Category: Uncategorized

An Unnecessary Proof by Contradiction

Until Covid, I was meeting up with a friend once a week for some recreational abstract algebra. This was an early problem in the textbook we used:

Prove that for any positive integer m there exists a sequence of m consecutive positive integers that are all composite.

My gut instinct was to try to prove this by contradiction, using the pigeonhole principle somehow. But before I could work that out, I saw that it’s easily provable by construction.

Proof by construction

Let n = m!. Then, for all integers k such that 0 < k \le m, we know k \mid n. So we can say that n + k = k (\frac{n}{k} + 1). Since k \mid n, the quantity \frac{n}{k} + 1 is an integer. Therefore n+k has k as a factor, making it composite. So, the sequence

n+1, n+2, \dots, n+m

are all composite. \blacksquare

Pretty trivial. But I still wanted to know if my gut instinct was right; could this be proved by contradiction?

It turns out that it can, but it’s much more complicated, and took me a couple of days to figure out.

Proof by contradiction

Assume that every set of m consecutive positive integers contains at least one prime number.

The positive integers can be expressed as a union of sets

\Z^+ = A_0 \cup A_1 \cup A_2 \cup \dots 

where

A_n = \{nm+1, nm+2, \dots, nm+m\}

Since \vert A_n \vert = m, our assumption requires that every set A_n contains at least one prime number.

Let a_i(n) be the i^{th} element of A_n, where 1\le i \le m. Call i the “position” of a_i(n) within A_n. Since there is only one even prime number, the maximum number of positions in A_n at which there can be a prime is \lceil \frac{m}{2}\rceil.


Lemma: If A_j has a prime p in position i then A_{j+q} cannot have a prime in position i when q is a multiple of p.

Proof: Let A_j and A_{j+q} be sets with prime numbers in position i, and say that a_i(j) = p and a_i(j+q) = p^*. Then

p^* = p + (m-i) +m(q-1)+i

In the sum on the right hand side of this equation, the term (m-i) represents all the positions in A_j after p, the term m(q-1) is all the positions in all of the sets between A_j and A_{j+q}, and the term i is how many positions in A_{j+q} we must move through to reach p^*.

This equation simplifies to

p^* = p +qm

If q is a multiple of p, then for some k \in \Z^+

p^* =p+kpm=p(1+km)

which contradicts that p^* is prime.


Consider a set A_j such that a_i(j) is prime. Let a_i(j) = p_1.

By the above lemma, we know that a_i(j+p_1) is composite. By our assumption, the set A_{j+p_1} must contain a prime number. There remain \lceil \frac{m}{2}\rceil - 1 positions in A_{j+p_1} at which there can be a prime. Let a_{i^*}(j+p_1) be a prime, and call it p_2.

Now consider the set A_{j+(p_1p_2)}. This set has two positions, i and i^*, at which there can’t be a prime, leaving only \lceil \frac{m}{2}\rceil - 2 positions at which there can be a prime. By our assumption, one of these positions must have a prime, which we will call p_3.

Continue this pattern until we find the set A_{k+(p_1 p_2 \dots p_{\lceil \frac{m}{2}\rceil})}. This set contains zero positions where there can be a prime. But this contradicts our assumption, which requires every A_n to contain a prime number. Thus our assumption is false, and there exists an A_n (a set of m consecutive positive integers) with only composite elements. \blacksquare

If there’s a simpler way to prove this by contradiction, I’d like to know about it.

Armadillocon 2021 Schedule

Image

I’ll be in Austin for Armadillocon this weekend, Oct 15-17. Here’s my full schedule.

Reading: Sat. 11:00 am – 11:30 am, Conference Center

Ethics of Magic: Sat. 12:00 pm – 1:00 pm, Southpark A

Making your Novel more Drool-worthy for Filmmakers and Producers: Sat. 5:00 pm – 6:00 pm, Ballroom E

Pro-Level Writing Workshops: Sat. 8:00 pm – 9:00 pm, Ballroom E

Genre Fiction as Activism: Sat 9:00 pm – 10:00 pm, Ballroom E

A Contract Is A Wish Your Heart Makes

The Science Fiction Writers of America had a press conference today, calling out the Walt Disney Company for failure to pay a writer’s royalties. News that a quarter-trillion dollar company is doing something that screws over individual artists isn’t particularly shocking, but the nature of Disney’s justification in this case is wild.

Background: Lucasfilm and 20th Century Fox used to be big, independent media companies. Now they’re part of Disney. Since Disney bought them, they now own the Star Wars and Alien media franchises, each of which contain a teeny-tiny crumb made up of novels.

Alan Dean Foster wrote the original novelization of Star Wars, and a sequel novel, Splinter of the Minds Eye. He also wrote the novelizations of Alien, Aliens, and Alien 3. Apparently, all five of these books are still in print. But, while the books are still for sale, Alan Dean Foster is no longer receiving royalties. Disney has explained to the Science Fiction Writers of America that this is all fine:

Disney’s argument is that they have purchased the rights but not the obligations of the contract. In other words, they believe they have the right to publish work, but are not obligated to pay the writer no matter what the contract says.

SFWA president Mary robinette kowal

I thought I was already maximally cynical about the habits of megacorporations, but I’m actually astonished at how brazen this is. I mean, I’m sure there’s a deep, rich body of legal theory that I’m about to ignorantly stomp all over, but: isn’t the entire point of a contract that you don’t get to pick and choose after the fact which parts apply?

I’d be shocked if this went to court. I suspect #DisneyMustPay will result in Foster getting what he’s due, if only because I can’t imagine the legal fees being less than the royalties. If this did go to court, though, I would assume it was because Disney has purchased “the rights but not the obligations” of so many contracts that it was worth the expense. And Foster’s experience does make one wonder: has Disney been paying royalties to any of the authors whose books came along with those acquisitions?

Also, is this a fight that Disney really wants to win? They license a lot of IP. Do they really want it to be possible for someone to just buy up a licensee, then carry on selling Mickey Mouse-branded red wine or whatever without paying any more fees? Because, by their own argument, that seems like it should be a-okay. Would that really be better than just paying authors the rounding error on Disney’s books that is their royalties?

Things I wondered after viewing Star Wars: The Rise of Skywalker

Emperor Palpatine:

What the heck was his long term goal anyway? I mean, literally what did he want to achieve? If he wanted to rule the galaxy again, he already had the tools necessary: a fleet of mini Death Stars and a loyal military to use them. Why did he bother waiting for Kylo Ren to show up before making a move? Why did he leave the ships hanging out on his planet where they’re most vulnerable? What was he trying to accomplish militarily/politically?

If Palpatine’s Plan A for regaining youthful vigor was for Rey to strike him down in anger, thereby making herself susceptible to the dark side and subsequent bodily possession by Palpatine, why did he tell her about it? She was there to kill him. If he just kept his mouth shut, or goaded her into avenging her parents, he would have won. Why did this arch manipulator galactic puppetmaster turn all Republic serial villain just in time to make Rey hesitate?

After Palpatine discovers that he can use the power of the Force connection between Rey and Kylo to heal his own body, why does Palpatine act like Rey is a threat? Sure, he has a Plan B now that doesn’t involve stealing Rey’s body, but Plan A should still work, right? The only thing that’s changed is Palpatine is stronger and Rey is weaker. So when Palpatine is attacking the rebel fleet with Force lightening and Rey pops up to try to kill him again, why doesn’t he just keep blowing up x-wings and let her kill him if she wants to, as per the original plan?

Having inexplicably decided to fight Rey, why does Palpatine keep shooting Force lightening after it starts ricocheting and tearing his face off? I mean, I get why he does this in Episode III; he is manipulating Anakin while keeping Mace Windu from decapitating him. But here he’s not trying to manipulate anyone by looking helpless, and, until the last five minutes, getting decapitated by Rey is what he has desired the entire movie.

Luke Skywalker:

If Luke knew that Rey was related to Palpatine, why didn’t he tell her? I seem to recall Luke was displeased when his own Jedi Master concealed his relationship with Darth Vader.

Why, during the many years Luke apparently spent looking for the two MacGuffins that point the way to Planet Sith, did it never occur to him to look in Emperor Palpatine’s room on the Death Star? Shouldn’t combing through that wreckage been one of the first things the New Republic did after the Battle of Endor? Isn’t the Emperor’s sock drawer the first place someone searching for Sith artifacts would think to look?

Kylo Ren:

Speaking of the MacGuffins, if Luke couldn’t find one despite years of looking, how does Kylo Ren track one down so easily? He makes it seems effortless. (Apparently the official companion publications say that Kylo finds MacGuffin 1 on planet Mustafar, inside what used to be Darth Vader’s fortress. So Luke spent years hunting for these things, but didn’t look in Emperor Palpatine’s room or Darth Vader’s house.)

Why does Kylo Ren bring MacGuffin 1 to Rey when he’s trying to stop her from finding MacGuffin 2? His whole deal at that point is forcing her to rely on him to find the Emperor. He even destroys MacGuffin 2 so she can’t possibly use it. So why did he bring another one along with him?

This gets even weirder when you consider that he crashed his first ship earlier in the movie. Are we to understand that Kylo wired the MacGuffin into a ship, crashed that ship, retrieved the MacGuffin from the wreckage, wired it into a new ship, and then flew that ship straight to Rey?

(The two ships do look the same, so I suppose it’s possible that he just has, like, a garage of identical pointy black ships. But then why did he pick the one with a MacGuffin in it to chase Rey? At first I thought maybe he still needed it so he could take them both back to Planet Sith, but later it’s a major plot point that anyone who knows the way can fly there, no MacGuffin required.)

General Pryde:

When General Pryde is watching the rebels ride warthog-horses across the top of his spaceship to break his very important antenna, why does he not just tilt the ship and tip ‘em off?

Lando Calrissian:

Why was Lando on Pasaana, anyway?

How did General Lando recruit an enormous fleet of reinforcements, when all previous rebel distress calls for the past two movies have gone utterly ignored?

Finn:

How did Finn fail to notice an entire extra spaceship? Either the transport he watched Chewbacca get loaded into had already taken off while he was getting Rey and he missed it, or else the one that Rey accidentally blew up was the first to take off and Chewie’s ship was still just sitting there where Finn left it. Either way, how do you miss an extra spaceship in the flat open desert?

Sith Knife Maker:

How did they know the exact shape that the wreckage of the Death Star would make when viewed from a specific cliff with enough accuracy to make a knife with the same outline?

Dead Jedi:

Why wouldn’t they talk to Rey all the earlier times she was trying to get in touch with them? Seems like every Jedi who ever lived was willing to chat at the end of the movie. So what changed?

Rey:

Why symbolically lay Luke and Leia to rest on Tatooine of all places? Luke hated it there, and Leia’s experience of the place primarily consisted of being enslaved to a giant slug monster.

Other:

Why bother pretending to kill Chewbacca when you’re not going to allow the characters to respond to his supposed death before learning that he’s actually still alive?

What narrative purpose is served by taking us to Poe’s planet, introducing us to Poe’s old friends, blowing up Poe’s planet, then quickly revealing that all of Poe’s old friends miraculously survived?

Why tell the audience exactly how C3PO’s sacrifice is going to be reversed before he even makes it? Sure, C3PO thinks R2D2’s memory banks are notoriously unreliable, but the audience knows that’s not true, so it doesn’t even work as a misdirection.

Are there deleted scenes in which Rose gets to actually do anything?

That split-second lesbian kiss at the end lampshading the widely criticized lack of LGBTQ representation in the series, was that a Legend of Korra-type thing where the parent company literally forbade the creatives from doing more, or was that the extent of directorial efforts to address the issue?

Was the entire script reverse-engineered from a checklist of fan service moments?

Is this the most perfunctory event movie I’ve ever seen?

Armadillocon 2019 Schedule

If you’ll be in Austin this coming weekend, you can find me at the Omni Southpark hotel for Armadillocon. Here’s my schedule:

Friday, Aug. 2, 5:00 pm — Art of the Short Story — Ballroom F

Saturday, Aug. 3, 8:00 pm — What Sciences Haven’t Been Used in Science Fiction? — Southpark A

Sunday, Aug. 4, 11:30 am — Author Reading — Southpark B

More People Abused by Bryce Milligan Come Forward

There has been a great deal of local news coverage of Hailey’s initial accusation, as well as her many classmates who have come forward to corroborate her accounts of Bryce’s behavior. Now two more women have published their stories of Bryce Milligan physically assaulting them. Courtenay Martin shares an account that, disturbingly, indicates people in the San Antonio literary community knew about Bryce’s inclinations well before he was hired to teach high school students. Denise McVea shares an experience that is both a confirmation of Hailey’s account and yet another portrait of Bryce using professional contexts to make unwanted physical advances on women. (I notice a particular similarity in Denise’s story—Bryce, when challenged, explaining away his actions by saying weight issues had made his a sexless marriage—with my own family’s experience of Bryce attempting to excuse his relationship with Hailey by saying he was impotent due to alcoholism. When called out for his improprieties in private settings, he seems to have a repeated tactic of oversharing personal sexual difficulties in an attempt to garner sympathy.)

These additional accounts undermine the feeble attempts that Bryce and those few speaking in his support have made to dismiss Haley’s story as a fabrication. Taken together, these stories depict a pattern of behavior going back decades.

Cover reveal for Nueva Madre

Here’s what the Spanish edition of “The New Mother” is going to look like. It’s the work of Cecilia García, and I adore it. It’s the first time I’ve ever seen an image of Tess, and she looks so much like she did in my head. I love the press badge, and the little picture of identical GDS siblings on her phone, and her wavy hair, and I especially love the uncertain look on her face. The bustle and blur of the city rising behind her is perfect, suggestive of the complex interplay of social forces Tess tries to navigate as she moves through the story. And that bright, full moon dominating the human skyline and framing the main character is symbolic of the themes in ways I’m sure don’t require elaboration. I’m also very pleased to see not just my name on there, but also Arrate Hidalgo’s, without whom Nueva Madre would not exist.

This beautiful thing will be available from Editorial Cerbero in November.

Release Day for The New Voices of Fantasy

It’s here! It’s here and it’s gorgeous!

Today is the release date for Tachyon’s anthology The New Voices of Fantasy, edited by Peter S. Beagle and Jacob Weisman, with fiction from some of the most exciting authors around. It has been getting a glowing reception, including starred reviews from Publishers Weekly, Kirkus, and Booklist. That mildly thumb-obscured picture above is of my contributor’s copy; this anthology is the only place to read my latest publication, “My Time Among the Bridge Blowers.”

If getting this book for free sounds cool to you, then you should know that, to celebrate the release, Tachyon is partnering with Goodreads to hold another giveaway. It’s running through Aug. 29, and you can enter here. If you think you might want to buy a copy, but would like to know a little more about it first, Tachyon has been aggregating reviews on their Tumblr page. They all seem to agree that the book fully achieved its goal of capturing the vibrant and diverse work that newer writers are contributing to the field, “authors who are ready to expand the definition of what fantasy can be, and what fantasy will be,” as Jacob Weisman writes in his introduction. Beyond the individual stories–all published since 2010, all stellar–this anthology serves to document an exciting time in the literature of the fantastic. As of today, you can hold that excitement in your hand.

Buy The New Voices of Fantasy from Tachyon Publications.

After I saw him kill the dog I was more afraid of being alone with my father than I'd ever been of anything. But over the course of months every fear, however strong, ebbs or changes. My father treated me with the same flustered abstraction with which he always had. –from This Census-Taker by China Miéville

The Most Annoying Thing About the New iOS Music App

Apple has updated the music app for the iPhone to include their new subscription music service. That’s fine. I don’t personally want to use a subscription music service, preferring to buy my music, but it’s easy enough to deactivate in the settings menu. They also gave the app a visual makeover, another fine thing to do. My problem is, the visuaI makeover seems to have introduced UI inconsistencies. That is not fine.

There are now an unpredictable number of hierarchies between me and my music. If I have my music sorted by artist, I’ve currently no idea what I’m going to see when I tap on an artists name. For example, if I tap Simon & Garfunkel, I get a collapsed list of albums that I have to tap on to see track names:

IMG_6276

But if I tap on The Submarines, I get a scrollable list of tracks with the albums as floating headings:

IMG_6274

Personally I would prefer it to always work as it does for The Submarines–a hybrid album and track view–which is how the old app worked. But more than anything I want it to be consistent. Not knowing what UI I’m going to encounter from artist to artist keeps me from developing muscle memory, and creates occasional frustration when I’m trying to play a specific song. If I can’t remember which Simon & Garfunkel album “Cecelia” is, I used to be able to just scroll until I saw it. Now I have to search, either by hand through the albums or by typing in the name. That delay between deciding what I want to listen to and being able to find it is new, and unwelcome.

Anyone know how to fix this?