Tuesday, December 7, 2021

Beal's Conjecture Counterexample

 I'm a math geek. Every once in a while I like to attack big problems as an intellectual workout. The problem I picked this time was known as Beal's conjecture. Why should you care? Aside from the sheer beauty of formulas and ramifications to fields like math and cryptography/cryptocurrency, it has a million-dollar prize. https://www.bealconjecture.com/

Everyone is familiar with the Pythagorean Theorem: A^2+B^2=C^2, but Fermat claimed (recently proven after 357 years of attempts) that this formula can't work for any power over two. 

A^n+B^n =C^n has no integer solutions. In the process of trying to prove Fermat's Last Theorem, a mathematician by the name of Andrew Wiles determined that in the general case A^x + B^y = C^z has no solutions for x, y, and z above 3 unless A,B, and C have a common factor. The American Mathematical Society has offered the big prize to anyone who can prove or disprove the conjecture.

I found how I could take any prime and use modulo arithmetic rules to construct examples where that prime is the common factor.

2^3 + 2^3 = 2^4

3^6 + 3^6*2^3 = 3^8

(5^2*41)^10 + (5^2*41*2)^10 = (5^2*41)^11

7^3 + 7^4 = 7^3*2^3

11^5*3^5 + 11^5*3^5*2^5 = 11^6*3^6  ... ad infinitum.

If there had been even one prime I couldn't do this with, it might have been a chink in the armor. No such luck. After playing with this for a week, I became convinced it's true, and nobody will be able to find a counterexample. Why? Because in an attempt to disprove it, I made the rules even weaker. I got rid of the common exponent condition and just looked for a counterexample where A, B, and C have no factors in common but all the exponents are above three. A^x + p1^m*p2^n*... = C^z

My reasoning for this was that if I generated a few dozen weaker solutions, I could then generalize a pattern for the counterexamples, and I could finagle one that also worked in the stronger Beal case. But that's never going to happen. In the looser construct, I found exactly one solution:

271^3 + 73^3*3^5*2^3 = 919^3

This example cannot be converted to the Beal case without increasing the power of 3 to a multiple of three. However, to do that, I would theoretically have to multiply every term in the equation by common factors, violating our rules and thus proving Beal's conjecture.

Of course, to make this an official proof I would have to show conclusively that this is the only solution, not just assert that it was the only one that doesn't overflow the integer variables on my computer and I didn't make a mistake in my programming. This is close enough for me. It adds to the body of knowledge without me having to lock myself in a room alone for months. These sorts of problems will eat you alive if you let them.

Tuesday, November 2, 2021

Should I Try Pubby

Pubby is a website where authors rate each others’ books. Membership has a monthly fee, but it bypasses Amazon’s stricture against paid reviews. Is a good idea to join? The short answer is: in moderation and in limited circumstances for a limited time.

Why I Tried

Indie authors should never pay for reviews. First of all, Amazon would remove your account. Although, a few Vine Voice reviewers hit me up for payment, but nothing happens to them. Second, nobody would believe the honest reviews you do have. While browsing on Amazon, few people will buy a book with under ten reviews, and some advertisers won’t help you until you reach that threshold. Normally, my mailing list of fans can provide 5 to 20 starter reviews, depending on how good the cover is and how close the new novel is to my normal style. If those fail, things get ugly. Sending out personalized invitations to 500 reviewers may get me another 5 reviews, but new reviews risk knocking off earlier unpaid reviews—three steps forward and two steps back. Giving away a hundred e-books on Goodreads doesn’t do much. I got one Amazon review and 4 Goodreads ratings. The read-to-review bulletin board hasn’t yielded results in a couple years. I won’t give away free copies on Amazon again to spur reviews. If you get a competing author or an offended customer who gives you an early one-star on Amazon, your book may never recover. Even free, it counts as a verified purchase and weighs as much as 8 older reviews from bloggers. What can an Indie author do in these circumstances?

Trading reads on a site like StoryOrigin is slow. I think I received six reviews for two books in the few months I belonged, and half were from Amazon.uk. One caused Amazon to remove a previous free review. My net gain was two reviews for three months of effort, not a great payoff. The site was valuable for meeting other authors and growing my mailing list. Although, site membership is now $10 a month.

 

What’s left? Desperate to repair my books that started with a bad cover or an early one-star, I turned to the Pubby website.

The Price

You get a free 10 day trial membership and a few points to spend. You get more points by reviewing the books of other authors on the site. I would recommend the trial membership to anyone with a few caveats. Though some are free, most books are only available to select if you belong to Kindle Unlimited or pay for them on Amazon. If you want reviews for your books to have full Verified Purchase weight, you have to pay Pubby an extra fifteen bucks as a one-time fee. The Kindle Unlimited option is another ten. Why would you want this? Because Amazon purchases from Pubby reviewers can actually earn you more than your subscription fees, and reviews don’t vanish due to the mysteries of the Almighty Amazon Algorithm. Anecdotal evidence has informed me when under ¼ of reviews are verified purchase, old ones can slip off when new ones are added. I opted to get both options and continue for two months at the limited 10-book level at $17.99 a month. For a beginner without a mailing list, it’s a no-brainer.

Summary

The site interface is crude but navigable. In theory, you can gain as many reviews as the work you put in, taking out two books at a time and waiting six hours after purchase to post a review. In practice, they will only allow 10 reviews per week, and your progress is limited by the speed at which the moderator confirms reviews (up to ten days). In case Amazon someday ends up invalidating all those reviews or the books themselves, I didn’t choose my best-rated/most-reviewed novels. Over a span of 64 days, I spent $205 (61 to Pubby and 144 for buying other authors’ novels) and got 57 reviews, with an average outlay of $3.60 each—a financial bargain.

Most of my novels only needed 2 or 3 reviews to repair a stain or boost them to the noticeable level. For books already rated below 4.0, I started cautiously to see how the community would react to them. Reviews for free copies were half the price (1500 vs 2500) and counted for less than verified. Amazon only allows 7 free reviews a week. Though I got a three-star for my lowest ranked older book, it didn’t sting or make an impact, and I stopped when I reached ten reviews. These novels didn’t change ratings at all, just count. The only permanent and significant ratings gains came because of volume and purchases. I’ll talk about my larger scale experiments later.

Though the Pubby-written reviews averaged two lines and were vague, about a third of them were thoughtful and deeply appreciated. A few reviewers obviously didn’t even crack the book or read the summary; their reviews could have applied to any work of fiction. A little resentment leaked through from other people who felt forced to read the books of others because they began with the phrase “normally I don’t read this kind of book.” So much so, the word “normally” now appears in my keyword list for reviews for “It Takes an Oni.”

Because verified reviews and longer books are more expensive, I gave more reviews than I received. For the 60 reviews I provided, I started with nice things above the More click line and then limited negatives to the worst four problems. I gave points for formatting, editing, meeting the promise of the genre, and whether I liked it or not, with examples. Fourteen of them had serious formatting issues. Forty-two of them had persistent grammar and punctuation issues. Most fiction authors I could contact through Goodreads (those I was giving 4-5 stars and wanted to help) admitted that they had no editor and no wish to spend that money. Reviews from kind peers are no substitute for a professional product. If I couldn’t say anything nice, which happened about three times, I put it back as soon as possible, returned my payment points, and got my real money back from Amazon. Of the eight organic paid novels I rated a three-star, one author gave up collecting reviews afterward. Another removed their book from Amazon entirely. However, half of my critical reviews became the top review for that book, so others thought they were helpful.

The biggest cost to me was time, an average of four hours per book for reading, add one each for note-taking, spreadsheet-accounting, and e-mails to remove website obstacles. That’s over 300 hours of effort, something I wouldn’t have time for unless I was between projects.

Fairness

The ratings sci-fi books receive on Amazon (sampling several authors) are typically about half a star higher than Goodreads. The ones I received on Pubby averaged 4.587, about .25 stars higher than my usual Amazon rating. Then again, when I had a choice between which of the ten novels on my shelf I would submit for the next review, I spent my hard-earned points on my best, most popular, and most sale-worthy novels first.

Books with fewer than 30 reviews and actual organic participation didn’t seem to be significantly skewed by the Pubby additions. However, fifteen of the books I reviewed relied solely on Pubby (fewer than 5 Goodreads reviews and over 30 Amazon). Those averaged 72 reviews with 4.458 stars (standard deviation of .19). We have no clue whether those were realistic, and my ratings also fell inside that narrow band. The outliers were more credible, giving some indication of quality range; however, 102 ratings that averaged 4.8 with no 3-star reviews doesn’t seem authentic. Even Jim Butcher’s latest triumph, “Battle Ground” (same star rating) had 4 percent of its ratings 3 and below. Someone in cycling competition would call performance like this sus. Since a quarter of those authors who received bad reviews dropped out before reaching the 30 mark, this might affect the average rating curve.

Hidden Benefits

Keep track of every review you have on Amazon before and during the experiment. If one of your Pubby reviews vanishes mysteriously, they will refund the points you paid. Second books in a series are notoriously hard to collect reviews for ahead of time, but you can get them done here! An epic fantasy book from a decade ago received favorable attention, which let me know I’d underestimated it and encouraged me to spend more effort on the series. If I have one paid review a day, the position on my genre list goes way up from 1.5 million to 2000, which adds a little respectability too.

The text you place in your book description often ends up in reviews, so choose wisely. Once you add a book, it’s there forever. You can also pay extra to have your book reviewed by people who rate your genre or your books higher than average, which we’ll come back to later. Children’s books were easy to review, but many weren’t suitable for Kindle.

The biggest hidden benefit was seeing how good my cover is by how fast people snap up the title after I request a review. If it’s a couple minutes, you have a winner. If it takes a day, invest in a replacement. I could tell how good a novel was by how long it took on average to go from “reading” to “review awaiting approval.” My book “It Takes an Oni” almost always came back with rave reviews the next day, whereas few people wanted to finish my genetic-engineering hard science fiction in five. On this site, you can tell quickly which of your products are marketable, where you should invest future efforts, and where you should cut your losses. A word of caution, any novel over 70k words will be considered “long” by this audience and make garnering reviewers more difficult.

Hidden Costs and Pitfalls

The word count in the book description page Pubby gives you is often wrong, especially for nonfiction. They don’t care. Don’t bother to email the site; just put it back because they aren’t paying you enough.

Often, nonfiction people game the system to a ridiculous level. They put themselves in an obscure subcategory that they’re really nowhere close to the subject matter to just to get higher on a chart with less competition. Please don’t commit fraud like this. I deducted a star and called it out in the reviews. If I could have complained to Amazon, I would have.

In one case, a person reading my book didn’t post a review in the week after, so the site put my on the market again at no cost to me. Unfortunately, someone grabbed it that normally doesn’t read sci-fi and rated it a 3 with a one-line review: “This looked like it would be a fun read, but not really. At least for me.” This was absolutely no help, with no specifics. Did they stop after the first page?

I tried to be super nice my first month—reminding myself of a young bride who won’t fart in front of her husband for the first year. I wouldn’t take a book assignment if I couldn’t post an average rating a four. In nonfiction, logical errors and misinformation from non-experts abound. Some writers only have one title and post it nonstop over a hundred times. As a rule of thumb, a book on Amazon will have about half the number of organic reviews you see on GR. If they only have three on GR and over 100 on Amazon, it feels like a sock puppet. I didn’t feel right lying to people about these and tried to avoid them. The worst side-effect of perpetuating these is that Amazon detects common purchasers, even from Pubby. With every review you ask for, you risk being lumped in with them because people who bought the sock puppet also bought yours. If you’ve ever downloaded your own book during a free day, you could infect yourself by touching this social network.

After receiving my the three-star review “not for me”, I stopped caring and became more honest. When I read a nonfiction book that lied on several levels and lacked basic formatting and editing, I issued my first two-star rating. Once the average score I posted dropped below four, the number of options I had to do reviews on dropped sharply. Why? People pay extra points to get the friendly reviews. The ones remaining were usually a lower quality, with no editing. As I gave more 3s, even though I kept the rules about starting positive, I was excluded by genre after genre and my review approvals slowed ever more so I couldn’t request reviews as often. By two weeks into my second month, I could only see nonfiction candidates once or twice a day. Since I wasn’t a woman of color into Raki massage subscribed to Kindle Unlimited, the options were even scarcer, only travel books or nothing for days. I was forced to drop out 10 days before my last month completed.

Case Studies for Larger Review Counts

What about the two special cases where I gathered more reviews? My medical thriller “Preconceptions” started with 3 reviews (2 fives and a one-star), sitting at around 3.5 average. Either readers didn’t like the topic of eugenics/near-future genetic engineering and the rules governing science, or I didn’t wrap it attractively enough. Once I accrued 7 mor reviews so it met the minimum 10 threshold, it had risen to an average of 4.1 with a believable bell curve. My first Book Barbarian sale last April only sold 15 copies. The second sale in mid-October with a comparable venue only sold 8. In a light-bulb moment, I figured out that email/social media sale venues don’t show the number of reviews, only the cover and a short pitch. Having a hundred reviews doesn’t matter if my book isn’t marketable. A sale can get you noticed on the first page of a genre where reviews matter, but only if you can garner enough interest on day one to make the leap. Pubby can’t help a novel nobody wants.

Lastly, I did a larger scale test for “It Takes an Oni,” my supernatural heist novel with a new cover. It started at 5 initial reviews (all five-stars) and actually went down to a more realistic 4.7 by the time it reached my goal of 30 reviews. It sold 51 on its opening sale in March, including my own mailing list. However, I hoped that seeing a sharp new cover and a lot of reviews would sway more of my existing fans to try it. If this happens, I could reach the fifty review mark organically, and Amazon might send a few emails on my behalf. At the very least, bloggers might be more receptive. The sale Halloween week sold over 100 copies, more in two days than the previous year! It also encouraged two additional organic reviews by the third day.

Conclusion

I got my money’s worth on this experiment but won’t be able to repeat it with my next book. Not only would I have to pay $25 a month to go from ten to eleven novels on my shelf, but once you’re completely honest, you’re no longer welcome on the site. Therefore for future novels, I have to make sure my book is marketable, has a good cover, and is supported by my base fans before I post it to Amazon.

Monday, January 25, 2021

Preconception Cover Reveal

 This past November, I wrote a 48K word medical thriller about a breakthru in treating common genetic disorders ethically before conception. It poured out in three weeks. As a result, I spent seven weeks rewriting it, but Preconceptions has more symbolism and raw feeling than any book I've written in the last decade. The main character is a gentleman journalist with few clues about the topic, ala Sam Spade. Other than the main invention, a tool to detect the rapid cell division as an egg tranforms into a baby, everything in it is as scientifically accurate as I could make it. It will be on sale for 99 cents for the first few weeks in order to stimulate reviews. click here for Amazon page

I'm very pleased with the way the cover came out: