ElijahJones

Yeah its easy you cannot have a bijection between sets of different sizes. That doe snot mean you cannot find maps that have useful structure though. On pumices problem let me see what I get form his original question. Ok Pumice you want [MATH]f_1(A,B,C,D)=g_1(A,Z,D)[/MATH] [MATH]f_2(A, Z,D)=g_2(B,C)[/MATH] I think you are generalizing your problem too much and by restricting you numbers to 0-1000 are actually making the problem harder. Just let everything be integers any integers what soever or if you have a fetish say let them be any natural numbers. If you would give me more details about your specific problem or possible insight I can saymore but here is somethin gthat satisfies your requirements from the first post. [MATH]A+B+C+D=A+Z+D[/MATH] This satisfies both of your conditions because you can substract A and D from both sides to get B+C=Z. Or multiplicatively you could go [MATH]ABCD=AZD[/MATH] Again the same thing is implied now just divide both sides by AD to get BC=Z. What was your reason for asking this question if I can ask?

I agree with Dave. Way to leverage cardinality in that proof Matt. I agree that the scalar multiplcation need not be the same as the operation used on elements of the vectors, but since it has to map into the same set its standard to have them be the same. In fact I cannot think of an example where those two operations are different. You construct a vector space by taking a field as scalars, but that field also provides the elements of your vectors (typically). Its the only way it works (as far as I know). If your vectors are formed on a subset X then F.X must equal X or else you do not have closure. But look X must be a subset that is invariant under multiplication by F. The definition of vector space makes no assumptions about the multiplicative structure of X, it need only be an abelian group under vector addition and that may not necessarily be derived from addition on X(though I cannot think of an example where it is not). So the question seems to imply more restrictions on a vector space than the definition itself. To get away from that apparent ambiguity lets say that the addition of vectors is derived from an addition defined on X an abelian group and that X is a subset of a field F that we will use as our set of scalars. The implication of this if we want a vector space is F.X=X. But if F.X=X then a.1 is in X for all a in F which contradicts the fact that X is a subset of F. Suppose on the other hand X contains F, then again we require F.X=X. Now this might be possible. It might be possible to take your vectors over C and have the scalars be in R and have a vector space. I mean multiplying a complex number by a real number r simply increases its modulus by sqrt®. Hmm I never thought of that before. I do not know where it would be useful but it does seem to work. So it works if the field F is a subfield of the set X you take your vector elements from but not the other way around. But trying to force Z to be closed under action of R is futile. Just one counter example will do the trick or a proof which I gave above. Suppose you did the opposite. What if we take complex numbers as the vector elements and use Z as the scalar ring (remember Z is not a field but it is an integral domain which is a commutative ring among other things). Well then of course we get this: scalar multiplication can only take vectors to discrete locations meaning we have established a very interesting partition of C^n, n is the dimension. Now since Z is not a field we cannot call our construct a vector space it is properly called a Z-module (Dummit and Foote 2nd Ed p388) The question turns out to be interesting though IMHO. BTW the vectors only need to be an abelian group under vector addition adding an inner product can yield a normed vector space. Adding a cross product can produce a field of vectors, whether you allow scalar multiplication or not.

Well T here is just the unit circle (right its all points one unit from the origin). Lets cal lyou set M. M is dense in T means that the closure of M is T. So you want to show that among points one unit from the origin, the closure of M is T. What is the closure? Its the intersection of all closed sets containing M in T. It is the union of M and its limit points in T. I cannot find your function cis(x) even in Knopp. But suppose the limit points are in fact {-1-i,1+i}. Well look these cannot be the limit points because [MATH]||-1-i||^2=2[/MATH] so your limit points are wrong. Anyways I would be happy to help further. The outline of the prooof is not difficult. What exactly is your function cis(x) and how is it constrained to T?

There is no mathematical definition of a wave. Waves are physical phenomena. There are functions that gets used to model waves, sine and cosine functions, functions of complex variables, fourier series etc. A vector V in its most general form is simply an ordered n-tuple whose elements are taken from a space S. Since vector operations are usually only meaningful when constructed from the operations of S, we choose S to be a set with some operations that give algebraic structure usually a field. An n-dimensional vector space can be contructed from the set of n-tuples taken over a field F by defining vector addition in natural ways from the operations of F. You must also include F as the set of scalar multipliers satisfying certain conditions. On top of this is usually defined an inner product (the dot product on multivariate calculus). If you want a really hard but excellent text try Dummit and Foote "Absract Algebra". I have 2nd edition but it may be into a third or fourth by now.

Light appears to move more slowly through certain crystals. If photons have no mass how can their path be altered by gravity? Answer, its not gravity warps space and time and light is just following equivalent of a "straight" line. I'm glad someone pointed out that E=mc^2 is for a particle at rest because it explains nothing. But looking at this thread reminded me that I have never seen a good derivation of the formula or an explanation of its context. I think I'll try Wikipedia and see what they have. They claim E-mc^2 is the basis for the bomb. It must have something to do with binding energies.

Parsimony is basically the same as Occam's Razor, which says: all things remaining equal the simplest solution is usually correct. It is sort of like saying that every system takes the shortest path to equilibrium. In classical physics this principle explains pretty much everything you see but it does not explain how anything could exist in the first place. Notice though that the lowest energy state is only one of many that a system can be in and there can be many paths of equal energy between two points (conservative systems). So when I read about evolution and they just casually use parsimony I say "ok what the hell it seems reasonable." Math modelers use parsimony all the time. But it does not prove that the tree of life is as easily explained as we want it to be. More realistically it gives us a starting point, namely simplicity.

Well I was gonna say something like what snail said but it was already said. Let me posit a theory and some possible tests of your hypothesis. You may have gotten electrocuted and it caused you to black out. If you had had 120V 15AMP AC on that battery for any amount of time it would have gotten very hot and possibly exploded or at least started leaking. Your dumping [MATH]P=IV=1800[/MATH] joules per second into a device filled with chemicals. The spark would not have produced a very large pulse. We could calculate how big of a gap this power source would jump. But remember saying it jumps is an illusion it actually ionizes air molecules first and then the sudden movement of these ions along a plasma path cause the spark. The light you see blue tells you something about how fast the ions are traveling when they collide with other air molecules. The net EMP from this should not be much larger magnetically than the field around the wire. The electric field was stronger for a moment and may have caused sympathetic electric currents in your brain. I think it can give a noticeable real effect but it is likely that you just got damn scared and so now you know not to play with electricity. I am not an expert on EM but I did have an EM fields course in college. Have you ever seen a lifter? Google lifter, now those things can ionize some molecules let me tell you. Ok anyways some ways to test your ability. 1) A Blind Test- erect a piece of construction paper. Sit on one side and place your arm next to the paper. Now making sure you cannot tell when they are moving have a friend move a magnet towards and away from the paper. Then at several points have him say "now" and you have to say whether the magnet is close or far away from the paper. That would test sensitivity to static field. If you don't get over 66% on three consecutive trials its probably not significant. You can also have you say "moved" if you feel anything odd, he will be able to verify if he was actually moving the magnet when you said "moved". This would test sensitivity to the time dependent field and is more like an EMP. A sensitivity to magnetism: you may have some metal in your blood, magnetite in non lethal levels or something. Who knows? Anyways because you are so clever and perceptive it is important that you not be able to see your friend and not hearing him may also be good. To do that you must cover your eyes and ears, make sure there are no breezes moving as he moves. In this version you agree to start the experiment at a certain time and both of you log your movements. He/she might write 7:32 movement towards and away 15 seconds. And you might right 7:45 felt something. Then you can time correlate. It would be interesting if you really have this ability because some hypothesize that most people who claim to be psychic are sensitive to EM fields. 2) Electric fields: We are all sensitive to static electric fields through our hair follicles. Time dependent e-fields induce magnetic fields in objects. Unless you are properly trained I would stick with rubbing your socks on the carpet and reaching for the door knob. You should test your theory though to see if it holds true. You will likely learn something about yourself and have some fun.

Yeah Richard is pretty sexy as a human being of any gender. To be so at ease with his own genius and apparently so free. Its too bad we all are'nt as well adjusted.

Nice, yeah I am in grad school now, environmental science and policy. My thesis is on modeling ecosystem change (ecological succession). Nice to meet another math major. Your profile said you like mathematical physics, any topics in particular?

Well you are probably going to say I am pulling a slights of hand but there are lots. One way to get more is simply to let [MATH]a[/MATH] be whatever you want it to be greater than one. Base 11 (11+2)(11+3)=13*14 Base 9 and Base 10 (9+2)(9+3)=(10+1)(10+2)=132 Suppose we did not know this we would have written [MATH]132 = a^2+C_1a+C_0 = a^2+(r_1+r_2)a+r_1r_2=(a+r_1)(a+r_2)[/MATH] We would look for [MATH]a[/MATH] such that [MATH]r_1r_2<a[/MATH] and [MATH]r_1+r_2<a[/MATH]. The point is we need a way of determining if such an [MATH]a[/MATH] exists for every natural number greater than three and a good algorithm for finding it if it exists. Can I ask how far along you are in your math training? I don't want to be talking about ring homomorphisms and Euler's phi function if you have not been exposed to those things. EJ

Extremes happen on both fronts so I am going to use a ® and (l) system here for the right and left extremes. ® and (l) if you fixate on the conduct of others even though it has no real effect upon your life (l) if you dress up as a condom for two consecutive Halloweens ® if you clutch your purse whenever a black man walks by (l) if you insist that everyone's child should read the story "My Two Dads" ® if you think that only people in your denomination are going to heaven (l) if you think that giving birth control to poor African women improves their standard of living ® if you think that by forcefully removing governments you are promoting democracy ® and (l) if you think that a giant corporation will simultaneously save and destroy the world

Cosmologists, like Quantum Theorists, and Economists are sensitive about their assumptions. I can accept an expanding universe, but I did hear a theory recently related to quantum field fluctuations that the universe sort of emerged at a certain 'bigness' and then took off from there. I don't now. Does it matter? Will it ever matter? I am a math guy so the question of whether observing that alot of things are moving away from us means everything is, well thats a statistical issue. I mean you know they have not checked every visible galaxy. I leave it be, because its an irrelevant question when it comes to human happiness. If that ever changes I will pay more attention to it. Dems my thots.

I like iglac's point. The forums are replacing many real community links. I like it because I live in a town that is a good place mostly but not very educated and there just is not much here for me. I come to a place like this and well as he said I find things I like to talk about and people who share common interests. There is a need for some regulation on the net regarding porn, spamming, cyber-crime like identity theft. ICANN is working on these issues. Prosecution of cybercriminals is an intersting problem because it requires multinational jurisdiction. So in that way the problems of the internet are really leading us towards a more interconnected world and not away from one. The full potential of the internet is still not appreciated by most casual surfers. imagine if you will a Library of Alexandria on line. An interactiv heuristically organized database of human knowledge, both historical and cutting edge with links to communities that are talking about whatever you want to know. It gives me tingles just thinking about it. The internet is a wonder of the modern world. My life would be less satisfying without it.

Can you describe the offending substance? Is it sticky like syrup, or is it more like stickky brown grey dust? What type of tower does she have or is it a workstation. If its sticky syruppy, you won't get it off the components without hurting something. Be careful open the PSU, there are high voltages. Wait at least five minutes after turning off the machine and be sure to not let your fingers become connectors for a spark.

The argument for tens of millions of intelligent life forms in the galaxy gets beat up by the reality that ecosystems are constantly being perturbed by things. There was a time around 60,000 years ago when the human population is thought to have dropped to 2,000 people. In fact Spencer Wells just found that every living human today is descended from one man who lived in Africa around 50,000. Now that is precariously close to extinction. So its hard to say that "certainly" life like us exists elsewhere because even on very suitable planets survival is a roll of the dice. Well unless you think that there may be other forces at work, metaphysical forces, but I am talking material science here.

Well, this is an active topic. Might as well throw in a few thoughts. I feel strongly that the existing arguments for an old earth (~4 billions years old) are in fact rational and sound. Also thinking old earth, Genesis 1-3 shows alot of insight if you read it right. I am also certain from studying mathematics and formal logic that you cannot prove or disprove the existence of God(omniscience, omnipresence, omnipotence). Since we somehow were "given" rational minds with which to find our way through the world, God cannot be in the business of sending people to hell based upon their sincere belief in rational arguments as tools for revealing truth. Its a fallacy that has invaded modern American Christianity that not believing the Earth is 6,000 years old causes the whole Bible to be meaningless. Its just not true. The wisdom of Proverbs stands all by itself. The vision of the endtimes written of by the prophets are inspiring to read and await fulfillment, thus they cannot be proven wrong at this point anyways. The message of Christ centered upon the golden rule is however in direct conflict with some of the ideas that Dawkins is throwing around. I believe that it is scientifically, socially and ethically more valid to create communities that bridle extreme self interest with an ethic of cooperation. We have lost that sense of really needing each other in America and the place is worse off because of it. People exist day to day in competition with each other, this is fine for producing pre-championship Michael Jordans. But even the great one had to learn how to empower his team to reach the pinnacle (those championships owed as much to Rodman, Pippen, Kerr, Paxon as they did to Jordan). Yes it is true that one can cooperate from selfish ambition. Personally I never get the warm fuzzies from using people like that. Of course I want to succeed, but I really want you to succeed too. Because when both of us are fulfilled we really have something to fight for, something to cooperate for. Sadly we have created a world where too many prizes go to only one person. But don't you realize that the food you get at the supermarket is produced by the work of thousands of others just like you? The gas that you pump, thousands of human hours went into that. How can we only be in it for ourselves when we need each other so badly? No, in my opinion humans will never see their real potential as a species until they know how to love. And with the world going the way it is, that is getting harder and harder each day.

The famous Last Theorem of Fermat (proven by Andrew Wiles in the early 1990's) http://en.wikipedia.org/wiki/Andrew_Wiles says There are no positive integer solutions for [MATH]n>2[/MATH] to the equation [MATH]z^n=x^n+y^n[/math] We will not prove this theorem here (Wiles method is very advanced). But we will play around with the implications a bit. Something fun for us mortals to do si assume the negative of FLT and then start messing around a bit and see what happens. It remains entertaining only because some people believe Fermat actually solved it with only his knowledge of number theory and that someday the "real" proof will re-emerge. It seems unlikely but stranger things have happened. So assuming the negation, suppose we have found an integer solution to FLT. Notice that we can easily write [MATH]x^n=z^n-y^n=(z-y)\sum_{i=0}^{n-1}z^{n-1-i}y^i[/MATH] It can be shown that if [MATH]n[/MATH] is even then [MATH]\sum_{i=0}^{n-1}z^{n-1-i}y^i=(z^s+y^s)\sum_{i=0}_{n/s}z^{n/s-i}y^i[/MATH] If [MATH]s[/MATH] is odd we can factor further since [MATH]z^s+y^s=\sum_{i=0}^{s-1}z^{s-1-i}y^i[/MATH] In all there are about four different factorization formulas that (no matter what [MATH]n[/MATH] is) let you peel this thing down to irreducible polynomials in [MATH]z[/MATH] and [MATH]y[/MATH]. Is there anything special about irreducible polynomials in this situtation? One thing it would mean here is that [MATH]x[/MATH] is not prime, other wise every irreducible polynomial factor would be a power of [MATH]x[/MATH] contradicting the fact that they are irreducible. We can follow a similar construction for [MATH]y[/MATH] and [MATH]z[/MATH] showing that they cannot be prime. Now somewhere in the dusty literature is a theorem that says that any allowable solutions to FLT must have at least two of the bases being prime. Have we found a contradiction? Why or why not? I'll get back to this eventually. Suffice to say that the theroem is proven (by Wiles) and now the result can itself be used to prove theorems.

In this method we will assume that every integer can be written in the form [MATH]N=az+r_0=(ax+r_1)(ay+r_2)[/MATH] We consider here only the case when [MATH]a=6[/MATH] and [MATH]N[/MATH] and at least one factor pair are all congruent to one modulo six. We proceed by [MATH]N=6z+1=(6x+1)(6y+1)[/MATH] [MATH]6z=36xy+6y+6x[/MATH] [MATH]z=6xy+y+x[/MATH] [MATH]y=(z-x)/(6x+1)[/MATH] Since [MATH]z[/MATH] is known we have reduced the size of our search area from all numbers under the square root to something nearer to the number of primes less than the square root. However just like in Method 3, [MATH]x[/MATH] is a substitute for knowledge of these primes. Example: Let [MATH]N=91[/MATH]. Then [MATH]91=6*15+1[/MATH] [MATH]15=6xy+x+y[/MATH] [MATH]y=(15-x)/(6x+1)[/MATH] Starting with [MATH]x=1[/MATH] we get [MATH]y=(15-1)/(6*1+1)=14/7=2[/MATH] It follows that [MATH]91=(6x+1)(6y+1)=(6*1+1)(6*2+1)=7*13[/MATH] And this is correct. This method leaves us alot to think about. In the end the general efficiency approaches knowledge of every prime under the square root and in many cases it is better. Enjoy!

This method builds on Method 1. Suppose we have [MATH]N[/MATH] an integer we wish to factor. Use the base [MATH]a[/MATH] algorithm to obtain [MATH]N=a^2+C_1a+C_0[/MATH] Suppose now that [MATH]r_{1}r_{2}>a[/MATH] then we arrive at a different more general system of equations than in Method 1. [MATH]C_1=r_1+r_2+t[/MATH] [MATH]C_0=r_1r_2-at[/MATH] We now have two equations in three unknowns leading to a search for [MATH]t, r_1[/MATH] or [MATH]r_2[/MATH] depending on our preference. Searching for [MATH]t[/MATH] is quite straightforward. We simply check each reasonable value for [MATH]t[/MATH]. That is we pick a value for [MATH]t[/MATH] and then solve the system [MATH]C_1-t=r_1+r_2[/MATH] [MATH]C_0+at=r_1r_2[/MATH] Determining bounds for [MATH]t[/MATH] requires making assumptions about the distribution of factors. Here is an algorithm for investigating this new system. 1) Suppose [MATH]C_1[/MATH] is small. Then it is routine to simply check [MATH]t[/MATH] from zero to [MATH]C_1-2[/MATH] 2) Notice that if [MATH]r_1[/MATH] and [MATH]r_2[/MATH] are residues of [MATH]a[/MATH] then [MATH]r_1r_2<(a-1)^2=a(a-2)+1[/MATH] so checking for [MATH]0<t<a-2[/MATH] completes the task if we are focusing on the equation [MATH]C_0+at=r_1r_2[/MATH]. Given the way the residues of [MATH]a[/MATH] multiply you are likely to find [MATH]t[/MATH] long before you get to [MATH]t=a-2[/MATH]. This means that the algorithm is not much worse in many cases and is often better than knowing all the primes less than the square root of [MATH]N[/MATH]. Only you don't need to know them in this algorithm [MATH]t[/MATH] is a substitute for them. 3) If we use one equation to eliminate one of the variables. We get [MATH]C_1-t=r_1+(C_0+at)/r_1[/MATH] [MATH]r_1^2+(t-C_1)r_1+C_0+at=0[/MATH] When you solve this for [MATH]r_1[/MATH] using the quadratic formula you end up with [MATH]r_1[/MATH] as a function of [MATH]t[/MATH]. We could also have solved for [MATH]t[/MATH] and obtained [MATH]t=(-r_1^2+C_1r-C_0)/(a+1)[/MATH] The crucial part of even looking at this method is understanding how these systems and the attendant residues and substitutes change with [MATH]a[/MATH] for a given natural number [MATH]N[/MATH]. The prime number theorem suggests that in general it is hard to get an algorithm that will do better than if you just divided by every prime less than the square root. That said, novel ways of factoring integers may still exist undiscovered and this is something that gives RSA some pause (read their website).

Yes of course. Let [MATH]N=143[/MATH] Then using base [MATH]10[/MATH] [MATH]143=10^2+4*10+3[/MATH] Where [MATH]C_1=4[/MATH] [MATH]C_0=3[/MATH] From the theorem if the factors of [MATH]N[/MATH] are linear over base [MATH]10[/MATH] we have the invertible system [MATH]4=r_1+r_2[/MATH] [MATH]3=r_1r_2[/MATH] Substitution gives [MATH]4=r_1+3/r_1[/MATH] [MATH]r_1^2-4r_1+3=0[/MATH] And that we know how to solve. Notice though that in this simple example we get the answer very quickly if we realize that under these circumstances everything must be a natural number so factor [MATH]C_0=3[/MATH]. Three is prime and has only the factors one and three. It is easy to verify that one plus three equals four so that the pair [MATH](1,3)[/MATH] must be a solution to the factoring problem. The result is [MATH]143=(a+r_1)(a+r_2)=(10+1)(10+3)=11*13[/MATH] The questions that become important then are these. 1) For sufficently large natural numbers with factor pairs near the square root does such a base [MATH]a[/MATH] always exist? 2) Is there a straightforward method for finding these special bases given a number to be factored? 3) Looking at base [MATH]a[/MATH] polynomials in general are there other novel factorizations? NOTE The base [MATH]a[/MATH] algorithm is the process used to find the digits of [MATH]N[/MATH] base [MATH]a[/MATH]. The coefficients of the base [MATH]a[/MATH] polynomial for [MATH]N[/MATH] are it's a-nary digits. I have more to say about this if needed. Enjoy!

No if you pressurize ice it will fracture at some point it is a lattice. Ice becomes brittle at very low temperatures also (crunchy snow). But then it does have some viscosity less than one (glaciers flow). I would not bet against some odd things happening with ice that gets really cold like 5 K or so. Ice will melt under pressure, just applying the pressure causes it to warm and melt somewhat. Water is incompressible but ice is not. Here is a thought experiment. Take a sphere of perfect ice one foot in diameter. Put it into a spherical vice that applies pressure evenly. Now increase the pressure. What do you think happens? The thing to ask yourself is this. Is there an energy state that the volume of ice can move to that helps to reduce the pressure. The system wants to ease the pressure. Here is the tricky part, I see two possible states. 1) A crushed ice cube may actually take up less volume under pressure than an intact one, it has to do with the lattice structure. So if fracturing can produce a state where the volume of ice is minimized then fracturing may occur. 2) If however this is not the case, it may still fracture but the pressure gradient will produce heat which initially reduces the volume. Water at 32 is more dense than ice, so after the phase change the pressure is reduced. Can there be such a thing as super cooled water? If I were building an experiment I would expect to find fracturing (mostly because real ice is not perfect and not spherically symmetrical) and some amount of melting. The specific amount will be hard to get at but it will be a change that minimizes the forces that are acting. The system will find its lowest energy state. This may not be a static state it may begin to produce flows of energy (heat and kinetic energy) that are cyclic or even chaotic. Perhaps a thin sheet of water will cover the sphere and the sphere will rotate slightly before coming to rest. Is'nt physics great!

Theorem If [MATH]n+1=(s+1)(m+1)[/MATH] then [MATH]\sum_{i=0}^{n}a^{i}=\sum_{i=0}^{s}a^{i(m+1)}\sum_{i=0}^{m}a^{i}[/MATH] for any natural number [MATH]a[/MATH]. Suppose that [MATH]N=\sum_{i=0}^{n}a^{i}[/MATH] where [MATH]n+1[/MATH] is composite. Then [MATH]N[/MATH] is factorable by the theorem above. [MATH]N=\sum_{i=0}^{s}a^{i(m+1)}\sum_{i=0}^{m}a^{i}[/MATH] Notice that the theorem establishes a correspondence between the factor pairs of [MATH]n+1[/MATH] and some subset of factor pairs of [MATH]N[/MATH]. Although the result is not generally useful I include it because of the elegance of this correspondence. I define the fermat divisors of [MATH]n[/MATH] to be one less than the divisors of [MATH]n+1[/MATH]. The advantage here is that we can factor [MATH]N[/MATH] by simply factoring [MATH]n+1[/MATH] which is typically much smaller.

Thank you Shelbyville!

This one is going to be a little tough unless you understand how the base [MATH]a[/MATH] algorithm produces a polynomial in [MATH]a[/MATH] for every integer [MATH]N>2[/MATH]. Using the algorithm for a suitably chosen base [MATH]a[/MATH] we can obtain [MATH]N=a^2+C_{1}a+C_{0}[/MATH] (1) Let us assume that we can pick [MATH]a[/MATH] so that not only is this true but that at least one pair of factors of [MATH]N[/MATH] can be written as [MATH]a+r[/MATH] where [MATH]r[/MATH] is a residue modulo [MATH]a[/MATH]. Then we can also write [MATH]N=(a+r_{1})(a+r_{2})[/MATH] [MATH]N=a^2+(r_{1}+r_{2})a+r_{1}r_{2}[/MATH] (2) Equating coefficients from 1 and 2 above yields the system [MATH]C_{1}=r_{1}+r_{2}[/MATH] [MATH]C_{0}=r_{1}r_{2}[/MATH] The conditions are [MATH]r_{1}r_{2}<a[/MATH] and [MATH]r_{1}+r_{2}<a[/MATH]. It is needed to show that for sufficiently large [MATH]N[/MATH]with factor pairs near the square root of [MATH]N[/MATH] there always exist values for [MATH]a[/MATH] that allow this construction. If one could devise a scheme for finding these values of [MATH]a[/MATH] quickly any RSA number would be within reach. Have fun!

I will try to introduce several methods here. There is security sensitivity to this topic but nothing we do here is likely to approach a real RSA caliber factorization method so we should'nt get visited by NSA.