md65536

20740059257 considers only the first few equations in the puzzle. The full answer I have is 73 digits long. I believe it's right (go wolfram alpha!) but if it's not, the answer will be smaller than that. I suspect that OP says the answer is 80 digits because the web page input allows up to 80 characters. However, brute force calculations and hardware-precision numbers aren't going to work. Note that considering ONLY the last equation, you know that x mod 800009 = 438462, so the smallest possible answer is 438462, and the next smallest is 800009+438462, so it is a waste to check every number in between. Even so, with optimizations like that I think you'll end up running a brute force algorithm for many magnitudes of time greater than the expected age of the universe.

Wolfram has no trouble with these equations and 80-digit numbers. On second thought I don't think it matters that 2 and 600008 have a common factor. "x mod 600008 = 318753" already implies that "x mod 2 = 1". They're consistent and the latter is superfluous. I don't see how its inclusion could screw up the answer.

It should only take a few minutes of computation. You'll need to do "multiple precision" or "big int" calculations, eg. gmp c library. However you can also offload the calculations to wolframalpha! I came across CRT, I think that's essentially the way I tried... Does the CRT apply here, since 2 and 600008 are not coprime? Is that okay, or perhaps 600008 is a typo?

This was posted in Homework Help... is it an assignment? Why do you want just the answer? If it's for homework, please explain what you've got so far. How do you know it's 80 numbers long? I came up with an answer but it's fewer than 80 digits and the website says its wrong...

The video is nothing but pseudo science that uses some science jargon but doesn't actually explain anything. 2:30 some physical phenomena are describes as "mysteries" that "don't fit" in the clueless scientists' view of the world. 2:50 Equations are only a vague attempt to explain stuff. Throwing out old assumptions and constructing "a new blueprint for reality" is a bold step that should be taken. 3:20 "Scientists believe" ... that "nature just doesn't make sense. Hmm..." 5:00 "We need to assume that space is literally and physically quantized" -- why? -- "That it's made of interactive pieces." Oh god, this is another "Space is made out of tiny particles" theory... 7:50 Space is now a medium. 8:05 Curvature is "explained" in terms of the density of these quanta. Denser quanta = less resonating = "they experience less time". 9:30 it somehow explains quantum tunnelling, 10:22 and "where the constants of nature come from" (pi) 12:20 and of course it explains dark matter and dark energy. 12:40 scientists can't explain it, but 13:40 a change in the density of the imaginary particles of space "is going to cause a gravitational field" explains it. So it is just another "scientists don't know anything (as far as I understand)", "my theory explains everything, by analogy, if you just use your imagination", and "I can fit it into some scientific jargon you may have heard, like 11 dimensions, quantum mechanics, Einstein curvature!, dark matter and dark energy!" attempt to sound legitimate.

A person would experience passing the event horizon, only there wouldn't be any nearby effects that indicate its happening. By analogy, if you walk between two markers, you experience passing the halfway point, but you wouldn't know you're there except through calculation or observations of distant locations. "Infinitely dilated" is only relative, according to a distant observer. A local clock still ticks at 1 second per second as the observer falls past the event horizon. Except for gravitational gradients ("spaghettification") in practical cases, the in-falling observer doesn't notice anything weird nearby, even while passing the horizon. You're speaking of relativistic mass, and that's just the mass equivalence of the total energy of a thing. Mass is "rest mass". The Schwarzschild radius event horizon is based on mass, not total energy, and it doesn't change with relative motion of the observer.

Interesting idea but I think you have to state your assumptions, and there are a few, including: - A part of the system can't be predicted without simulating the whole system (everything is connected, and there is no way to begin from a known intermediate state). - The universe is perfectly deterministic to computable precision (which contradicts current theory as others mentioned), but also cannot be computed in parallel. Computing "more" requires "faster", which might not hold true. You would need to specify the limitations of your imaginary simulator, not just assume it is the simplest of systems. Why would the computer have to simulate the results of its own calculations? In your calculator example, the calculator calculates that 2+2=4. It would not need to simulate the workings of itself to know that the simulated calculator will also produce 4, since it already knows it calculated that. Or to put it another way... you're saying that suppose a machine is able to predict a future state X, you want to show that this will derive a contradiction. Suppose state Y is the state of the present after that prediction has been made, and we assume that X depends on Y (simulating the results of the simulation is required). Well that gives a solution to the paradox right there: We must also assume that the simulator is inside the universe and is much smaller than the universe (or otherwise there's no point). Thus to simulate the simulator should require much less processing than to simulate the rest of the universe. So all you need to do is make the simulator more powerful than it needs to be to simulate up to the present. For example say that 90% of the simulator is simulating the rest of the universe, and 5% is simulating the simulator, and the rest is idle, then if could set it up so that the idle part of the simulator has only an easily predicted effect, then that part doesn't need to be simulated. However I think that there are too many assumptions that are left to choice. Since you're not describing the mechanism for simulating the universe, it doesn't make sense to both claim that it can be done and make arbitrary restrictions. You could as easily make up a rule that the idle parts of any simulator all affect the future as much as anything else does and needs to be simulated, but then this is all just a bunch of made-up rules about what's possible and what's not possible. Edit: Trying to clear my brain fog, I think perhaps you might be right. In my example, even if "simulating the simulator" can be done efficiently, it still assumes that in simulating state X, it first simulates the state Y, at which point the prediction of X is already known in the simulation. So it doesn't make sense that the simulator doesn't yet know state X, but the virtual simulator does. If you also assume that no state can be simulated without knowing what that state is, then I think you're right. On the other hand if you suppose that it's possible to simulate the simulation of state X, and to know all possible effects of that without having to know what X actually is! (OMG confusing), then it could be conceivable to simulate the effects of simulating X on the virtual computer, and then continue on to calculate X on the real computer. Well at the least I think you've proposed a good mind bender of a thought experiment.

This is no longer completely on-topic but is a misunderstanding of a related idea. No it doesn't take energy proportional to distance, to move sideways in a gravitational field (or along the width of the accelerating train). Suppose you roll a heavy sphere along a smooth frictionless horizontal surface. It will take energy to get it moving, but then the ball will keep rolling due to inertia (not requiring extra energy to keep moving). The energy put into its momentum is kinetic energy. Meanwhile if you lift the ball, the energy goes into gravitational potential energy. It won't keep moving. Each meter you lift it requires additional energy. If you roll a ball up an incline (or forward in an accelerating train), each meter moved will require additional energy. They don't, in the inertial frame where "exact same time" is determined. From the track's perspective, the clocks could be coordinated as you say, and remain in sync. This is not the case being discussed in other recent posts. The accelerating train is not an inertial frame, and "at the exact same time" will not mean the same thing for the differently located accelerating observers (they won't agree on simultaneity of events). After the train has completed accelerating, it has an inertial frame, and the differently located observers would from then agree on simultaneity of events, and could arrange for their clocks to be synchronized.

I don't think that's correct. I think by the equivalence principle you could have the clocks in a uniform gravitational field but at different heights (ie. different gravitational potentials) corresponding to their different x positions on the train. Therefore if you do set up the train so that it accelerates uniformly for some time, you still have the clocks ticking at different rates according to GR and can end up out of sync depending on the rest of the setup. As studiot mentioned, this might not describe the original setup. Edit: Just thinking to myself... I understand the instinct to put the clocks at the same height around a massive body in order to get the same field strength, but that's the wrong way to think about it. A uniformly accelerating train is like a uniform gravitational field. It will still take effort to "climb" your way from the back of the accelerating train to the front. The effort is proportional to distance climbed. Equivalently it takes effort proportional to distance climbed, to climb out of a gravity well with a uniform field. The location on the train and in the equivalent gravitational field matters.

I'm trying but I'm hopelessly lost, I don't think I have the basic math skills :/ Where I'm going with it: Am I heading in the right direction? I'm stuck and might not work on this more.

I tried, thinking there must be a simpler solution than I gave...

I think most people who post here are amateurs, as I am. I don't think anyone is offended by questions about trying to understand relativity (as long as they don't involve purposefully ignoring answers people have already given and don't include "Relativity must be wrong, because..."). Also, no one understands it perfectly or has considered every aspect, so questions can be helpful to everyone.

Did my answer in steps to not spoil it all at once...

Edit: I agree with Janus' reply, some of my post is redundant. Figuring it all out first using only inertial frames, without specifying the timing of the train's acceleration, would probably be helpful. That answer applies to the case where the train is accelerated in such a way that the rest length of the train remains the same after acceleration. You have the trains remaining the same only in the track frame. Someone else had the backs of the trains synchronized. All of these situations are possible. People will fill in their own details and interpretations unless everything is unambiguously specified. If they're moving in unison in a particular frame, they will maintain their distance in that frame. Remember you have to specifically coordinate them to do this, and it is not always physically possible (Bell paradox; a rope or train that can't be infinitely stretched will break at a high enough gamma). Note: It is only the lengths between synchronized points that remain the same. If you synchronize all points (fronts and backs of both trains all synchronized) in the track frame then the trains and gap remain the same length (but must be physically stretched). If you synchronize only the middle points (as I think you might have specified) and let the trains maintain their rest length (this was not specified, so is left open to different configurations), then the trains will contract and the gap will indeed become greater in the track frame. Move inertially, I mean maintain constant velocity, or uses only one rest frame. Yes switching frames means changing relative velocity or accelerating. Err... I don't think this line of reasoning will help you until you've figured some other things out first. Frames don't have a single clock. Clocks measure proper time (the time at the clock, not the time at other locations). An event is a single point in space and time, and has a definite proper time according to a clock that passes through the event. A point of the train passing through a certain point on the track is an event, and all observers agree on the proper time that that happens. Now, you can have other clocks on the train and on the tracks, and different observers can disagree on which clocks are ahead or behind relative to the proper time of the given event. But again to say which is ahead or behind I think you'll have to specify the details of how you've coordinated the clocks... and where those clocks are located relative to the event. I'll work through an example to help myself figure out what I'm talking about... Suppose a train is 100m long in its rest frame and is heading East, and passes a station that's 100m long in the track's frame. Have 3 clocks on the train, front middle and back, synchronized in the train's frame. Have 3 clocks at the station, West end, middle, and East end, sync'd in the track's frame. Say the middle clocks pass each other at exactly 12 noon according to the station's middle clock, and 10:00 according to the train's middle clock (edited to avoid introducing misconceptions). According to the train, all 3 clocks on the train read "10:00". The station is contracted, so the front of the train has already passed the East end of the station, and the back of the train has not yet reached the West end of the station. According to the station, all 3 clocks in the station read "12:00" at the moment the middle clocks pass. The train is contracted, so the front of the train has not yet passed the East end of the station, and the back of the train has already passed the West end of the station. To sort out this situation, note that the front clock of the train reads 10:00 only after it has passed the East of the station. According to the station, this hasn't happened yet: the train's front clock does not yet read 10 and is running behind. Similarly, the train's rear clock had struck 10 earlier before reaching the station; it is ahead. Symmetrically, according to an observer on the train, the station's West clock is behind, and the East clock is ahead. Eg. the West clock reads 12 some time after the back of the train has passed, and that hasn't happened yet.

If you force various points to remain synchronized in the track frame, the distances between those points will remain the same in the track frame. Effectively you'll be stretching the distance between points (the gap will be increased in the moving train's rest frame) to exactly counteract length contraction. This is set up like Bell's paradox. The answers to your questions can probably be found in an explanation of the paradox. If you don't want to deal with the details of relativity of simultaneity, just give the train a fixed rest length, and let it remain moving inertially throughout the experiment. Don't worry about how it accelerated. If you want to have the train switch inertial frames, I think you are going to have to factor in the details of relativity of simultaneity, and you may need to decide on a few more details than you're giving. I think it's fairly common that people want to figure out one aspect of relativity that they don't get, and they completely avoid another aspect like it's too complicated to consider. It's like trying to figure out how 2+3=5 without considering the 3... "how does 2 add up to 5, relativity makes no sense!"

Which I think is equivalent to two clocks at different gravitational potentials in a uniform gravitational field, which also run at different rates relative to each other.

I say the clocks don't stop. The clocks were synchronized in the track frame before the train started moving. They won't be sync'd in the moving train frame unless the parts of the train are started in a peculiar way to achieve this result, with either parts of the train having different velocity profiles or the train being deformed in its own frame when moving. The clocks were synchronized in the track frame. They'll stay synchronized only if they accelerate at the same time in that frame. However if that is the case, the moving train will not be length-contracted shortened in the track frame (it will be stretched in its own frame as the front started moving too long before the back did, but stretched plus length-contracted to maintain its rest length in the track frame, as in Bell paradox). If this is the case, the problem can be solved without even considering relativity, just delay of light will tell you that in the track frame the simultaneously emitted signals won't reach the moving middle of the train simultaneously. In the track frame, the middle of the frame closes the distance to where the front signal was emitted, to intercept it before the rear's signal. In the train's frame, the front clock is ahead and the signal is emitted before the rear signal.

Strange is right. Remember the twin paradox. When the accelerating twin changes inertial frame, the coordinate time of the distant twin that it accelerates toward jumps forward (only relative to a local clock). The coordinate time of a twin that it accelerates away from jumps backward. The "jump" is merely a change in relative simultaneity. From the other thread... if a train is a certain length at rest on the tracks and all parts of it accelerate similarly and it is again the same length in its new rest frame, it must be that from the track's perspective, the back of the train accelerates first and the front last. But from the moving train perspective, the front accelerated first and the back last. The observer in the middle was in both frames... so how can that be? Just before accelerating, the middle observer understands that she will accelerate first before the front of the train. Suppose with sync'd clocks the middle observer will accelerate at time t=1, and the front at time t=2. The middle will accelerate toward the stopped front of the train, so the coordinate time of the front of the train will "jump" ahead relative to the accelerating middle, let's just suppose it jumps ahead 2 units of time to t'=3. Now an instant after time t=1, the middle observer is now in the moving frame, and the front of the train has already started moving at its time t'=2, and continued to move up until its current time t'=3. The clocks are no longer in sync. The distance to the location on the tracks where the front of the train started from is now length contracted and is no longer half the length of the train... how can the train still be the original length? Well, the front of the train has been moving between its time t'=2 and t'=3, which occurred in an instant for the instantly accelerated middle, and the front is now ahead of that point, at the right distance. To get the exact details you'd have to do the math. As always all the details work out consistently. Meanwhile the observer at the front of the train observers local time passing normally, while other coordinate times jump (backward). Since the clocks are no longer in sync after the train accelerated, they won't emit the flashes at the same time.

It sounds like you understand it. There's no "center of contraction" similar to how there's no center of the universe yet it is expanding in all directions. The contraction (expansion) is uniform... the same at any point. Yes, I would say that picking a center is similar to picking the time that an arbitrary point on an infinitely long train passes an arbitrary point on infinitely long tracks. You could choose different points and it happens at a different time. You could as easily say that relative simultaneity is a consequence of length contraction and time dilation, or that all three are a consequence of the constant speed of light (or vice versa). They all fit together, no matter which one you start with.

Careful because there is no universal meaning of "at the same time". It will be different for someone beside the tracks compared to someone on the moving trains. For example... This is true for an observer at rest relative to the tracks, if the trains are 100 m long with 100 m between in the moving trains' frame. If all parts of the trains start at the same time according to an observer on the tracks, they'll remain 100 m long with 100 m between them in the track's frame. Is this helpful or confusing? Unfortunately you can't investigate all the details of a thought experiment like this one without being precise about whose frame measurements are specified in. The question can be simplified by considering only inertial frames and ignoring when the trains start moving. If the trains are moving with gamma=2 and are 100 m long with 100 m in between them in their own frame, then they'll be 50 m long and 50 m between them in the track's frame. The answer is that all lengths in the direction of motion are contracted. (Just for complication, how would a train have to begin moving in order to get this situation? You can imagine the trains starting at rest in the track frame, and an inertial "ghost train" that eventually lines up with the moving trains. From the track the ghost trains are 50 m long and separated by 50 m, so the end of the last train will have to start moving first to line up with the ghost train, then the start of the second train, etc until finally the start of the forward train begins moving last. Meanwhile from the perspective of the moving trains's frame, the ghost trains are 100 m with 100 m between, while the trains at rest relative to the track are 50 m long with 50 m between. From this frame, the front of the forward train must begin moving first, and the end of the trailing train must begin moving last. Fun! Also: It would be a frame in which the tracks and the moving train have the same speed but in opposite directions, in which the trains started moving at the same time. They're length contracted the same whether at rest relative to the tracks, or traveling in the opposite direction as the tracks.)

Not speaking from experience, I can imagine some of the major draw is fantasy and a sense of belonging. With fantasy, you can escape reality and make your own. In your mind you can make your alternate reality as important and fascinating and magical as you want. So even if the cartoon is not a richly developed world, or aimed at your demographic, you can build it out in your mind and get attached to what you've imagined. Then once you've escaped into this alternate reality---which you might have done partly out of not feeling a sense of importance or belonging in the real world---and you find like-minded people who accept you and your interests, the sense of importance and community in the fantasy world becomes a real feeling. Maybe for some it starts as a slight interest, and then they get pushed away from people who think they're weird toward people who embrace and accept them. Then, the more you get in to a fantasy world, the bigger it gets for you, and the more you feel disinterested or out of place in the "real" world. This applies to any fantasy world, I think. What bronies see from inside that world is a lot bigger than what we see from the outside. It's probably very different for different people, but I see the two aspects of a fantasy world and of seeking acceptance of quirky interests as appealing.

You can experiment by holding weights. Swing them and your body will sway in the opposite direction. This already happens even without any additional weight. It's not that one part of you moves while the other is fixed; both move, but the massive parts not as much. You can detect slight backward motion of your body when punching forward. Here's an experiment: Stand steady with your back to a wall, as close to touching it as possible. Punch both hands quickly straight forward, and feel the force with which your back pushes against the wall.

But all evidence is consistent with "universal gravitation", that all masses attract each other. If you introduce another massive body, a test mass will gravitate toward it the same as it would to any other similar mass. Does your speculation predict that a mass will gravitate differently to its "source", contrary to universal gravitation? If not, how do you differentiate a "source" and everything else that effects gravitation?

I disagree. The proper conclusion is that there was thrust measured with both setups. You can conclude that there is no net thrust due to the configuration of one setup vs the other, but you can't dismiss the measurement entirely until you identify what's causing it, even if you assume it must be due to some bias or error.

I'm trying to follow this conversation but I don't get it. Is there an example of what you'd want it to do? I can't see why anyone would expect a GPS by default to give altitude relative to the tide level. If it did, would that mean that the readings on land would also fluctuate with the tides? Or that your altitude coordinate would be different depending on if you were on sea or land? I can see nautical GPS applications for that. A lot of GPS devices run custom software, and I wouldn't doubt that someone has incorporated tide data into some software (even something general, might have some kind of "sea mode" setting). If not, it should be possible to automate that. But that should be handled after getting the position from the satellites, it shouldn't be built in to the satellite system.