Sept 16, 2025 was Pythagorean Day

Computational Complexity 2025-10-21

Several people emailed me that September 16, 2025---written as 9-16-25 in the US---represents the integer side lengths of a right triangle.

9-16-25 is the only such triple that is also a valid date. This kind of mathematical alignment only happens once every 100 years. The next occurrence will be September 16, 2125. Since this is such a rare event, let's explore some more math-themed dates. 1) Pythagorean Triples That Work as Future Dates Note that 9-16-25 is not a Pythagorean triple; however, 3-4-5 is. Here are some future dates that are both Pythagorean triples and valid calendar dates: March 4, 2105 is 3-4-5 May 12, 2113 is 5-12-13 June 8, 2110 is 6-8-10 July 24, 2125 is 7-24-25 (Darn---July 24, 2025 was recent and I missed it!) August 15, 2117 is 8-15-17 I think that's it. Recall that we need the month to be in \(\{1,\ldots,12\}\) and the day to be in \(\{1,\ldots,31\}\) with some exceptions: Thirty days has September, April, June, and November All the rest have thirty-one, Excepting February, fun! And that has twenty-eight days clear And twenty-nine in a Leap Year There are 24 versions of this poem at a website which is here. 2) Why Didn't Anyone Email Me About Earlier Dates? I wonder why nobody emailed me on, say, March 4, 2005 (3-4-5). That's a Pythagorean triple, but maybe it just looked like three consecutive numbers. Oh well. And what about May 12, 2013 (5-12-13)? That's a really cool Pythagorean triple. Oh well. 3) Other Math-Related Dates Using Month, Day, and Year. So dates like Pi Day don't count---we want the full date to be interesting mathematically. Side note---I looked up how Pi Day is referred to and its Pi Day, not \(\pi\) day. Probably because not all typesetting systems can easily produce \(\pi\).

a) Square days:

Dates where the full 8-digit number (MMDDYYYY) is a perfect square.

a1) September 27, 2025 is 9-27-2025 and \(9272025=3045^2\).

Bonus: if you write it as 27-09-2025 then: \(27092025=5205^2\).

a2) Feb 2, 2084 is 2-02-2084 and \(2022084=1422^2\).

b) Palindrome days

b1) March 11, 2030 is 03-11-30 might be the next one.

b2) I was hoping that Feb 2, 2022 (2-2-22) would be a Tuesday (Twosday) but alas, it was not. I asked ChatGPT what is the next year that ends with 22 where Feb 2 is a Tuesday. It gave me incorrect answers four times. When I pointed this out it thanked me for checking its work and then gave me a later incorrect answer. It then gave me a python program that I could run to find out myself. I found out that between the years 1622 to 9922, only looking at years ending with 22, the following pattern happens:

Feb 2, 1622 is a Wednesday

Feb 2, 1722 is a Monday

Feb 2, 1822 is a Saturday

Feb 2, 1922 is a Thursday

Feb 2, 2022 is a Wednesday

Feb 2, 2122 is a Monday

Feb 2, 2222 is a Saturday

Feb 2, 2322 is a Thursday

This pattern repeats as far as I computed, which was 9922.

I started in 1622 since the Gregorian calendar started in 1582. I stopped in 9922 because of numerical Python issues. I do not know if the pattern goes on forever, or if one of the leap year exceptions will cause that pattern to change. Recall that

Every year divisible by 4 is a leap year.

EXCEPT if the year is divided by 100 but not 400 then its not a leap year. I have not found any other exceptions on the web but there may still be some.

b3) I was hoping Feb 22, XX22 (2-22-22) was sometimes a Tuesday. Here are all the years after 1622, ending in 22, before 10,000, where Feb 22 is a Tuesday: 2022, 2422, 2822, 3222,...,9622 and I assume in general (2022+400x). Again, the pattern might not hold if there are leap year exceptions I don't know about.

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That is the end of my post. A bonus for my readers: a mini meta post. Long time reader and fellow SUNY Stony brook math major David Marcus (he was class of 1979, I was class of 1980) has been proofreading some of my posts before I post them. Lance suggested I used chatty instead. Instead of using chatty instead, I used chatty and then used David. David still found mistakes. I give an example here by pointing to all three versions:

Original Version: here

After ChatGPT: here

After David: here (this is not the final version since I read it and made some minor changes)

I may at some later point look at several ORIG/AFTER CHAT/AFTER DAVID and see what errors chatty is not catching.