The Almost Perfect Discovery

This is the story of my greatest mathematical “discovery” to-date.

In his entertaining video on leap years (link below), Numberphile contributor Matt Parker mentions towards the end of the video that the next “power year” will be in 2048.

That is to say, the next year that is a power of 2 will be 2048:  2¹¹ = 2048. The last one was the year 1024 (2¹⁰) and the next one won’t be until way in the future – the year 4096 (2¹²), to be exact.

A little quick math revealed that I will turn 84 in April of 2048 (assuming I’m still around to annoy people), and I thought it would be a mildly interesting blog post if I could associate 84 and 2048 somehow, preferably mathematically.

I spent about 15 minutes tackling this from different angles. They’re both divisible by 2, and their greatest common factor is (only) 4, which is surprising considering they both have 12 factors (including themselves):

Factors of 84: 1, 2, 3, 4, 6, 7, 12, 14, 21, 28, 42, 84

Factors of 2048: 1, 2, 4, 8, 16, 32, 64, 128, 256, 512, 1024, 2048

No luck comparing prime factors, either:

Prime factors of 84: 2X2X3X7

Prime factors of 84: 2¹¹ (2x2x2x2x2x2x2x2x2x2x2)

Then I remembered Matt’s video on compatible pairs (also linked below), and I applied that system to these two numbers. Nothing special about 84, other than it is an abundant number, which is a number whose proper divisors (there’s another term for this, which I didn’t know about, and which we’ll get to shortly) sum to a number greater than itself. In this case, 84’s abundant (or excessive) number is 140 (1+2+3+4+6+7+12+14+21+28+42).

2048 was a different story. It immediately struck me that the sum of the proper divisors of 2048 came out to be 2047 – one less than the number itself. “How odd!”, I thought to myself, followed by “I wonder how prevalent this is?”.

I quickly threw together a spreadsheet listing the first 100 natural numbers along, with all of their factors. After the quick calculation of adding up all factors except the last one (the number itself), a pattern emerged. This appeared to be happening with every power of 2!

(At this point, I have to say that, had I been paying attention, what was happening would have been readily apparent. My number manipulation also revealed another pattern, one where the sum of the proper divisors of a number equaled the number itself. There are two numbers that this happens with in the set of natural numbers to 100…6 and 28. It honestly didn’t register with me what was happening…)

Here is the pattern that I discovered:

Sure enough, the powers of 2 less than 100, i.e. 2⁰ through 2⁶, followed this pattern, and they were the only numbers that were doing so! I did the same calculation for 2⁷ through 2¹², and:

…the pattern held! I was super excited, thinking I’d found some hitherto unknown, or at least unexplored, feature of the powers of 2.

So excited, in fact, that I sent Matt Parker and email about it, detailing my “discovery” much as I’ve done above.

At this point, I closed everything down and went about my day. I went to listen to a friend play guitar at a consignment market, grabbed something to eat, washed my car. Went home and began surfing the interwebs a little bit.

That’s when I came across the Wikipedia page for Powers of 2 (yes, there is one, unsurprisingly – I’ve linked it below as well).

In short order, I found that:

-Summing the proper divisors has a technical name, the aliquot sum. I couldn’t find a proper etymology of the term, but this function has been known for thousands of years. Well, of course it has.

-What I “discovered” is known as the set of “almost perfect numbers” – almost perfect because their aliquot sum is one less than the number itself.

Which brings us to 6 and 28, who’s aliquot sum equals the number itself. As many videos I’ve watched regarding Fermat and Mersenne prime numbers, I should have recognized that these are the first two perfect numbers, so named because of the very quality I “discovered” in my spreadsheet above.

After sending Matt a follow up “never mind!” email, and texting everyone else that I was, in fact, not about to become math-famous, I shut down my computer for the day and enjoyed the unseasonable beautiful weather we were experiencing.

I still haven’t found a blog-worthy, interesting connection between 84 and 2048 – I’m sure they’re out there, I just need to dig a little deeper. Like I should have done initially when I made this amazing “discovery!”

Better luck next time, I guess!

Links:

Matt Parker’s Leap Years video on Numberfile

Matt’s Compatible Pairs video

Powers of 2 Wikipedia page