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на картинке сверху — тождества¹ из заметки С.Маркелова в Мат. просвещении, и там предлагается придумать обобщения

¹ там только есть опечатка… найдите

программа в комментариях — говорит, суммы каких косинусов надо взять для произвольного p вида 3k+1, а также какому кубическому уравнению они удовлетворяют (и на всякий случай численно проверяет, удовлетворяют ли)

(upd) а также находит формулу для суммы S кубических корней из этих сумму косинусов, шоб было совсем как в заметке


p: 13
primitive root: 2
partition of cosines: [3, 11] [7, 9] [1, 5]
values of trigsums: -0.136945 -0.688601 1.325547
cubic polynomial: 8t³-4t²-8t-1
P(trigsums): -0.0 0.0 0.0
S³ = (3³√-13+7)/2



p: 73
primitive root: 5
partition of cosines: [13, 19, 25, 29, 31, 39, 53, 55, 57, 59, 67, 71] [1, 3, 7, 9
, 17, 21, 27, 43, 49, 51, 63, 65] [5, 11, 15, 23, 33, 35, 37, 41, 45, 47, 61, 69]
values of trigsums: -2.475085 2.40906 0.566026
cubic polynomial: 8t³-4t²-48t+27
P(trigsums): 0.0 0.0 -0.0
S³ = (3³√219-17)/2


теорема Рамануджана о том, как посчитать сумму кубических корней из корней данного кубического уравнения, обсуждается например в Кванте
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на картинке сверху — тождества¹ из заметки С.Маркелова в Мат. просвещении, и там предлагается придумать обобщения

¹ там только есть опечатка… найдите

программа в комментариях — говорит, суммы каких косинусов надо взять для произвольного p вида 3k+1, а также какому кубическому уравнению они удовлетворяют (и на всякий случай численно проверяет, удовлетворяют ли)

(upd) а также находит формулу для суммы S кубических корней из этих сумму косинусов, шоб было совсем как в заметке


p: 13
primitive root: 2
partition of cosines: [3, 11] [7, 9] [1, 5]
values of trigsums: -0.136945 -0.688601 1.325547
cubic polynomial: 8t³-4t²-8t-1
P(trigsums): -0.0 0.0 0.0
S³ = (3³√-13+7)/2



p: 73
primitive root: 5
partition of cosines: [13, 19, 25, 29, 31, 39, 53, 55, 57, 59, 67, 71] [1, 3, 7, 9
, 17, 21, 27, 43, 49, 51, 63, 65] [5, 11, 15, 23, 33, 35, 37, 41, 45, 47, 61, 69]
values of trigsums: -2.475085 2.40906 0.566026
cubic polynomial: 8t³-4t²-48t+27
P(trigsums): 0.0 0.0 -0.0
S³ = (3³√219-17)/2


теорема Рамануджана о том, как посчитать сумму кубических корней из корней данного кубического уравнения, обсуждается например в Кванте

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In February 2014, the Ukrainian people ousted pro-Russian president Viktor Yanukovych, prompting Russia to invade and annex the Crimean peninsula. By the start of April, Pavel Durov had given his notice, with TechCrunch saying at the time that the CEO had resisted pressure to suppress pages criticizing the Russian government. One thing that Telegram now offers to all users is the ability to “disappear” messages or set remote deletion deadlines. That enables users to have much more control over how long people can access what you’re sending them. Given that Russian law enforcement officials are reportedly (via Insider) stopping people in the street and demanding to read their text messages, this could be vital to protect individuals from reprisals. "The argument from Telegram is, 'You should trust us because we tell you that we're trustworthy,'" Maréchal said. "It's really in the eye of the beholder whether that's something you want to buy into." Right now the digital security needs of Russians and Ukrainians are very different, and they lead to very different caveats about how to mitigate the risks associated with using Telegram. For Ukrainians in Ukraine, whose physical safety is at risk because they are in a war zone, digital security is probably not their highest priority. They may value access to news and communication with their loved ones over making sure that all of their communications are encrypted in such a manner that they are indecipherable to Telegram, its employees, or governments with court orders. "We're seeing really dramatic moves, and it's all really tied to Ukraine right now, and in a secondary way, in terms of interest rates," Octavio Marenzi, CEO of Opimas, told Yahoo Finance Live on Thursday. "This war in Ukraine is going to give the Fed the ammunition, the cover that it needs, to not raise interest rates too quickly. And I think Jay Powell is a very tepid sort of inflation fighter and he's not going to do as much as he needs to do to get that under control. And this seems like an excuse to kick the can further down the road still and not do too much too soon."
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на картинке сверху — тождества¹ из заметки С.Маркелова в Мат. просвещении

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