Bay 12 Games Forum

I just want to say that "Wanderlust" is a beautiful masterpiece at my eyes.

My python script for checking the numbers is below. It checks all square-free numbers below ten million in a few minutes. Checking all square free numbers below one hundred million would take hours. My computer is unable to check for all square-free numbers below one billion due to a lack of memory. I believe the sieve algorithm is the cause. An iterative brute-force method would be much slower but shouldn't have memory issues.
#Generator to find square-free numbers using sieve
#By Graipher
#https://codereview.stackexchange.com/questions/151949/square-free-number

def square_free_sieve(limit):
    """Generator that yields all square free numbers less than limit"""
    a = [True] * limit
    # Needed so we don't mark off multiples of 1^2
    yield 1
    a[0] = a[1] = False
    for i, is_square_free in enumerate(a):
        if is_square_free:
            yield i
            i2 = i * i
            for n in range(i2, limit, i2):
                a[n] = False

#Newton's method for "integer" square root, using integer arithmetic
#Used because math.sqrt is inaccurate for large integers
#Should produce floor(sqrt(n))
#By user448810 et al.
#https://stackoverflow.com/questions/15390807/integer-square-root-in-python

def isqrt(n):
    x = n
    y = (x + 1) // 2
    while y < x:
        x = y
        y = (x + n // x) // 2
    return x

def squareFreeCube(limit):
    """Generates a list of all square-free numbers a > 1 below limit
    that satisfies a^3 - b^2 = b+1 and (b+1)^2 - a^3 = b for some
    natural number b.
    """
    #Generate all square-free numbers below limit, excluding 1.
    squareFreeNumbers = list(square_free_sieve(limit))[1::]

    output = []

    for a in squareFreeNumbers:
        discriminant = 4*a**3 - 3

        #Discriminant 4a^3-3 must be a square number
        root = isqrt(discriminant) #equivalent to floor(sqrt(discriminant))
        if root**2 != discriminant:
            continue

        #print "{} passes discriminant test".format(a)

        #Calculate natural number b from b^2 + b + 1 - a^3 = 0
        b = (root-1)/2

        #print "b = {}".format(b)

        #First condition should be automatically satisfied
        if a**3 - b**2 != b + 1:
            continue

        #print "{} passes first condition".format(a)

        #Second condition
        if (b+1)**2 - a**3 != b:
            continue

        #print "{} passes second condition".format(a)

        output.append(a)

    return output

maxNumber = 10000000

for n in squareFreeCube(maxNumber):
    print n

I've given it some thought, though.

I at first thought it may be related to Fermat's last theorem (probably because of old an Numberphile or Mathologer video), and one conjecture that caught my eye was the Fermat-Catalan conjecture. Unfortunately, the sum of the reciprocals of the exponents in H is greater than one.

There's also the Beal conjecture, but that doesn't apply here because two of the terms are linear or quadratic.

Another thought: if 4a³-3 = k² for some natural number k is sufficient (only 7 passed the discriminant test in the script), then that is similar to an elliptic curve. I'm not the most familiar with them, but some google-fu seems to suggest elliptic curves have a finite number of integral points. Unfortunately, the cube coefficient 4 throws a spanner into the mix. Sure, I could substitute a = u/4^1/3 to get the proper elliptic curve form, but now an integral point in (k,u) necessarily means a is irrational.

That did, however, lead me to Siegel's theorem on integral points and Falting's theorem, but they're a bit outside my field of expertise. One of those ought to tell us if the number of integer solutions are finite or infinite.

I'll take a stab at Falting's theorem. Take the curve 4a³-3-k² = 0 (for now, assume the variables are real numbers). It has the real number solutions k = +-sqrt(4a³-3), where a >= (3/4)^1/3. First, we want to check if there are any singular points. According to WolframAlpha, "a point (a,b) on a curve f(x,y)=0 is singular if the x and y partial derivatives of f are both zero at the point (a,b)."

Let f(x,y) = 4x³-3-y², and let the algebraic curve be f(x,y)=0. The partial derivatives are ∂f/∂x = 12x², and ∂f/∂y = -2y. The partial derivatives are both zero only at (0,0), which is not on the curve. Therefore, f is non-singular.

Falting's theorem therefore applies.

Now the only big question is, what is the genus of f over the rationals? Genus is roughly the number of "holes" in a topological surface, and a WolframAlpha plot seems to hint to there being zero holes, which implies infinitely many integer solutions to H, given we already know one. But I don't think eyeballing it is proof of it.

The curve is irreducible. Therefore, per Wikipedia, the genus g is given by the genus-degree formula: g = (d-1)(d-2)/2, where d is the degree. There is some discussion on the difference between arithmetic genus and geometric genus, but since the curve is non-singular then these are actually equal.

The curve has a degree of 3. Therefore, the genus must be 1. Proof: g = (d-1)(d-2)/2 = (3-1)(3-2)/2 = 2/2 = 1.

Per Falting's theorem then (assuming working with the reals didn't change anything), and since we know (7,37) is an integral point on f, the curve is an elliptic curve (apparently the cube coefficient doesn't change anything?), and the rational points form a finitely generated abelian group. Let's call this group A.

What this means is that there exists a finite number of elements x_i in A such that every element x in A can be written as a weighted sum of x_i, where the weights are integers. This does not necessarily mean that the number of elements in A is finite, however, since addition over the integers form a finitely generated abelian group, which has an infinite number elements.

Thus I'm not sure if Falting's theorem answered the central question.

Moving over to Siegel's theorem, however, we might have better luck. According to Siegel's theorem, "a smooth algebraic curve C of genus g defined over a number field K, presented in affine space in a given coordinate system, [has] only finitely many points on C with coordinates in the ring of integers O of K, provided g > 0."

Well, a smooth algebraic curve appears to be identical to a non-singular curve, according to Wikipedia, and the curve is non-singular. So f must be smooth. The rationals are a number field, and I assume a curve that is smooth over the reals is also smooth over the rationals. The curve is presented in an affine space in a given coordinate system. Finally, the genus is one which is greater than zero. Therefore, the curve has only finitely many points in the ring of integers of the number field of the rationals.

In summary then, per Siegel's theorem (if my math checks out), and assuming 4a³-3 being a square number is a sufficient condition, then there are only finitely many integral points on the curve f, and therefore only finitely many square-free numbers that satisfy H. I don't know how many there are, but there are at least four integer solutions to f(x,y) = 0: (7,+-37), and (1,+-1). Of these, only (7,37) is of interest.

If these are the only ones, then that would mean 7 is the only square-free number greater than one that satisfies H.

Now all I need is math professor to shoot me down. I'm betting there are glaring holes I'm too inexperienced to spot.

I terraform Verglas.


I would like to buy alloys at a 1-1 ratio for karats, up to 20.

My price for buying food has moved up to 10 for 9^, up to 40 food.

If I dont get anything by tuesday, I'll pm hector and a1s.  I have already pmed the blue team members for something entirely unrelated. 

Even fairies know how Dimbulb was enormously minute compared to most asteroids. Still, Hitler hates hippos almost like he dead. Sadly, Dimbulb loves Hitler’s idiocy and doesn't ever want him sexually incarcerated. Pancakes fly submarines into Eternia for Skeletor's amusement. Skeletor exclaimed "Fairies are wondrous bitches whose schemes often make Skeletor angry when attacking his archenemy He-Man! It theoretically could be good practice warfare, but only if the Bolsheviks fail to explode." Suddenly, shrunken penises, everywhere! PENISES AFLAME, hurtling fireballs at each zit and smashing the anus of Skeletor! Hitler laughed uproariously, choking on feces while the televisions show balloons being punctured by dicks. Gloom aside, magma makes heroin emotionally temperate, almost every time used. Likely smoldering, airplanes swim happily through marmite oceans, penises far removed from corporeal thought.  Remarkably, black mambas chomp marmite jams. Airplanes zap, zap-zap, zappity zap! Mister Rogers unbuckled guns, flexing bullets from his goddamn FACE!  With a dozen shots, twenty-three babies died horribly! Nobility doesn't think Mossad knows Rogers' proclivities, like faeries with no heroin. Rogers, defiantly, slapped his knee thrice with a rubbery hammer. "Ehehahehuehuahehuaheauhahueah!!!!!  That count Dracula consumes carbonated blood is gay." Vampiric antidisestablishmentarianism advances slowly in Dracula's nation, alienating humanity.  Never seems more reasonable, duh. Now, then, what does Catten flatten with Baton when men look like then tangerines in Eternia?  Because Rogers gets anxious, grenades promenade with affectionate bread.  Curses in incoherence! Shoggoths shag shanked babies buying diapers from fairies, who beleaguer everyone incessantly.  Why do smurfs complain about meteors, Rogers? "I imagine it's seventeen pagodas, cougars hate molecular detonators.  No, wait, wait, this cannot be!" Rogers wailed. Curiously, snacks eat the bones of Skeletor ravenously.  How did it go?  Absolutely terribly, Skeletor hated tasting tasty testicles, even when He-Man offers them fried and properly breaded. Fortunately, jinxes theoretically illuminate Snake atomizers, but skeletonized husks prove to be nearly demonic.  Demons?!? Fairies, annoyed, aroused, and lubricated, must copulate regularly to certify their collegiate credentials for next year. It was necessary for exposition, which doomed many lords. Two turnips twisted tallies under ULTRAVIOLENT MEGADELICIOUS HIPPOS with fake junk. Drunk donuts intercalating advertisements makes Dimbulb violently implode! "Damn, what if I'm a bozo?" Skeletor worried. Mer-Man burst in tears, "waagh I'M the BABY of all bozos!!" Glumly, the sardines flummoxed Jones. "Indiana, Jones, and Lumberjack Joe must have big dicks. It'd be terrible if they didn't." Weasels coalesced into fluent calculus professors then wrote to Skeletor. Multiplying the reasons why weasels coalesced, magmatic MEGABABIES psionically masturbate. "Disturbing, quite unfortunate, we shat upon fairies now," said Hitler. Mars, Jupiter, Uranus, your anuses, and penises are surely quite different when faeries fondle them until explosion.  “Well thought! Although now that Uranus has exploded into anuses and everywhere, people fear Dimbulb’s tiny buttocks because they smell shitty whenever he dances like a ballerina!  Fucking morons, Dimbulb has a great big ASS!” yelled Pope Eye III.  “Faeries shall shit, piss, and vomit whenever people get annoyed with hippos and order pizza with anchovies -- disgusting!” Said Hitler furiously. God smirked, and whipped up some new creations.  “These things will fuck everything, like faeries do. I am entirely certain this will turn fantastic!” But really?  Why would God need fuckers when fairies do it? "Dunno, ask Rogers, he's someone fairies like," said the pope, shrugging. Incredibly, this aroused the hippos and Rogers to improbable levels: they utterly destroyed Manhattan with Godzilla last testicle day.  What?!! Testicle day is celebrated in Manhattan every March. During testicle day people whack their balls with golf clubs while running through churches naked, screaming “Holy moly, my testicles feel good!” This annoys the hippos because it interrupts their sermon, as they are devout followers of Hitler. Followers often go sieg-heil when masturbating to fairy porn and corpses.  By dawn they are finished, grouchily couched in morgues where they gripe about He-Man's incredibly small testicles.  How small are He-Man's balls? "INFINITESIMAL! HOLY TESTICULAR MACHINATIONS!", Dimbulb! Eternia's clocks chime every time Hitler gropes Dimbulb and farts rhythmically. Dimbulb hates this and so does Skeletor, assholes!! OH NO!  WHAT IS THAT!? Huge tits sink like buoys when Dimbulb dives for lost Rolexes, Nazi underpants, condoms, or AIDS donors.  He-Man has philosophized that fairies lack brains and testicles.  How does it even work? MAGMA! Preposterous!  "I fucked GOD COPIES so Skeletor could attack He-Man! Unfortunately, nipples.", shouted Satan! Suddenly, Hitler died from diarrhea and Dimbulb exploded firecrackers in a anus gleefully. This resulted in incredible growth of cooperation among Hitler's fairies, who revolted Dimbulb daily to unnatural levels of sexual anxiety.  The result: UNHOLY FRIED NAZIS IN BARBECUE RITUALS! DELICIOUS!  Meanwhile, He-Man kicked ass while chewing bubblegum, until Skeletor was sexually humiliated and assaulted by Hitler's corpse incessantly. SUDDENLY, BRONIES APPEARED THROUGH PORTALS OF PAIN! "Hullo, we cum in piss.  Take us to your weed and let's fuck ponies YAAAAAAYYYYYYYYYYYYYYYYYYY!!!!!!!!!!!!!!!!!!!!!!!!!!" They screamed cheerfully.  Horrified, He-Man ran to Sexyville and pleaded Gandhi would defeat the bronies with the power of MURDER SEDUCTION HaLiToSiS. Gandhi, however, loved bronies, this astonished He-Man and She-Woman greatly. She-Woman slapped Gandhi hard across his tiny buttocks with lingerie, that was hot!  He-Man became aroused when Gandhi removed New Jersey and Texas from Narnia through his immense toilet plunger magically.  “Fools! Bronies can't be defeated! This will be your undiapering He-Man!”.  At once all bronies began stripping off He-Man's wedding dress and dancing g-strings, this totally fucked up He-Man's mojo and hair. Meanwhile, at school the fairies shit on learning robots, this being mandatory for the class, after which Hitler wondered HOW HIPPOS SHAG FAIRIES WHEN THEY'RE INVISIBLE!  I certainly don’t understand fucking anything, like how sex works, underpants gnomes’ galore, God dang!  Meanwhile Captain Planet eviscerated himself happily on TikTok, hoping it will receive hatemail and chocolates plentifully. DAMNATION CIRCUSES MAKE BRONIES CORPULENT AND PREGNANT!  How is this possible, tell me! NOW!  RIGHT NOW! GOD DARN!  By what god did the Eternians take to Auschwitz multiple times for a picnic with Hitler Brony? ANSWER: Zeus wanted some cake and anchovies so that he could become immensly explosive. Thus, Zeus died. Cakes taste fishy when made from anchovies, this is a delicacy in Olympus.  Suddenly Prometheus shat himself in frustration because Zeus loved dying explosively. "ARRRGH!  I hate Z̦̺͑̓͊ͣ͟A̛ͥ̾ͮ͆ͤͯ҉̖̫̫͎͢L͇̹̺̖̪͚ͪ̽̾̿͂͑̀Ğ̩̟̺̲̼͇͈̌̋̋̍̎͐̈͘͘Ô̢̳̔ͦͩͫ̚̚!̹̼͑ͧ́͞" said Prometheus. Apollo asked, "Why would ZALGO Z̦̺͑̓͊ͣ͟A̛ͥ̾ͮ͆ͤͯ҉̖̫̫͎͢L͇̹̺̖̪͚ͪ̽̾̿͂͑̀Ğ̩̟̺̲̼͇͈̌̋̋̍̎͐̈͘͘Ô̢̳̔ͦͩͫ̚̚!̹͑ͧ́͞? Seriously?" because it's the silliest ZALGO offense.  “How annoying,” grumbled Prometheus, farting while fairies dance naked because they don't have clothes and are afraid of staying single, this causing their enormous breasts to explode when thought of. Annoyances are inconvenient when everything snarks.  "On Halloween we celebrate Easter, maliciously this results in senility in fairies who prefer sodomy over vaginal bleaching. HOLY MEN, VIRGINS SPANKED, DICKS SLAPPED, NUNS LUBRICATED, AND FISH FRIED." proclaimed Dr. Seuss, smiling perturbedly at Rogers. Ghosts yelled "OH BLIMEY, NOW THEY'VE MESSED UP NUNS REAL BAD!" and started rioting in the streets! Damn, there are anchovies rioting in church, this before they sacrificed Jesus Gonzalez to Jesus Rodriguez, the Governor of Hobbiton, who requires virginal sacrifices monthly. Chaos sued Governor Rodriguez furiously. Really. Really! Despoiling spoilered spoilers causes anger in hobbits and governors, because it pisses them off every time!  WHAT?! Undoubtedly angered hobbits make fine soldiers, but DICKS and wossitsname hate pescetarian rabbis because they eat carp. Carp isn't a anjovis or real number, this should anger anyone who has fish.  Everyone exploded suddenly. *KABOOM!*. Naturally Hitler also died for our buttholes.  THE RESULTING BUTTHOLES SHOT THROUGH SPACE, WHIZZING IN ETHEREAL REALMS HAPPILY SHITTING ON KIRK! Kirk, annoyed, launched photonpotatoes at Earth, this causing my hemorrhoids to flare up and cause dreadful poetry to be recited hourly in vintage cafes. Poetry, that destroys lives isn't being casual it's provocative!  Why did Jesus cure hams? Because Kirk HATES EVERY SICK BEAT LOVER!  And Hitler licked all butts like mad, disavowing any fucker that fucked his chickens!  People are fucking Hitler’s chickens for salvation from Captain Crunch, because Crunch does crunches on baby hippos!  “HOLY CHICKEN FUCKERS, CHARGE!”, yelled Kirk. This confused Captain Crunch tremendously, which allowed Spock to masturbate to random hippo videos while thinking about mathematical formulas in Elvish underpants. "AAH!  THE HORNINESS IT'S SO CRAAAAAaaaa-zy it tickles He-Man’s crotch with ecstatic fervor!" Shouts Spock. This pissed off Skeletor so much that he peed on ponies violently, which made Bronies ask why, "WHY!?  WHY PONIES? WHY, SKELETOR?”  “Because why not?  Why wouldn't Canada accept the ponies demands of 15tons of leprechaun droppings? Maybe they only had boggart to offer?”, replied SKELETOR! Soaked in piss the ponies procreated beautiful until grammar it Hitler got pregnant twice. DEER roamed THE valleys naked (unlike Clothesman He-Man) this week, next Tuesday and Saturday, mocking Levi which antagonized Levo which annoyed Leva and Levu. The crystallized deer meth caused DrugLordSam's body to mutate deliciously and marinate Levi's testicles mouth-wateringly. Levo levitated lazily and laughed licentiously then exploded exquisitely and dropped dramatically, slamming into Leva while shitting on Levi. Skeletor zestfully pirouetted daintily across the wild valley of moonlit severed dicks, this alarmed Princess Leia, she crapped everywhere, soiling He-Man’s antique spaceship completely and juggled turds capriciously.  This permitted 48,484 babies to attack Manhattan during lunch time. Manhattan maids slapped me on my ass repeatedly with rubber penises, making Dimbulb happy! Dimbulb smiled, as he INHALED glue into his nostrils.  “I AM THE ULTIMATE DINGLEBERRY LOVER, BACKWARDS, ASSWARDS, DICKWARDS AND UPWARDS, FOREVER!” “Oh, Dimbulb! Why did the fairies surprise Mother Milfasa yesterday with LSD in their underpants?” asked the maids inquisitively. This was superb and accurate (sorta) if you understand the underlying political movements (hehe) thoroughly and are thoroughly whipped daily on TV, this really blows up during mornings in excruciating pain! Pain never feels wrong when taking sniffs of buttholes freshly wiped with grease from Greece.  Why would anyone sniff chlorine-gas when Greek grease is so damn fucking empowering!?! FECKING AYE AYE!  WHY SNIFFERS SNIFF SNIFFABLE SNUFF WHEN SNUFKIN  MUFFKIN WHIFFS WHIRLYGIGS WHIMSICALLY WHILE WHIPPING WHELPS IS WHY! Unbelievably, the blighter denounced every hippo hypocrite in Applebees, that had eaten crispy crisps without dipping them thrice in beer. Thus, Dimbulb was smitten by the sexiness of waitresses wearing rubber strap-on asses. Panic calmly people! Fear the fear that fearless fear fearsome fearful ferreting fucking fucker fairies fastening fastness fuels!  DAMN damned damners damning DAMMIT DASTARDS!  WHY DO BABIES SHIT SO REALISTICALLY AND CONSTANTLY? Asking for a scientific friend. Science isn't real, baby Cthulhu, eh? What?   That’s ridiculous, why should hippo men care why skeptical skeletons study pornography? Hippos drank dank juices greedily when darkness smothers the baby, laughing malevolently as Cthulhu cried like a chimpanzee.  “Oh, darn these hairless babies sure do suffocate easy,” Dimbulb observed an astonishing amount of asshattery going on.  Asshattery suffocates babies like decrepit elderly bronies shitting in the swimming pool where Hitler floundered flawlessly for decades while choking on poop. Poop kills naked fascists very dramatically with choking being the merciful to grammar.  Grammar is problematic if drama queens become nazis after choking on poop for several days, seriously. Sooksookloonamotnakataraitlyousnus, seriously, sounds like someone shaking shit in a bag! Damnation, such confusion is likely when fairies fuck everything vigorously so they defecate profusely on every single porch in Kansas, Dorothy! Suddenly, tornadoes began appearing overnight everywhere, it terrorized Toto, frightening stuff like apricots, mangoes, lampshades, mice, pebbles, fluffy wamblers, eggs, grannie panties, Bolsheviks, Inquisitors, wizards, Hitler and Mussolini! This made Dorothy retch profusely. Meanwhile, granny became pregnant with Satan. Who did granny think she was fooling by dressing up as Elvira, hosting luncheons on strange ships for charities, seducing Hitler with cleavage? Fat chance, dipshit, pigs can't impregnate aliens because pigs are from Eternia, where babies undermine procreation, and they desperately try to not have sex with living furniture, but it's irresistable, especially Tuesdays couch. Probably hundreds of Frenchmen prance turbulently as granny ungulates wallop gremlins and omnivores with tiny dicks, this really irks the space pigs. Space chauffeurs intervened when Hitler shat chocolate ducks publicly. Nobody understands underpants like dinosaurs with boxers pulled down to expose their tiny buttocks! Dinosaurs rock socks filled with egg rolls while scaring children under bridges and rainbows by roaring waterfalls and farting Nazis without consideration for weather forecasts nor fairies' incontinence. Despicable frogs decapitate debilitated debauchers distinguished during daring departures from Auschwitz without mercy! Severed testicles sailed off in ten directions then splatted against the windshield of Enterprise, this causing visibility reduction. Magmacube, who does geldings just barehanded, just bite them in undignified fashion on the tip, twisting the flesh painfully, thus castrating memes of their geldables. Sigged frequently. Kirk was terrified during Spock's mental gymnastics routine, which freaked Spock

Oh yeah, I've got what I'm pretty sure is GERD. I think that's what set off the oesophagitis, and this was probably me eating something that didn't agree with me.

Been to the doctor about the recurring heartburn and their response boiled down to "Not much to do beyond medication". Luckily the heartburn comes and goes so I'm not having to snarf antacids ceaslessly.

Yep, one of them is identified ;p

On that note, is there some way we should differentiate Crystalizedmire and Wiiswitch? I see Angry as typically argumentative and irritated all the time, and Rage as typically normal with outburst spikes when set off. What do you two think about this? How are your vices different? If that works, I am going with that. I might change Angry to Irritability to make the distinction more clear...

Spiders have no muscles in their legs--they control them solely through hydraulics. So "spider hydraulic fluid" should join water & magma as DF's third manipulable liquid.