Recent content by Charles Link

a copy and paste of mine from the thread "a trigonometry problem of interest": I'm finding it somewhat amazing how the ai overview is able to summarize for me the steps that are needed for the method of finding rational roots of a conic section with linear parametrization using the rational...

I'm finding it somewhat amazing how the ai overview is able to summarize for me the steps that are needed for the method of finding rational roots of a conic section with linear parametrization using the rational root theorem. I would like to copy and paste their summary which I think even did...

Just last night, I put an Edit 2 to post 48 that some may find of interest. Finding positive rational solutions to the problem ##x^2+x+1=y^2 ## turns out to be rather easy for this rotated ( 90 degrees) and translated hyperbola, using the method of posts 41 and 46. The starting point is the...

additional numbers to complete the set with 5 in the denominator for ## m ##: ## m=9/5 ##, ## a=59 ##, ## b=227 ##, ## x=151 ## ## m=11/5 ##, ## a=39 ##, ## b=327 ##, ## x=201 ## ## m=12/5 ##, ## a=26 ##, ## b=383 ##, ## x=229 ## ## m=13/5 ##, ## a=11 ##, ## b=443 ##, ## x=259 ## Notice as is...

Going back to post 56, I now tried, just for the fun of it, to see if I could get another solution using @daverusin 's method of post 41. I tried ## m=7/3 ##, and I very readily got ## a=11 ##, ## b=131 ##, and ## x=79 ##. I tested the solution to see if we have that ## x^2=(a^2+ab+b^2)/3 ##...

In post 43 @Gavran cited it on his very last line in parentheses. It is so easy to overlook something like that though, because this thread has a lot of posts, and it can also be difficult to go back to the previous page of posts. The solution we have though, written out in post 52, is a...

@hutchphd It gets harder as we get older. I am 70 now. We sort of have an excuse. Too many of the younger ones though I see are relying too much on google and want instantaneous answers, and don't want to take the time to wrestle with math problems the way our generation did. :)

Looking back over things, I do think @daverusin in post 41 has something of considerable merit. With his method, which I figured out in posts 45 and 46, he was able to spot a solution that @renormalize missed in his post 32. The algebra of finding where the line with slope of ## m ## that...

@Gavran I presume in your solution that you plugged in ## R=x ##. You then get ## a^2-b^2+3x^2 =2a^2+ab ## so that ## 3x^2=a^2+b^2+ab ##. Very good. :) You have a simple solution also, and these last two algebraic steps that I included show that your method here is also in complete agreement...

Looking back to the other posts, I see @Gavran also used Ptolemy's theorem previously in post 43, but the post 52 method gets the answer of ## x=7 ## as well as the arithmetically simple result of posts 30 and 32 without the extra calculations of post 43.

@hutchphd Here is the solution: ## (ac)(bd)=2x^2+(11)(2) ## from Ptolemy ## (ac)^2=4x^2-2^2 ## from Pythagoras and Thales ## (bd)^2=4x^2-11^2 ## from Pythagoras and Thales This gives ##(ac)^2(bd)^2=(4x^2-2^2)(4x^2-11^2)=(2x^2+(11)(2))^2 ##. We get ## 16x^4-4x^2(2^2+11^2)+11^2...

@hutchphd a super result. :) With a couple algebraic steps one gets ## 3x^2=11^2+2^2+(2)(11) ##. I'll let you write it out, but if you don't get to it in a day or so, I could post the 3 equations and their rather straightforward solution.

Here is an aside item, and in a way a little simpler. The problem of finding integer solutions for ## x^2=(a^2+ab+b^2)/3 ## reminds me of something I tried a few years back. We know that ## x^2+2x+1 ## is always a perfect square, but could the factor that appears from the difference of two...

and a follow-on to the above: The case of ## m=1 ## gives the point ## (1,1) ## for ## (x,y) ##. It may be worth mentioning that I tried using the point ## (1,1) ## as a starting point on the ellipse for the known rational solution, but it seemed to give poor results. The resulting ## m's...

I think I figured out now what @daverusin did in post 41. One begins with ## x^2+xy+y^2-3=0 ##, and to find the rational solutions, you first pick out one rational solution. In this case ## (-1,-1) ## makes a good point to use. You then have a line of slope ## m ## though that point, and...