From: david.brown@hesbynett.no   
      
   On 14/02/2026 22:22, MitchAlsup wrote:   
   >   
   > David Brown  posted:   
   >   
   >> On 14/02/2026 19:48, MitchAlsup wrote:   
   >>>   
   >>> Thomas Koenig  posted:   
   >>>   
   >>>> Some people are throwing large amounts of computing power at trying   
   >>>> to disprove the Collatz conjecture.  Graphics cards have sped this   
   >>>> up enormously.   
   >>>>   
   >>>> (A reminder, if any is needed:  From a starting number n, the   
   >>>> transformation, recursively applied,   
   >>>>   
   >>>>   
   >>>>      f(n) = 3*n+1, if n is odd; =n/2 if n is even.   
   >>>>   
   >>>> is conjectured to always reach 1 for any positive integer 1).   
   >>>   
   >>> While n/2 remains even, n/2 is guaranteed to converge to 1.   
   >>> So, only the odd cases are interesting.   
   >>>   
   >>> 3*n+1 is always even (since n was odd, and odd*odd is odd and   
   >>> odd+1 is even). So, a double iteration is (3*n+1)/2.   
   >>   
   >> Yes, that is often a micro-optimisation used.   
   >>   
   >>>   
   >>>> All the work is done on general-purpose hardware, which has many   
   >>>> capabilities that are not needed, and consume area and power that   
   >>>> special-purpose hardware would not need.  Also, the hardware   
   >>>> is limited to 64-bit integers, and the range of tested numbers   
   >>>> is now up to 2^71, so   
   >>>   
   >>> Induction would say it has been proven by then.   
   >>   
   >> Would you like to re-think that?  "Induction" says nothing of the sort.   
   >>   
   >> The conjecture has been proven true for all numbers up to 2 ^ 71   
   >> (assuming that is the current figure) - it has most definitely /not/   
   >> been proven to be true for all numbers.  There are other things proven   
   >> about it, such as the asymptotic rarity of numbers that are exceptions   
   >> to the pattern, but no one has any notion of how to prove it in general.   
   >   
   > By proving up to 2^71, you simultaneously prove that it cannot be proven   
   > by computer-like arithmetic processes. What if the number was 2^71,000,000   
   > you still have not proven it? Thereby, it is not provable by simple computer   
   > arithmetic processes.   
   >   
      
   All that has been proven is that if the conjecture is not true, then the   
   smallest exception to the pattern is greater than 2 ^ 71.  Maybe it is 2   
   ^ 71 + 1.  Current computer technology could take the search a lot   
   further than just 2 ^ 71 - if someone wanted to pay for it!  Future   
   computer technology could perhaps greatly increase the bounds.  Maybe   
   Thomas's "useless machine" could lead to far greater bounds without   
   excessive costs - and maybe research into that could lead to useful new   
   technology or techniques with applications beyond an interesting maths   
   problem.   
      
   There are lots of examples of conjectures where the first exception is a   
   high number, and all testing of low numbers fitted the conjectures.   
   Sometimes such counter-examples are far beyond the reach of any   
   brute-force computational testing, but sometimes not.   
      
   We have not proven that the Collatz Conjecture can be disproven by   
   brute-force testing - we have merely shown that doing so would be   
   difficult, and seems unlikely to be successful.  And we have known all   
   along that computational methods can never prove a conjecture like this   
   to be true - that much should be obvious.   
      
   Testing conjectures like this can be helpful to mathematicians.   
   Sometimes they can show patterns that can be used to get more insights -   
   maybe there is a pattern in the numbers that lead to peaks of growth of   
   the Collatz function before dropping towards 1.  Such patterns can   
   inspire proofs - or more targeted searches for exceptions.   
      
   > Ergo, you need a different means.   
      
   You don't imagine that the only thing mathematicians have done with the   
   Collatz Conjecture is test it to large numbers, do you?   
      
   >   
   >> Being true up to 2 ^ 71 makes it reasonable to believe it is always   
   >> true, but that is very different from being /proven/.  There are plenty   
   >> of mathematical conjectures that are true up to very large numbers, but   
   >> fail later on.   
   >   
   > You need cannot even /prove/ it using BigNum computer arithmetic in finite   
   > time.   
      
   Surely everyone who has done high-school mathematics already realises   
   that a conjecture like this cannot be proven by trail and error alone?   
   Brute-force testing is about looking for exceptions to /disprove/ the   
   conjecture, or gaining insights that can lead to its proof or disproof.   
      
   >>   
   >>>>   
   >>>> Comments?   
   >>>   
   >>> Interesting problem,   
   >>> waste of power.   
   >>   
   >> It is a mathematically very interesting problem.  Whether there is   
   >> anything to be gained by continuing the search for exceptions by brute   
   >> force, is a different matter - but /if/ it is worth doing, then doing so   
   >> more efficiently is a good idea.   
   >>   
   >>   
      
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