rendered":"https:\/\/www.johndcook.com\/blog\/?p=248036"},"modified":"2026-10-14T05:03:15","modified_gmt":"2026-10-14T10:03:15","slug":"poe-bessel-and-drones","status":"publish","type":"post","link":"https:\/\/www.johndcook.com\/blog\/2026\/10\/14\/poe-bessel-and-drones\/","title":{"rendered":"Poe, Bessel, and Drones"},"content":{"rendered":"Related posts<\/h2>\n
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\n\t\t\tHome<\/a>\n\t\t<\/span>\u00bb<\/span>\n\t\t\tMath<\/a>\n\t\t<\/span>\u00bb<\/span>\n\t\t\tPoe, Bessel, and Drones\n\t\t<\/span><\/div>","aioseo_breadcrumb_json":[{"label":"Home","link":"https:\/\/www.johndcook.com\/blog\/"},{"label":"Math","link":"https:\/\/www.johndcook.com\/blog\/category\/math\/"},{"label":"Poe, Bessel, and Drones","link":"https:\/\/www.johndcook.com\/blog\/2026\/10\/14\/poe-bessel-and-drones\/"}],"_links":{"self":[{"href":"https:\/\/www.johndcook.com\/blog\/wp-json\/wp\/v2\/posts\/248036","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/www.johndcook.com\/blog\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/www.johndcook.com\/blog\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/www.johndcook.com\/blog\/wp-json\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"https:\/\/www.johndcook.com\/blog\/wp-json\/wp\/v2\/comments?post=248036"}],"version-history":[{"count":4,"href":"https:\/\/www.johndcook.com\/blog\/wp-json\/wp\/v2\/posts\/248036\/revisions"}],"predecessor-version":[{"id":248040,"href":"https:\/\/www.johndcook.com\/blog\/wp-json\/wp\/v2\/posts\/248036\/revisions\/248040"}],"wp:attachment":[{"href":"https:\/\/www.johndcook.com\/blog\/wp-json\/wp\/v2\/media?parent=248036"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.johndcook.com\/blog\/wp-json\/wp\/v2\/categories?post=248036"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.johndcook.com\/blog\/wp-json\/wp\/v2\/tags?post=248036"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}},{"id":248028,"date":"2026-10-13T08:45:18","date_gmt":"2026-10-13T13:45:18","guid":{"rendered":"https:\/\/www.johndcook.com\/blog\/?p=248028"},"modified":"2026-10-13T08:45:18","modified_gmt":"2026-10-13T13:45:18","slug":"harmonic-triangle","status":"publish","type":"post","link":"https:\/\/www.johndcook.com\/blog\/2026\/10\/13\/harmonic-triangle\/","title":{"rendered":"Harmonic triangle"},"content":{"rendered":" 1\r\n 1\/2 1\/2\r\n 1\/3 1\/6 1\/3\r\n 1\/4 1\/12 1\/12 1\/4\r\n 1\/5 1\/20 1\/30 1\/20 1\/5\r\n \u2026\r\n<\/pre>\n
![\"L(r,c)](\"https:\/\/www.johndcook.com\/liebniz_harmonic_triangle1.svg\")
<\/p>\n![\"L(r,c)](\"https:\/\/www.johndcook.com\/liebniz_harmonic_triangle2.svg\")
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\n\t\t\tHome<\/a>\n\t\t<\/span>\u00bb<\/span>\n\t\t\tMath<\/a>\n\t\t<\/span>\u00bb<\/span>\n\t\t\tHarmonic triangle\n\t\t<\/span><\/div>","aioseo_breadcrumb_json":[{"label":"Home","link":"https:\/\/www.johndcook.com\/blog\/"},{"label":"Math","link":"https:\/\/www.johndcook.com\/blog\/category\/math\/"},{"label":"Harmonic triangle","link":"https:\/\/www.johndcook.com\/blog\/2026\/10\/13\/harmonic-triangle\/"}],"_links":{"self":[{"href":"https:\/\/www.johndcook.com\/blog\/wp-json\/wp\/v2\/posts\/248028","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/www.johndcook.com\/blog\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/www.johndcook.com\/blog\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/www.johndcook.com\/blog\/wp-json\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"https:\/\/www.johndcook.com\/blog\/wp-json\/wp\/v2\/comments?post=248028"}],"version-history":[{"count":5,"href":"https:\/\/www.johndcook.com\/blog\/wp-json\/wp\/v2\/posts\/248028\/revisions"}],"predecessor-version":[{"id":248033,"href":"https:\/\/www.johndcook.com\/blog\/wp-json\/wp\/v2\/posts\/248028\/revisions\/248033"}],"wp:attachment":[{"href":"https:\/\/www.johndcook.com\/blog\/wp-json\/wp\/v2\/media?parent=248028"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.johndcook.com\/blog\/wp-json\/wp\/v2\/categories?post=248028"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.johndcook.com\/blog\/wp-json\/wp\/v2\/tags?post=248028"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}},{"id":248025,"date":"2026-10-12T08:04:09","date_gmt":"2026-10-12T13:04:09","guid":{"rendered":"https:\/\/www.johndcook.com\/blog\/?p=248025"},"modified":"2026-10-12T08:04:09","modified_gmt":"2026-10-12T13:04:09","slug":"categorical-models-of-logics","status":"publish","type":"post","link":"https:\/\/www.johndcook.com\/blog\/2026\/10\/12\/categorical-models-of-logics\/","title":{"rendered":"Categorical models of logics"},"content":{"rendered":"![\"\\begin{tabular}{ll}](\"https:\/\/www.johndcook.com\/curry_howard_lambek.svg\")
<\/p>\n
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\n\t\t\tHome<\/a>\n\t\t<\/span>\u00bb<\/span>\n\t\t\tMath<\/a>\n\t\t<\/span>\u00bb<\/span>\n\t\t\tCategorical models of logics\n\t\t<\/span><\/div>","aioseo_breadcrumb_json":[{"label":"Home","link":"https:\/\/www.johndcook.com\/blog\/"},{"label":"Math","link":"https:\/\/www.johndcook.com\/blog\/category\/math\/"},{"label":"Categorical models of logics","link":"https:\/\/www.johndcook.com\/blog\/2026\/10\/12\/categorical-models-of-logics\/"}],"_links":{"self":[{"href":"https:\/\/www.johndcook.com\/blog\/wp-json\/wp\/v2\/posts\/248025","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/www.johndcook.com\/blog\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/www.johndcook.com\/blog\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/www.johndcook.com\/blog\/wp-json\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"https:\/\/www.johndcook.com\/blog\/wp-json\/wp\/v2\/comments?post=248025"}],"version-history":[{"count":2,"href":"https:\/\/www.johndcook.com\/blog\/wp-json\/wp\/v2\/posts\/248025\/revisions"}],"predecessor-version":[{"id":248027,"href":"https:\/\/www.johndcook.com\/blog\/wp-json\/wp\/v2\/posts\/248025\/revisions\/248027"}],"wp:attachment":[{"href":"https:\/\/www.johndcook.com\/blog\/wp-json\/wp\/v2\/media?parent=248025"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.johndcook.com\/blog\/wp-json\/wp\/v2\/categories?post=248025"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.johndcook.com\/blog\/wp-json\/wp\/v2\/tags?post=248025"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}},{"id":248022,"date":"2026-10-08T08:43:02","date_gmt":"2026-10-08T13:43:02","guid":{"rendered":"https:\/\/www.johndcook.com\/blog\/?p=248022"},"modified":"2026-10-08T08:43:02","modified_gmt":"2026-10-08T13:43:02","slug":"privacy-policies-and-modal-logic","status":"publish","type":"post","link":"https:\/\/www.johndcook.com\/blog\/2026\/10\/08\/privacy-policies-and-modal-logic\/","title":{"rendered":"Privacy policies and modal logic"},"content":{"rendered":"![\"box,](\"https:\/\/www.johndcook.com\/ltl_operators.png\")
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<\/p>\n![\"\\pi(x)](\"https:\/\/www.johndcook.com\/pnt_qrh.svg\")
\nIf RH holds, \u03b8 = \u00bd.<\/p>\n\n\t\t\tHome<\/a>\n\t\t<\/span>\u00bb<\/span>\n\t\t\tMath<\/a>\n\t\t<\/span>\u00bb<\/span>\n\t\t\tConsequences of progress toward the Riemann Hypothesis\n\t\t<\/span><\/div>","aioseo_breadcrumb_json":[{"label":"Home","link":"https:\/\/www.johndcook.com\/blog\/"},{"label":"Math","link":"https:\/\/www.johndcook.com\/blog\/category\/math\/"},{"label":"Consequences of progress toward the Riemann Hypothesis","link":"https:\/\/www.johndcook.com\/blog\/2026\/10\/07\/consequences-of-qrh\/"}],"_links":{"self":[{"href":"https:\/\/www.johndcook.com\/blog\/wp-json\/wp\/v2\/posts\/248019","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/www.johndcook.com\/blog\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/www.johndcook.com\/blog\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/www.johndcook.com\/blog\/wp-json\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"https:\/\/www.johndcook.com\/blog\/wp-json\/wp\/v2\/comments?post=248019"}],"version-history":[{"count":1,"href":"https:\/\/www.johndcook.com\/blog\/wp-json\/wp\/v2\/posts\/248019\/revisions"}],"predecessor-version":[{"id":248020,"href":"https:\/\/www.johndcook.com\/blog\/wp-json\/wp\/v2\/posts\/248019\/revisions\/248020"}],"wp:attachment":[{"href":"https:\/\/www.johndcook.com\/blog\/wp-json\/wp\/v2\/media?parent=248019"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.johndcook.com\/blog\/wp-json\/wp\/v2\/categories?post=248019"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.johndcook.com\/blog\/wp-json\/wp\/v2\/tags?post=248019"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}},{"id":248015,"date":"2026-10-07T16:41:47","date_gmt":"2026-10-07T21:41:47","guid":{"rendered":"https:\/\/www.johndcook.com\/blog\/?p=248015"},"modified":"2026-10-07T18:35:18","modified_gmt":"2026-10-07T23:35:18","slug":"faster-fourier-transform","status":"publish","type":"post","link":"https:\/\/www.johndcook.com\/blog\/2026\/10\/07\/faster-fourier-transform\/","title":{"rendered":"Faster Fourier Transform"},"content":{"rendered":"Related posts<\/h2>\n
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<\/p>\n