four impossible math problems

Problem One: draw or illustrate how you would draw a line of infinte length enclosed in finite space.

normally its the simplest problems in mathematics that provide us with the best problems. or worst, depending on what your stand point is. take for example arithmetic, and the multiplicative structure of the integers 1, 2, 3......... from this comes the idea of the prime number (a number that can not be obtained by multipling two smaller numbers, except for itself and one.) yet even with the discovery and development of mathematics like chaos theory or knot theory, there is still no known formula to determine if a number is prime.
the largest known prime is 2^13466917 - 1 (that is - two to the power of thirteen million four hundread and sixty six thousand, nine hundread and seventeen minus one) found on 14-november-2001.
Problem Two: write a program or equation to find the largest prime number...?
Problem Three: write a program or equation to find primality, for all x.
...or just go playing about with it and make your processor work for the money you paid for it.
the greeks used a method known as 'the sieve of Erathosthenese' which for all x|Z, basically boils down to multiplying all combinations of integers <x and not getting x.
heres the source for more primes. click here if you want to see a previous largest prime (2^6,972,593 - 1) written out in decimal?

as you all know, a compiler converts a program written in high level language into an equivalent program in machine language then the linker resolves symbolic refferences and generates a file that can be loaded into memory and executed. therefor the program exists in the hardware electronic signals that are interprited as one's and zero's. when a computation is performed its the threads of binary that are run through the processor.
Problem Four: write a program whose binary output is random or write down a binary string that is random and prove its random.
note: a random sequence has no pattern. hence a not random sequence has a pattern. so a sequence has a pattern if it can be computed by a shorter sequence. so by definition a random sequence can not be computed by a shorter definition because it has no pattern.


one of the above was actually done i believe.eof(max)

OpenBSD as pretty darn good pseudo random number generator (not that I can personnaly prove it).

This article on PRNGs and TCP ISNs was pretty interesting (disclaimer: might fry your brain ;)
http://razor.bindview.com/publish/papers/tcpseq.html

Ammo

Problem 1: There is no sutch thing as infinte, you can have infinatly growing, but to messure something's length it must be static, and once the line start's to fold onto it self, it become's a solid shape, and not a line......so the question is impossible my consept, not by math's....
Problem 2: I'll get around to it....shouldn't be that difficult, the only problem is it would consume huge amount's of resources....

Nice Post....
- Noia

um, no. a random number dosnt not have patern...... a random number has........ that is random....... how do i say this. um what your note describes is an inrealistic number, not a random number, by your definition pi would be random but it is a mathmatical formula and can be repeted an infanite number of times and you will get the EXACT same answer, thus pi is not random, thus that isnt the definition

once i wrote a vb program that find prime numbers from X to Y, it worked well but i have some ideas about how to make this program even faster.

Oh and btw my ICQ number is a prime number :)

in linux theres a program called primes that takes input of numbers and pulls up all the prime numbers between those.......

steroid > cool, my ICQ number is even, so there. infact ill bet you that about half of the ICQ UINs are even..... he he he, im a master of the obvius!

"Problem One: draw or illustrate how you would draw a line of infinte length enclosed in finite space. "

i'm not clear on the definitions used, so i'm just throwing out some possibilities:

- a circle if the line can exist as a shape
- if not, then a decaying radius that never completely expires.
- decimal increments represented between 0 and 1.0, or 0 and .1, or 0 and .00000000000000000000000000000000001, etc.

IF they're impossible then why did you post them? :)

heres an infinite line drawn in a finite space


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problem 1. is not impossible... if you take any two number say 1 and 2 there are an infinite number of interger(numbers) between them. the numbers 1 and 2 represent finite space and the numbers inbetween them .5, .80, .75 are infinite.
just my way of looking at it

ironNsilk

What is believed to be impossible is something that has been tried many times, but still yet very difficult to find an answer, concluding that's where the phrase "nothing is impossible" came from. I don't know how to find the answer itself, but i can give a tip that when looking for an answer, look for the simplest one of all and base your answer from there. There's a missing link somewhere that could help someone find the answer, but that surely isn't me.

:--------------------: - not infinite, but still it stops somewhere
<-------------------> - infinite yes, but the numbers in integers can be considered infinite because of the numbers being able to go neg. or pos.

A fractal curve can be considered as a line of infinte length bounded by a finite space. As the detail gets smaller the length gets longer. Imagine measuring the Florida coastline. As the unit of measurement decreases (kilometers, meters, centimeters, millimeters, etc) the total length gets larger and larger.

HOW WOULD YOU KNOW THE LINE WAS INFINITE YOU WOULD NOT BE ABLE TO MEASURE IT BECAUSE IT IS IMPOSSIBLE TO LIVE LONG ENOUGH TO MEASURE IT (PERPETUAL MOTION)ENCARTA :D

The first one: fractals... they have infinite complexity but are in a finite space. If you want a line specifically then go with the Rossel or Lorenz attractors. They repeat forever, but if you look at it being drawn it is simply a line... and it never leaves a finite space.

Problem 2: I can write a program... but since there is an infinite possibility of numbers... there will always be one number at some point that is prime. As for a calculation to see if it's a prime number: take the square root of the number and store it somewhere, starting at 1, use modulus on the original number and everytime it returns 0, no remainder, other than at 1 will make it a non-prime number. You increment the integers until you reach the square root of the number. Once you get there all the numbers you hit afterwards will by byproducts of the numbers before that so you can stop there. How small a scale you want to use for the incrementing of the values is up to you, either use integers or use decimals to whatever precision you desire.

Problem 3: see problem 2

Problem 4: If it's randomly generated then there can be a pattern. It does not make it non-random because there is one, it just makes it less likely to have been random if someone thought of it. Since you can't prove something isn't random unless you made it yourself, because in all chaos everything exists, then no matter what the program returns so long as you didn't code it to do something based specifically on something else, it can be considered random. Then again, using the same argument you could say that nothing is random because something influenced it at some point and everything can be traced backwards if you tried hard enough...

Problem 1: can I use more than three dimensions?

cool link ammo, thanks. i had no idea that strange attractors were used in

networking

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Problem 1: There is no sutch thing as infinte, you can have infinatly growing, but to messure something's length it must be static, and once the line start's to fold onto it self, it become's a solid shape, and not a line......so the question is impossible my consept, not by math's....
Problem 2: I'll get around to it....shouldn't be that difficult, the only problem is it would consume huge amount's of resources....

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unfortunately infinity does exist. not as an actual number composed of digits, but it does exist. it might be easire to understand if you think about the other end of the scale first. for example, whats the smallest step you can take for 0 to 1. you can't say 0.0001 or .0000000001 or even 0.0000000000003150001, because all you have to do to take a smaller "step" is add more digits to the right of the decimal place. infact the smallest step you can take from 0 to 1 is infantesimal. so wheres this going? hmmm, if you subtract infantesimal from one, your answer is for all intents and purposes 1 (fecking engineers). but mathamatically speaking it NEVER actually reaches 1. this would be an infinity, because its infinately close to 1. infinity is a variable that describes not how big the number is, but the way it gets bigger hows problem two coming along?

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um, no. a random number dosnt not have patern...... a random number has........ that is random....... how do i say this. um what your note describes is an inrealistic number, not a random number, by your definition pi would be random but it is a mathmatical formula and can be repeted an infanite number of times and you will get the EXACT same answer, thus pi is not random, thus that isnt the definition

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i guess i was kind of unclear when describing my note. to clear it up i'd like to point out that i wasn't refering to a number, hence use of the word sequence. this was because i was refering to binary strings. not numbers, or not directly directly. i suggest you think along the lines of the Turing Machine, or Undecideability Theroy. for example the sequence "11111111111111111111" can be described as "twenty 1's". note the second
description only uses ten characters, hence is easier to say( one, one, one.....!). as another example consider the string "11111111110000000000" or "ten 1's plus ten 0's" which is described in 18 characters. so by these descriptions the second string is almost twice as complecated as the first. this leads to the idea that a binary string has complexity or information content "c" if there is a program (think of the program in binary) which computes the string from suitable data, the total length of the string and program data being no more in length than c. suppose for example the complexity of a string is n-5,
where n is the length. then it can be computed by a program whose code, plus data, had length n-5 or less. there are at most 1 + 2 + ... + 2^(n-5) = (2^(n-4) -1) such programs, or just 2^n - 4 to keep it easy. therefor there are 2^n - 4 strings of length [i]n[/] and complexity less than or equal to n - 5. since there are 2^n strings of length n the proportion having complexity <= n - 5 is at most 2^(n - 4) / 2^n = 1 / 16. Only one string in 16 has complexity only 5 greater than its length. Similarry only one string in five hundread will have complexity <= n - 10. most strings are random, or pretty close to random. the irony continues, there is a definate number K such that it is
impossible to prove that any string of 1's and 0's has complexity K. so while almost all long strings of 1's and 0's are random, it is impossible to write down an arbitrarily long string and prove it to be random. because to "write down" an arbitrarily long string you have to
state a general rule for its entries, or write the program, but then this rule is shorter than suitabely large sections of the string, so the string can't really be random. One way to think of it is to use the Solomonof approach and consider the phrase "the smallest string is not definable in fewer than 62 characters" which by definition is defineable in a phrase
of 61 characters. so are you sure your random number generator is generating random numbers?

and uuh, isn't pi a number, and not a formula, like an irrational number, or its an infinate non-recuring decimal fraction, just like e and sqrt(5).

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I can write a program... but since there is an infinite possibility of numbers... there will always be one number at some point that is prime.

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...and so the problem is impossible, case closed. buahaha:)
or if you want to be more optimistic, you could write your program, but you would probably have to rent time on a supercomputer so it can do enough calculations to just find the next biggest prime (as in, just bigger than the current largest) and then theres always going to be one bigger. the real impossible would be to come up with an equation (a single mathematical equation) that on given a number x would tell you if it was prime or not.

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a fractal curve can be considered as a line of infinte length bounded by a finite space.

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that pretty much raps up the first 'problem'.

Pi is a traanscendental number which means it won't solve any algebraic equations. Like e, it is an infinite decimal number with randomly occurring digits. There are a number of series equations that calculate pi and e, but they are only approximations as good as the number of terms in the series.

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Originally posted here by Pecosian
As for a calculation to see if it's a prime number: take the square root of the number and store it somewhere, starting at 1, use modulus on the original number and everytime it returns 0, no remainder, other than at 1 will make it a non-prime number. You increment the integers until you reach the square root of the number. Once you get there all the numbers you hit afterwards will by byproducts of the numbers before that so you can stop there. How small a scale you want to use for the incrementing of the values is up to you, either use integers or use decimals to whatever precision you desire.

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you would have a much better program if you kept a list of primes as you got them and incremented through those instead(assuming you were trying to make a list of primes). that way you would eliminate redundant testes(such as testing 2, then 3, then later testing 6)

I haven't read anyone else's responses and my thoughts are probably duplicated (?), but...

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Originally posted here by (V)/\&gt;&lt;
Problem One: draw or illustrate how you would draw a line of infinte length enclosed in finite space.

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Well, technically (?) infinite is a finite number... one of the simplest proofs would seem to be that, given any two points you can bi-sect them - that is, go from point A, half-way to point B... now go from that point, half way of the remaining distance... rinse and repeat ad-infinitum. If you can repeat this action an infinite number of times, that original distance must also be infinite. The fact that we, in time and space, can pick a point and transcend it would seem to indicate that these "infinite" distances are obtainable and, thusly, finite.

[QUOTE]the largest known prime is 2^13466917 - 1 (that is - two to the power of thirteen million four hundread and sixty six thousand, nine hundread and seventeen minus one) found on 14-november-2001.[QUOTE]

Two to the power of anything should still be divisible by two and, thusly, isn't a prime number.

Oh, nevermind... there's a "minus one" hidden in there that this font makes hard-to-read. Oops. LOL

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Problem Two: write a program or equation to find the largest prime number...?
Problem Three: write a program or equation to find primality, for all x.

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These sound like an assignment I had back in college and, well... I'm not going to try to repeat them here. LOL

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There are a number of series equations that calculate pi and e, but they are only approximations as good as the number of terms in the series.

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i don't think there is an equation for getting them. pi is just a ratio, the ratio the circumference fits into the radius or something, and its the same for all circles, coincidence, i think not! and e is just like a naturally occuring phenomenon, like, its the rate that a cup of coffee will cool (or is that e^-x) and its the rate the universe is expanding and loads of other stuff im sure.

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you would have a much better program if you kept a list of primes

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now THAT sounds like a predefined order of events, and a pretty good one at that. (a plan)

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go from point A, half-way to point B... now go from that point, half way of the remaining distance... rinse and repeat ad-infinitum

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thats pretty much the idea of it alright, technically (in simpliest form) you should have said circle, not line....Problem One: draw or illustrate how you would draw a line of infinte length enclosed in finite space. one of the earliest is the von Koch snowflake curve, which is enclosed in a finite space (erect on each side of an equilateral triangle, a triangle one third as large, and repeat infinately). functions with no derivatives, buahahahaa.

max

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Originally posted here by (V)/\&gt;&lt;

i don't think there is an equation for getting them. pi is just a ratio, the ratio the circumference fits into the radius or something, and its the same for all circles, coincidence, i think not! and e is just like a naturally occuring phenomenon, like, its the rate that a cup of coffee will cool (or is that e^-x) and its the rate the universe is expanding and loads of other stuff im sure.

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um, if there were no equations to get them how would we have the value? anyway, there are equations for atleast pi, but i'm pretty sure there are for e too.

Has anyone ever thought that our views on infinity or even mathamatics is faulty. Maybe the way we are thinking is like Windoze software filled with bugs and loopholes. I mean is there anything that we can think of that is truly infinite :confused: Some people claim proof that the Universe is infinite and I know what my Physics teacher said but I can't help thinking that it is our own mathematic laws that make it hard to understand these anomalies. Sorry a little off topic :D

Isnt the geometric RAY infinate?

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