Walking wrap up, 2023

My goal in 2023 was to walk less than in 2022, since my 2022 mileage (3,024 miles) required an obsessiveness I didn't want to repeat. 

Goal achieved! By 4pm on New Year's Eve, I had reached 2,813 recorded miles, which is indeed less than 3,024, but still enough to say "I walked a lot."

Then I thought, "but wouldn't it be cool to have a mile total that's divisible by 3? And by 2?" So at 10pm, after dinner with friends yet before sitting around a fire pit with them waiting for midnight, S and I excused ourselves to walk a tidy figure 8 around our block and an adjacent block, and I ended the year with 2,814 miles.

2,814 is indeed divisible by 3 and by 2--and by 7 and 67. Wouldja look at that--it's the product of four distinct prime numbers! What fun! I wonder how many of those are there in the vicinity of 2,800?¹ 

Had I been paying more attention, I might have happily stopped walking at 2,813 miles, since 2,813 is a semiprime--the product of just two prime numbers (2813 = 29 * 97). By definition, those prime numbers don't have to be distinct--so the squares of prime numbers are also semiprimes.

What about products of three prime numbers? A positive integer having three distinct prime factors is called a sphenic number (by definition, no squares are allowed). Once you go from three to four prime factors, you enter the territory of k-almost primes, where k is the number of prime factors. Squares are allowed, so 2,814, with its four distinct prime factors, is 4-almost prime, but so is 16 (16 = 2 * 2 * 2 * 2). I have mixed feelings about this.

I think for 2024 I should aim for my mileage total to be a sphenic number.³

I asked ChatGPT for a list of sphenic numbers between 2,800 and 3,200, and learned that ChatGPT is as incompetent at listing sphenic numbers as it is at reading roadmaps. This both gives me hope and stokes fear: AI will never replace people, but reliance on AI will probably end civilization as we know it.

To the sphenic list rescue: The On-Line Encyclopedia of Integer Sequences! Had I stopped at 2,810 miles last year, I would have nailed it.

For those of you dear readers who are still hanging in there, here's one last, exciting tidbit tangentially related to the number 2,814. I find it delightful how easy it is to see that 7 is a factor of 2,814, compared to e.g. the less immediately obviously 7-factorable number 2,226 (also a 4-almost-prime with four distinct factors). So I asked Google if there's an easy way to quickly determine whether a number is divisible by 7, AND THERE IS!² Oh my goodness, that extra figure-8 mile at 10pm on New Year's Eve was worthwhile!

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Here are the 2023 walking stats I'm most proud of:

Longest single walk: 26.81 miles. I finally walked a marathon! Twice, actually, on January 15 and January 27. The first was my annual American Tobacco Trail long walk. The second was a repeat for a friend who had wanted to go the first time.

Number of weeks topping 100 miles: 4 (2 in June, 1 in July, 1 in November)

Most miles in a single week: 108 (July)

Number of months topping 300 miles: 2 (June and July)

Most miles in a single month: 361 (June)

Favorite multi-day walk: In June, S and I went out the front door of the familial cottage in southern Bavaria and spent 11 days hiking to Italy.

Favorite photo: Hiking up from Hallerangerhaus to the Lafatscher saddle on Day 6 of that multi-day hike: 

Nothing else new to report. All of the walking philosophy from the preceding walking-intense years still stands.

What does 2024 hold in store? Another American Tobacco Trail marathon. Another car-free February. Hopefully completing our multi-day trek southward across the Alps, from our 2023 end point to where the mountains yield to flat terrain. 

Here's to great walking in 2024!

¹Between 2,800 (my minimum mileage goal last year) and 2,850, there are three almost-primes that have four distinct prime factors: 2,805 (5 * 3 * 11 * 17); 2,814 (2 * 3 * 7 * 67); and 2,838 (2 * 3 * 11 * 43).

²To quickly determine whether a number is divisible by 7: 

(1) remove the last digit of the number; 

(2) double that digit; then 

(3) subtract that from the truncated original number. 

If the result is divisible by 7, then the original number is divisible by 7. 

For example, 2,226:

(1) remove the 6;

(2) double it (6 * 2 = 12); then

(3) subtract 12 from the truncated number (222 - 12 = 210). 

210 is divisible by 7, so 2,226 is divisible by 7. 

My mathematician dad liked general proofs over anecdotal examples, so I'm including this link for him, wherein a general proof is offered.

(And about the above example, what fun: 210 = 2 * 3 * 5 * 7 -- the product of the four smallest prime numbers. 2,226 is likewise the product of four distinct primes -- 2 * 3 * 7 * 59. Clearly 2023 has given me food for thought for my 2024 walks.)

³I was interested in the etymology of the word sphenic, so I looked it up in the etymology bible, a.k.a. the Oxford English Dictionary--and there's no entry for the word at all. Tsk. Wikipedia to the rescue: the word is from Ancient Greek σφήν (sphḗn, “wedge”), + -ic.