261–269 of 269 posts
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Thank you for the Putnam; I did not know about it. For anyone else that did not understand this reference; https://news.ycombinator.com/item?id=35015 . Legendary.
That's an epic thread. Thank you very much for sharing it :) And now, looking back 17 years later, I'd say he succeeded. It's the tarsnap founder.
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4 and 5 too
How so? I'm too dumb to see it.
You can prove it by induction; just expand (n(n+1)/2)² – (n(n-1)/2)², the result is n³.
Happy new year as well, even though it already started for me on the 21. of December on Solstice, when the days were starting to get longer. That is a real turning point in the natural cycle and I like to follow natural cycles where possible. 1.1. is just arbitary numbers changing to me, but for the majority it means something, which also makes it real and I can appreciate the intent and celebrate anyway.. (And I did…
How can I be where you are
Otherwise check out ecstatic dance for a similar vibe. (If you were refering to the wild drugfree party)
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Some more (thanks to chatgpt-o1) 6) sum of the first 45 odd numbers: 1+3+5+...+89 7) is a Harshad number: https://en.m.wikipedia.org/wiki/Harshad_number
6 is kind of cheating. It's a restatement of 45^2. 3^2 is the sum of the first three odd numbers. 4^2 is the sum of the first four odd numbers. 5^2 is the sum of the first five odd numbers. Edit: sorry, don't mean to be a pill.
Actually, property 5) trivially implies 1) but also 2), as `(1+2+...+n)² = n²(n+1)²/4` and either n or n+1 must be divisible by 2 hence one of the squares divisible by 4 hence it is a product of squares. But also property 4) as `(1+2+...+n)² = 1³+2³+...+n³` (easy to show by induction).
45^2 = 2025 Happy perfect square year, everyone. The previous one was 1936 and the next one will be 2116.
2025: 1) is a square: 45² 2) is the product of two squares: 9² x 5² 3) is the sum of 3-squares: 40²+ 20²+5² 4) is the sum of cubes of all the single digits: 1³+2³+3³+4³+5³+6³+7³+8³+9³
I created an account in 2022, but had been a regular reader much earlier. From Slashdot in 2002 to Digg in 2004 to Reddit in 2010 and now HN. Happy New Year (+30). Here's to a personally productive 2025!
I wish you a year covered in hot grits in Soviet Russia, being spied on by old people via email.