A Proof of Quadratic Reciprocity:
We shall prove quadratic reciprocity using one of Gauss' methods, by way of the modified division algorithm, Gauss' lemma, and the Pretty Picture theorem.
1) Modified Division algorithm:
For all x, y in N, there exist integers q and r such that x = qy + er, with 0 £ r £ y/2 and e = (-1) ^ [2x/y]
Proof:
Because q can be any integer, we can assume WLOG that x y and prove for y x by subtracting from q.
We can find a q such that |x – qy| £ y/2 because if it is more than y/2, then adding or subtracting 1 from q will switch the sign of x – qy and place it within y/2 of 0.
If x – qy 0, then x = qy + r, and 2x/y = (2qy + 2r)/y = 2q + 2r/y. 2q is an even integer, so it can be ignored, so there is only 2r/y left. r £ y/2, so 2r/y £ 1. Ignoring the case where r = y/2 for now, 2r/y 1, so [2x/y] = 2q, which is even. Thus e = 1 and it fits. The same argument works backward, so if e = 1, x = qy + r.
If x – qy 0—
Drey-chan: Layra-chan, what are you doing?!
Layra: I'm proving quadratic reciprocity for our readers.
Drey-chan: But this is supposed to be the last chapter of Fourth Impact! And I'm supposed to be the one writing it!
Layra: Really? *Pouts*
DreyonLegacy: Yes, Layra-chan. We made a deal. I write the chapter, you jump up and down and yell at the TV.
Layra-chan: TV! TV! TV! Ka-boom! *Runs off*
DreyonLegacy: sigh.
Disclaimer: Wow, I've done so many disclaimers. Perhaps I could skip one? No, I guess not now that I've gotten so far. If I forget the disclaimer and they yank the fic off the site, where will all my hard work be?
So, in all regrets…screw that. See chapter 3
Fourth Impact
Chapter 56: Side Effects May Include…
Impact
For a single instant, the entire universe stopped.
Ten hovering disks suddenly formed around the building, connecting into a huge figure sprawling across hundreds of kilometers.
Roots and branches seemed to sprout from the building, extending into the air and ground. The shards of the Lance of Loginus flew up to drive themselves into the inverted tree growing in the middle of the city.
Then the tree shattered as a brilliant white light shot out from the hotel room, spreading out in all directions, across the planet and into space.
A rip appeared in the fabric of Reality as two figures appeared on the battlefield, ejected from the world of their own making. One of them roared in triumph as the other collapsed in despair. And then they were swept into battle once again.
The light pierced through stone, metal, flesh with ease, subtly altering everything it touched.
In that moment, billions of stars became supernovas, and huge portions of the universe went dark. The sun bulged outward, then sunk back into itself.
All across the world, everybody suddenly burst into full consciousness. The Sea of LCL spit out billions of people, then evaporated.
In NERV bases, Evas fell apart, disintegrating into LCL and dust. The Geofront exploded with new vegetation, everything in full bloom.
Billions of lights covered the Earth as each human's soul was bathed in the glow of life. A cry of infinite joy burst out, a song that rumbled in each human's heart, reverberated in the primitive and the primordial portions of each being.
And as suddenly as it had come, the light passed, leaving each soul feeling oddly fulfilled, as if some great task had been accomplished and respite was upon them.
Misa woke up with a slow groan. Everything was fuzzy, and she couldn't find the strength to move. On the plus side, she had found out why Rei always had that strange smile on her face…
She opened her eye to a slit and found silvery hair before her. She could see a bit of pillow through it. She must be lying on top of Adamu.
Finally getting her muscles to respond to her commands, she sat up a bit, and opened her eyes fully.
Bad idea.
She hadn't been lying on top of him as she had thought. She was lying in him. Shrieking, she pulled back, a sharp pop shooting through her as their cores were yanked apart. She sat up, with her hips sunk in his and her hands inside his chest, looking down at him.
He opened his eyes. "Good morning." He said blandly. "Why are you awake so early?"
"Adamu-chan, we're joined at the hips!" she screamed at him. He looked down and blinked calmly.
"I take it that you won't be getting up, then?" She pulled out of him completely, each feeling a slight jolt as their AT fields fell back into place and sealed up. Immediately the buzzing in Misa's head ceased.
Then she noticed that something else was missing.
"Adamu-chan, I can't feel Adam anymore."
She woke with a start, finding no memory of having fallen asleep. She glanced around, suddenly very worried. The trees were a lot bigger than they had been the night before…A lot bigger. And her body felt different. What happened…
Operator: Someone set up us the bomb
Operator: We get signal—
DreyonLegacy: Layra-chan! Stop that!
Layra-chan: what?
Stop that!!
Ariline slid out of bed, worried. She had felt Impact, and knew that something had happened, something that would make a big difference to her.
She slipped into her plugsuit and pulled a long coat over it, then slipped out the door without waking Samantha or Dai-syo. Or so she thought.
She looked around, tensed and frightened. The situation was completely alien, and she was not used to handling new situations. She wasn't the one who was supposed to think. She was supposed to act.
A cracking twig alerted her to another nearby. She spun to face the newcomer, when suddenly a familiar presence.
"Unit 16!"
Samantha glared at the boy beside her. "Look, she's a big girl, she doesn't need you looking out for her all the time."
"She took off in a big hurry. You felt what happened last night. Whatever it is, she's worried, and so we should be worried too." Samantha rolled her eyes at him but kept going, tracing Ariline's path through the forest.
"Well, just remember, now that Lillith's gone so are our powers."
"We'll be prepared enough."
Neither of them would ever have expected to see what they did when they got to the clearing, though.
Blink. Blink. Attempt to not stare…Attempt to avert eyes…attempt to block bleeding from nose…
Ariline had her arms around a completely nude girl with long, crimson hair and lips locked on hers. The new girl had long, talon-like fingers gripping Ariline's coat, and was dripping LCL.
The redhead pulled out of Ariline's embrace and shoved the blonde behind her, growling at Dai-syo. A third eye shot open in the middle of the girl's forehead, revealing nothing but crimson.
"No!" Ariline shouted. "Wait!"
The redhead continued to growl, but Ariline held her back. "Jurokume-chan…" Unit 16 turned to look at Ariline, her growl turning into a purr. "Don't worry. They're friends."
Then Ariline looked over at Samantha and Dai-syo. "Don't be alarmed. It's only Unit 16. I don't know how this happened," she said, gesturing to Unit 16's new body, "but I'm not objecting." she concluded with a smile.
Dai-syo blinked. "But…Unit 16 is…"
Samantha grabbed Dai-syo's collar and hauled him back into the forest. She sighed. "Look around you. Unit 16 the Eva is gone. The clearing is empty. That girl has Unit 16's eye, her claws, and if that weren't enough, she even has scars in the same places that Unit 16 got hit by Unit 19's thorns. To top it off, her core has the exact same hue and patterning as Unit 16's. And that's if the girl's aura isn't enough."
Dai-syo glanced out to the clearing to see Ariline and Unit 16 kissing again. "Dai-syo, when I told you that she was in love with her Eva, I didn't mean platonically. I mean full-blown, tooth-rot-fluffy, down-on-one-knee-proposal-in-the-moonlight-with-roses love." She shook her head. "You are blind, Dai-syo."
"Look, I need to talk to you, all right?" Dai-syo asked. "A lot just happened, and I'm confused."
She almost smiled. "Fine with me. I…I need to talk to you too. About something else."
Now's my chance. Now that he's seen that Ariline's taken…Misato looked down at the screen before her, blinking owlishly. "So, that was Fourth Impact?" she asked Ritsuko.
"No more Adam, no more Lillith, and the Sea of LCL is gone." Ritsuko replied. "Everyone in the hospital's been completely cured, and several stars went supernova last night. That was Fourth Imapct all right." She turned to Misato. "By the way, the UN called."
Misato groaned.
An hour after they got back to the hotel, Ariline walked in dragging Unit 16 behind her. Unit 16, or Jurokume as Ariline called her now, was dressed in Ariline's coat, while Ariline wore only her plugsuit. "Juro-chan, welcome to our home, at least until we rent an actual apartment or buy a house." Jurokume looked around, her third eye staring while her other two blinked rapidly. All three of her eyes were pure crimson, with no defined iris or pupil. She clung to Ariline's hand and shook her head. "Please, I know it's not what you're used to, but it's what we have."
Then Ariline turned to Dai-syo and Samantha. "Since it didn't work to well before, I'd like to introduce you to my love, Jurokume Evangelion." She pushed Jurokume forward. "Juro-chan, I know you remember these two from before, but do you want to reintroduce yourself? Start over without all the bitterness?" Jurokume glanced back at her for a moment, then turned to Samantha and Dai-syo, bowing.
"Well met, sir, madam. 'Tis a pleasure to meet ye. My name is Jurokume, though mayhap you hadst known me as Unit 16 before." She turned to look at Dai-syo alone. "I assure thee with all of my heart, artificial though it be, I would have no contention with thee if thou wouldst have none with me. I concern myself only with the health and happiness of milady, and would be at peace with thee if thou dost so wish."
Dai-syo could only blink in shock. Her Japanese was far better than his, though a bit antiquated, and her manners were very polished. Samantha answered for him. "Good to meet you too, Jurokume." She said in her blunt, American way. "Glad to know that you're wiping the slate clean. Dai-syo's happy too, he's just a bit dazed right now," she glared at him, "aren't you."
Dai-syo still wasn't responding, so she grabbed his head and pushed it forward as if he was nodding.
Ariline chuckled at the gesture. Dai-syo snapped out of his shock and grinned a bit foolishly. Samantha only threw her hands in the air and fell onto her bed.
"Well, now that we've gotten that out of the way, I'm going back to sleep."
Adamu held his wife in his arms, staring at the ceiling. For the first time since Yumana's death, he felt whole again.
Misa shifted, and Adamu felt a slight pain as her cross dug into his shoulder. She had insisted on putting it on right after they separated, and he hadn't the heart to object. Besides, she looked good wearing the necklace…and nothing else.
A heartbeat slowly thrummed in the back of his mind, a reassurance that he was not alone, that he was accepted and loved for being himself, Nagisa Adamu Ikari, Angel, hero, husband and in dire need of a drink.
And God is in His Heaven, and All is Right with the World.
AN: And that's it. That's Fourth Impact. 56 chapters, all building up to the most influential honeymoon in the history of mankind.
In case you were—
Layra makes a daring interrupt to present: End of Fourth Impact Omakes!
Omake:
Now's my chance. Now that he's seen that Ariline's taken, I get to have him all to myself and do what ever I want to him and his beautiful body. And he doesn't know that I still have my powers…
Omake 2:
Misato looked down at the screen before her, blinking owlishly. "So that was Fourth Impact?" she asked Ritsuko.
"Actually, that was Seventh Impact." She replied. "Impacts Four, Five and Six went completely unnoticed." She turned to Misato, taking in the look of horror on her face. "Just kidding!"
Misato groaned.
Omake 3:
An hour after they got back to the hotel, Ariline walked in, dragging Unit 16 behind her. Unit 16, or Jurokume as Ariline called her now, was dressed in Ariline's coat, while Ariline wore only her plugsuit. Both articles of clothing were in tatters, and the two girls wore the most twisted smiles on their faces.
They walked up to Dai-syo. "Dai-syo," Ariline whispered softly, "I know that you loved me, and though you have Samantha now, and I have Juro-chan, I think that we can still have something…" Ariline and Jurokume advanced on him.
Dai-syo backed away slowly. "Um…Samantha, help?" But Samantha only grinned at him.
"Foursome!"
And God is in his Heaven, and All is Right with the World.
And now the computer is returned to Drey-chan before Layra gets seriously hurt. Bye!
DreyonLegacy: *shaking head* where's a good exorcist when you need one…
Anyway,
Reviews!
ChibiIchigo: I feel for you. I'm always pissed when a good story comes to an end.
Invincible-Titan: Thanx! Yes, it is good to be god…even though I'm an atheist, and therefore I do not believe in god, and thus I don't exist…uh-oh.
RisingDragon: If you look carefully, you'll see that the summary is actually a quote from chapter…something, by Samantha. I felt it explained the inner plot well without giving anything away.
If anyone else sent me reviews for the last chapter, sorry, but I feel like posting this now. Otherwise, I hope that answered all your questions.
Layra-chan: And about Jurokume?
DreyonLegacy: Jurokume means 'sixteenth' in Japanese. So her name means literally Sixteenth Eva. See?
Layra-chan: I'm talking about the f/f. Do you know how many f/f couples you have in your fics?
DreyonLegacy: Um…Kay/Resala, Rega/Larana/Misell, Tisako-Emerald, Inara/?…
Layra-chan: At this rate, people are going to think you're just some perverted boy with a thing for fem-slash.
DreyonLegacy: And?
Layra-chan: You can't have them thinking that. I'm supposed to be the pervert. You have to show them that you're doing it for the romantic value and not the stimulation.
DreyonLegacy: And how am I supposed to do that? It's not like I'm going to admit that I'm actually a girl—