Happy Pythagoreas Day

8/15/17

Happy Pythagoreas Day! What does this mean and what is a Pythagoreas? Pythagoreas was a Greek philosopher and mathematician. Today is a "Pythagoreas Day" because 8²+15² = 17². In general a Pythagorean triple is a collection of numbers a, b, and c, such that a²+b² = c², the "Pythagorean Theorem".

How many Pythagoreas days are there moving forward? My first inclination was to use Excel since it was right in front of me. I made a column of all the days for a year in the form m/dd/yy, for example 4/15/17. Then I used the day(date), month(date), and year(date) functions, to get my a, b, and c. After that, I calculated a²+b² and noted if it equalled c² or not. For that I had a simple IF statement that returned a 1 if equality, and a blank otherwise.

Turns out, there's just today's date for the rest of 2017. In 2018, none. In 2019, none. In 2020 one, on 12/16/2020. In 2021, none. In 2022, none. In 2023, none. In 2024, none. In 2025, one, on 7/24/2025. In 2026, one, on 10/24/2026.

In fact there are no more. And there cannot be any Pythagorean days after 2033 or 2034 (except in the next millennium like 2117, etc., since this would again be year 17 like in 2017), since max(month)=12, max(day)=31, and 12²+31² = 1105, and sqrt(1105) = 33.24.

What about a complete list, for years 00 through 99 (for example, 2000 through 2099)? The exhaustive list is:

• 3/4/05
• 4/3/05
• 6/8/10
• 8/6/10
• 5/12/13
• 12/5/13
• 9/12/15
• 12/9/15
• 8/15/17
• 12/16/20
• 7/24/25
• 10/24/26

So now you know. Thanks for reading.