Here we are at Carnival #128!

In typical fashion, let’s first talk about the number 128, from fascinating number properties to fascinating numerology.

I have to state the obvious — 128 is a power of 2! (and 2!) Specifically \(2^{7} = 128\)

Ok, what else do we know about 128? Well, let’s check the Twitterverse.

@xdoublestar gives this

128 is a power of 2 whose decimal digits are all powers of 2 (if we count 1).

Are there any others? I suspect there are no more, but this seems difficult to prove.

— x** ⚔️🛡️ (@xdoublestar) May 25, 2019

This led to a nice thread with @jonathanavt

Single-digit ones. Other than that, it seems statistically unlikely, but I’m not going to work on it now.

— Jonathan👣🚲 (@jonathanavt) May 25, 2019

@xdoublestar gives another one to mull over

Here’s another one that depends less on numerology, if a bit contrived: 128 is a power of 2 that is the sum of a perfect number and a perfect square greater than the perfect number.

128 = 28 + 100

Another example is 64 = 28 + 36.

Can you find others?— x** ⚔️🛡️ (@xdoublestar) May 25, 2019

This leads to more inquiry!

Inspired by a convo with @shahlock: what powers of 2 have a decimal representation that consists entirely of even digits? 2, 4, 8, 64, 2048 are the first few; are there more?

Just like the previous problem I suspect this one is difficult

— x** ⚔️🛡️ (@xdoublestar) May 26, 2019

Since we’re on a power of 2 kick. @cardcolm gives this dad joke.

Old dad joke the owner of a famous Segovian restauant proudly told me in 2003:

Why is Ireland the richest country in the world?

Cause its capital is always doublin’

(works better when heard, not read) https://t.co/zgmhOj6mNt

— Card Colm Mulcahy (@CardColm) May 23, 2019

and speaking of jokes, here’s a doozy from @aap03102 …

And 4096 looks cute too, doesn’t it? (Geeky pun alert: 4096 looks cute 2 ᵈᵒᶻᵉⁿ it)

— Chris Smith (@aap03102) May 25, 2019

But back to 128. Here’s some more numerology.

- 1283 is the first prime number that contains “128”.
- Interestingly, 10000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000283 is a 128-digit prime, but notice that if you get rid of the zeros, we have 1283!!! WHAT!!

It gets better, check out these “1283” primes

- 12 digits: 100000000283
- 14 digits: 10000000000283
- 19 digits: 1000000000000000283
- 28 digits: 1000000000000000000000000283
- 34 digits: 1000000000000000000000000000000283
- 124 digits: 1000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000283
- 128 digits: 10000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000283
- 138 digits: 100000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000283
- 741 digits: 100000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000283
- 752 digits: 10000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000283
- 754 digits: 1000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000283

Finally, consider this. $$2^{168} = 374144419156711147060143317175368453031918731001856$$ what do you notice missing about this power of 2? There’s no 2!! I haven’t been able to find a larger power of 2 that is missing at least one digit.

### Ok, enough of this screwball-ery. Where is the content!

Our friend @aap03102 shares this video on doubling and noodle making.

Moving on, @DavidButlerUofA sends in three fun and interesting articles.

The first is about why 65536 is his favorite power of 2.

Next, he’s done some wonderful research on likeable primes! This is a great article with some fun conclusions! Before you read it, take a guess for yourself of what would make one prime number more likeable than another prime number.

Ever hear of the Sausage Stacking Theorem? Well now you have. This is a good lesson about division.

The natural pivot from division is multiplication. Here we have a write-up of a multiplication game from @findthefactors that’s worth checking out.

Leaping incredibly ahead from multiplication / division but still somehow staying in elementary mathland, we have an exploration of Martin Gardner’s hexapawn game from @benjamin_leis.

Next, from @letsplaymath, we have a reminder that about teaching math in a senseful way.

### Polite, Funny, And More Types Of Numbers

If there’s anything I like, it’s making up objects, naming them, and studying them. Again from @letsplaymath, we have an activity involving “polite numbers“.

Have you read my take on funny numbers?

Twin, cousin, and sexy primes are discussed in this post sent by @TopCat4647.

Do you know what semi r-primes are?

We have an exploration, authored by @evelynjlamb, of the Edward-Mullin sequence, a curious prime number sequence. There’s also a request to name a special (non-mathematical) temporal occurence … (no spoilers from me!). H/T to @icecolbeveridge for sending the article.

Speaking of primes and how this article started …

### Twitter Primes

In all the goofiness around putting together this carnival, I received a lot of fun stuff from the Twitter-sphere as you saw above. But the conversation turned to primes as well.

Check out this thread with what I will call a “Twitter Prime”, that is a prime number that is 280-digits long — 280 characters is the maximum character limit of a tweet.

1000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000037

— M Shah (@shahlock) May 25, 2019

@Very_Stable_G shows us a Golden Twitter Prime!

1618033988749894848204586834365638117720309179805762862135448622705260462818902449707207204189391137484754088075386891752126633862223536931793180060766726354433389086595939582905638322661319928290267880675208766892501711696207032221043216269548626296313614438149758701220340805887

— Dr. Sylow P. Subgroup (@Very_Stable_G) May 25, 2019

The always mathematical @daveinstpaul shows us the largest Twitter Prime.

9999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999461

— David Radcliffe (@daveinstpaul) May 25, 2019

and @SFrancismath pushes the question of Twitter Primes, even deeper

Both the prime M Shah and you posted use 4 distinct digits. I wonder if there is a 280 digit prime number that uses fewer?

— Steven Francis (@SFrancismath) May 26, 2019

Finally, today is 27/05/2019, which when written without the forward slashes is 27052019 — a prime!

I hope you enjoyed this Carnival. The next one will be hosted by @mathhombre at his website mathhombre.blogspot.com.

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