tag:blogger.com,1999:blog-3722233.post1093332795126035941..comments2024-04-19T18:30:53.405-05:00Comments on Computational Complexity: A Math Question and a Meta QuestionLance Fortnowhttp://www.blogger.com/profile/06752030912874378610noreply@blogger.comBlogger18125tag:blogger.com,1999:blog-3722233.post-39409124111386703512024-04-07T00:36:26.086-05:002024-04-07T00:36:26.086-05:00"Python (yes, Python has large integer arithm..."Python (yes, Python has large integer arithmetics out of the box!) says that these numbers produce 3745.00642."<br /><br />Python has large integers, but it also converts ratios to floating point. If you clear fractions and then do the calculation, you're more likely to be able to test solutions.<br /><br />(i.e. set a = y+z, b = x+z and c = x+y, then<br />xbc + yac + zab = 4abc)<br /><br />But, yes. The number immediately above are wrong. The numbers given earlier by another anonymous at 4:44 work fine in Python using the above cleared-fraction method.<br />Anonymousnoreply@blogger.comtag:blogger.com,1999:blog-3722233.post-72789346080652219792024-04-05T05:46:42.601-05:002024-04-05T05:46:42.601-05:00Stop using AI tools as if they're oracles, alr...Stop using AI tools as if they're oracles, already.<br /><br />Five seconds of Python (yes, Python has large integer arithmetics out of the box!) says that these numbers produce 3745.00642. At first, I thought "well, at least it's close to an integer" - then I realized it's off by three orders of magnitude. Anonymousnoreply@blogger.comtag:blogger.com,1999:blog-3722233.post-57347965021018361792024-04-04T11:21:31.249-05:002024-04-04T11:21:31.249-05:00I haven't checked this solution, but Perplexit...I haven't checked this solution, but Perplexity AI says the solution to the equation x/(y+z) + y/(x+z) + z/(x+y) = 4, where x, y, and z are positive integers, is as follows: <br /><br />The smallest positive integer solution to this equation is:<br />x = 36875131794129999827197811565225474825492979968971970996283137471637224634055579<br />y = 4373612677928697257861252602371390152816537558161613618621437993378423467772036<br />z = 154476802108746166441951315019919837485664325669565431700026634898253202035277999125Anonymousnoreply@blogger.comtag:blogger.com,1999:blog-3722233.post-22537214146212009182024-04-04T10:38:35.713-05:002024-04-04T10:38:35.713-05:00PS This https://pari.math.u-bordeaux.fr/gpwasm.htm...PS This https://pari.math.u-bordeaux.fr/gpwasm.html can be used to check answers :-).Andyhttps://www.blogger.com/profile/14405544863964795598noreply@blogger.comtag:blogger.com,1999:blog-3722233.post-80621489550712481932024-04-03T20:24:21.844-05:002024-04-03T20:24:21.844-05:00Just for the record, the above comment was mine: I...Just for the record, the above comment was mine: I forgot to set the from field. I was wondering how much faster clearing fractions in the original expression, coding the resulting integer inequality it in 64-bit ints in C++, and searching for solutions would be, but the solution shows search wouldn't find it, no matter how sneaky I got about finding shared subexpressions and calculating them first...<br /><br />David in Tokyonoreply@blogger.comtag:blogger.com,1999:blog-3722233.post-15080336237342243512024-04-03T16:24:40.080-05:002024-04-03T16:24:40.080-05:00I had fun with this a couple of years ago. It too...I had fun with this a couple of years ago. It took me a few weeks, working off and on. As noted above, I used some theory of Elliptic Curves. Here is a link to my solution: https://1drv.ms/b/s!AhtWOqognyHphyTQdcEXyyLNDEyq?e=mQ4fpu<br /><br />As for the meta question, I like it when people post an answer to puzzles they post. I have no problem if they post it a week later or something. I guess the posted answer shouldn't be "in your face" with the puzzle, so that one can choose to discipline one's self to work on it for a while.Andyhttps://www.blogger.com/profile/14405544863964795598noreply@blogger.comtag:blogger.com,1999:blog-3722233.post-29130449385706450342024-04-03T12:35:40.667-05:002024-04-03T12:35:40.667-05:0011 4 -111 4 -1Anonymousnoreply@blogger.comtag:blogger.com,1999:blog-3722233.post-44003986411298079702024-04-03T09:19:40.017-05:002024-04-03T09:19:40.017-05:00There's no need to exclude 0, since there'...There's no need to exclude 0, since there's no integer solution with 0. E.g. setting z=0 results in x/y + y/x = 4, where x/y = 2 +- sqrt(3) which is not rational.Anonymousnoreply@blogger.comtag:blogger.com,1999:blog-3722233.post-34778558762817495012024-04-03T09:03:44.959-05:002024-04-03T09:03:44.959-05:00(bill) The problem asked for positive naturals. Ho...(bill) The problem asked for positive naturals. However, I wonder if there is some way to, given an answer, find another answer with bigger values, so yours could be a starting point for finding a solution. Anonymousnoreply@blogger.comtag:blogger.com,1999:blog-3722233.post-22420289206529936232024-04-03T05:34:21.205-05:002024-04-03T05:34:21.205-05:00hint:
the smallest solution has 80 digits and requ...hint:<br />the smallest solution has 80 digits and requires advanced number theory (elliptic curves).<br /><br />solution:<br />https://www.agftutoring.com/x-yz-y-xz-z-xy-4/Anonymousnoreply@blogger.comtag:blogger.com,1999:blog-3722233.post-44707508302301179372024-04-03T04:44:32.477-05:002024-04-03T04:44:32.477-05:00a=437361267792869725786125260237139015281653755816...a=4373612677928697257861252602371390152816537558161613618621437993378423467772036<br />b=36875131794129999827197811565225474825492979968971970996283137471637224634055579<br />c=154476802108746166441951315019919837485664325669565431700026634898253202035277999<br /><br />https://mathoverflow.net/questions/227713/estimating-the-size-of-solutions-of-a-diophantine-equationAnonymousnoreply@blogger.comtag:blogger.com,1999:blog-3722233.post-33986923763289658432024-04-03T04:35:43.757-05:002024-04-03T04:35:43.757-05:009 11 -59 11 -5Anonymousnoreply@blogger.comtag:blogger.com,1999:blog-3722233.post-8060383819752240402024-04-02T09:05:45.529-05:002024-04-02T09:05:45.529-05:00(Bill) Austin says there are no Solutions and Anoy...(Bill) Austin says there are no Solutions and Anoymous says that there are. Anonymous- please email me your solution (gasarch@umd.edu)gasarchhttps://www.blogger.com/profile/03004932739846901628noreply@blogger.comtag:blogger.com,1999:blog-3722233.post-18230258591390735282024-04-02T07:11:16.337-05:002024-04-02T07:11:16.337-05:00The last time a fun problem like this came up, I c...The last time a fun problem like this came up, I coded up some Python to run a few numbers, and, as someone who grew up coding assembler on machines that run at most 1 MIPS, I was completely freaked out at how fast interpreted Python runs on a 3.6 GHz i7. Times have changed.<br /><br />That time, I got the answer real quick. This one doesn't fall to brute force so easily. (Or maybe my programming is rustier than I think...)<br /><br />But to answer the meta question: for problems like this, please don't give the answer (or a pointer to the answer), at least for a day or two. Let us agonize over it for a bit. At least here, an excuse to do some simple programming is appreciated, and maybe if I sleep on it, I'll think of something. Maybe.<br /><br />By the way, I read a Comp. Sci. math textbook recently that used N for natural numbers including zero (since it was comp. sci.), but my impression was that it had to use N+ (natural numbers excluding zero) in the vast majority of cases. This made me think that maybe the math blokes know what they're doing...<br />Anonymousnoreply@blogger.comtag:blogger.com,1999:blog-3722233.post-28971056991588767532024-04-02T07:05:03.754-05:002024-04-02T07:05:03.754-05:00Wolframalpha suggests that the problem has no solu...Wolframalpha suggests that the problem has no solution https://www.wolframalpha.com/input?i=x+%2F+%28y%2Bz%29+%2B+y+%2F+%28x%2Bz%29+%2B+z+%2F+%28x%2By%29+%3D+4Austin Buchananhttps://austinlbuchanan.github.io/noreply@blogger.comtag:blogger.com,1999:blog-3722233.post-26518949841741277752024-04-01T21:03:10.019-05:002024-04-01T21:03:10.019-05:00(Bill) Some textbooks define 0 as a natural. My on...(Bill) Some textbooks define 0 as a natural. My only concern for this post is to clarify that I want x,y,z to be 1,2,3,... NOW- go work on the problem rather than comment about Natural vs Whole Numbers.Anonymousnoreply@blogger.comtag:blogger.com,1999:blog-3722233.post-10232124379339228812024-04-01T21:01:17.220-05:002024-04-01T21:01:17.220-05:00Zero is not a natural number. A set of Natural num...Zero is not a natural number. A set of Natural numbers contains counting numbers. We extend the set of Natural numbers by including zero. Such extension is called the set of Whole numbers.Anonymousnoreply@blogger.comtag:blogger.com,1999:blog-3722233.post-69321097612119283682024-04-01T19:37:30.467-05:002024-04-01T19:37:30.467-05:00The two questions are not exactly the same if you ...The two questions are not exactly the same if you are a logician or a computer scientist since their natural numbers include 0 which is excluded by the fruit version of the problem. (On the other hand, my elementary school textbook did not include 0 as a natural number and used the "whole" numbers as stated in the question for that case.)Anonymousnoreply@blogger.com