   goldbach conjecture
  i'm aware this is an unsolved problem, but i've run into a situation in a simulator where knowing whether the conjecture is true is a crucial deciding factor in whether the algorithm i'm employing will always work.is it generally considered safe to assume the conjecture holds?i can't go into details, but the fact that the conjecture has been verified up to very large numbers is irrelevant to the algorithm in question.also, i won't know if the simulation is working properly. i can't ever really know, however if i knew the conjecture held, i'd know the simulator was functioning properly, if i knew the conjecture didn't hold, i'd know the simulator was functioning improperly.assuming there are no other problems, could i be confident enough that the simulation was working properly, to wager rather heavily on the validity of the simulation's operations? like say, my life?   
  If it helps any, i would trust YOUR life with it being true (;hehe   ;  It's pretty reasonable to assume it is true but don't expect me to tell you you can bet your life on it. On the other hand if you do bet your life on it and lose, you'll enter history books as the person to disprove a famous mathematical theorem in the funniest way ever.Care to give more details?   ;This site says that the Goldbach Conjecture has been verified for integers up to 12*10^17 -- or, integers around 57 bits long.  Assuming you are working with 32-bit integers in your simulation, you cannot get outside this range, and you have nothing to worry about.;  but dont forget about that thing where one out of every trillion operations on a CPU comes out wrong (if your life truely depends on it ;P)   ;  Besides the checking all the cases up to a certain value, there is at least one good reason why you might expect the Goldbach conjecture to be true: the prime number theorem gives an estimate of the density of primes (informally, the probability of n being prime is 1/log(n)). According to that, the number of pairs that sum up to a particular even number should be an increasing function that by the time you get to 1000000 is in the thousands. The data seems to show that the prediction is basically correct. Wikipedia has this explanation.It could be that there is some hidden structure in prime numbers that would make the density argument fail, but not a shred of evidence exists that would suggest that.As John Conway told me, if there is no good reason why some surprising result should be true, it probably isn't. Donald Knuth also seems to think that the Goldberg conjecture is true, even if it might be impossible to prove.By the way, I don't believe the OP's claim that the correctness of his algorithm depends on this. But I wouldn't bet my life on it. :)   ;Quote:Original post by alvaroBy the way, I don't believe the OP's claim that the correctness of his algorithm depends on this.Indeed, if the correctness of his algorithm depended on the conjecture, then a demonstration that his simulation is "functioning properly" (to use his specific terminology) is equivalent to a proof of the conjecture.  This is what makes that initial claim most suspicious to me.   ;Quote:Original post by Atrix256but dont forget about that thing where one out of every trillion operations on a CPU comes out wrong (if your life truely depends on it ;P)I'm skeptical of that number because, if it were the case, with modern CPUs such errors would occur many times per hour, and since those of us without IBM mainframes are not using redundant processing, data corruption would be far more rampant than it actually is.If you're really paranoid though and can't afford a mainframe, underclock your CPU, put the computer in a sealed lead case, and sink it to the bottom of a swimming pool.   ;Quote:Original post by PruneQuote:Original post by alvaroBy the way, I don't believe the OP's claim that the correctness of his algorithm depends on this.Indeed, if the correctness of his algorithm depended on the conjecture, then a demonstration that his simulation is "functioning properly" (to use his specific terminology) is equivalent to a proof of the conjecture.  This is what makes that initial claim most suspicious to me.Not necessarily; for example, there are some root-finding algorithms that depend on the Riemann Hypothesis being true.  In order for his simulation to prove the Goldbach Conjecture, it would need to function properly on all possible natural number inputs, meaning it would have to be observed to function properly for an infinite amount of time.   ;Quote:Original post by Captain_ThunderQuote:Original post by PruneQuote:Original post by alvaroBy the way, I don't believe the OP's claim that the correctness of his algorithm depends on this.Indeed, if the correctness of his algorithm depended on the conjecture, then a demonstration that his simulation is "functioning properly" (to use his specific terminology) is equivalent to a proof of the conjecture.  This is what makes that initial claim most suspicious to me.Not necessarily; for example, there are some root-finding algorithms that depend on the Riemann Hypothesis being true.  In order for his simulation to prove the Goldbach Conjecture, it would need to function properly on all possible natural number inputs, meaning it would have to be observed to function properly for an infinite amount of time.correct   