As a follow-up to yesterday’s post, here are some answers and steps toward figuring out the scarcity (or frequency) of Pythagorean triple days (like yesterday 8/15/17) in a century:
I will mention I did not get the answer right away but needed some prompting (from a math-teacher colleague on Twitter), so I’ll take you through my thought processes.
- I have certain primitive Pythagorean triples memorized (like 9/40/41) — perhaps from several years of teaching geometry, or from studying number theory in college where integer (or rational) values for algebraic relations are important (cf. Euclid’s formula for Pythagorean triples, Diophantine Equations, & Fermat’s Last Theorem), or perhaps even a few recalled from taking high school geometry. In my head I tried these, saw several had occurred already in the past (5/12/13), while others were not interpretable as months days and years, like 9/40/41 (since there are max 31 days in a month). To go beyond the primitive triples, I tried multiplying each times 2, times 3, etc. In this way I discovered yet to come this century: 10/24/26 (twice 5/12/13) and 12/16/20 (quadruple 3/4/5).
- I thought I had them all, but had forgotten some primitive triples, like 7/24/25! At this point, referring to this list of primitive Pythagorean triples, I tried to systematically enumerate all with a≤12, then go through multiples of those until all values were larger than 12 (and therefore none could fill the month slot). I did this in Excel, color coding as blue dates already past, yellow yet to come, and red invalid date formats:
- In creating that chart I did mentally transpose months and dates to see for example, that 12/5/13 could also work as a Pythagorean triple date in addition to 5/12/13. But this only yielded more in the past so I didn’t bother to write them out. My fatal flaw, though, was not permutating with the years also, so I neglected to find two more future dates: 7/25/24 and 10/26/24. After realizing this, I did add more to my Excel chart, going through each permutation that yielded a valid date in rows below the main ones:
- I do believe this is now a complete list of past, present, and future Pythagorean triple days, written month/day/year. By my count there will be a total of 28 this century, 23 now past and 5 still in the future. Let me know if I missed any!