But in all seriousness, we learned that if we have a number of the form 




where 
is an integer, then 








describes it's multiples as long as 


or 










. By adding 









or 












we are effectively multiplying the number by 











or 











(Sorry if I changed the name of certain variables).












































































Let us remind ourselves of the reason we are finding the multiples of 




. We were able to find arrow relations between Plato triples using

















































We wanted to know if we could change our values of 
and 
such that 




and 


don't change. For example, when 


, you can add any number to both














































This is the reason why every Plato triple has a right arrow going to the next term in the Stifel sequence. By finding more of these, we might be able to fully describe the arrow relations between two multiples
So we need to multiply two numbers by the same 











or 



































If we say 













, then we are also saying that 













. Since 
and 
are integers we see that the only solution is 






. (We started with 
because multiplying by 
doesn't make sense and the numbers between can never divide a number of the form 




)












































If we say 













, then we are also saying that 













. Since 
and 
are integers we see that the only solutions are 






and 


















































If we interpret these solutions, we see that the only numbers we can multiply by 
and 
and 
. (One of those solutions was basically a duplicate) We can see that 


















and that 





























































I will continue exploring this with numbers higher than 
, but here is a cool pattern I found related to 




. Firstly, here is a list of 




up to 


















1
7
17
31
49
71
97
127
161
199
241
287
337
391
449
511
577
647
721
799
881
967
1057
1151
1249
1351
1457
1567
1681
1799
We start at 
and move forward 

to 
. 




, the next perfect square after 
.










If we take 

and move forward 




to 


, we might notice that 







. And it just so happens that 







is the next perfect square.


























I haven't looked at why this is, but I suspect it is a side effect of 







being a Pellian equation.








Another interesting pattern occurs when we consider number we can multiply 




by





If we consider 


, the first 10 numbers we can multiply by are



7.0
23.0
41.0
73.0
103.0
151.0
193.0
257.0
311.0
391.0
If we consider 


, the first 10 numbers we can multiply by are



23.0
47.0
113.0
161.0
271.0
343.0
497.0
593.0
791.0
911.0
You may have noticed that 23 is second in 


and first in 


. This pattern actually continues, so hopefully I will be able to exploit it and discover more patterns






This week's calculator will be up shortly
s:
Even when your project started, most of this was over my head, but each time I read your blog, I'm reminded of this fact. You mentioned this pattern you've discovered where numbers appear to reappear in M = x and M = x + 1. Do you think you'd take any steps to prove why this occurs (so you can generalize it in a more abstract form)?
ReplyDeleteHello Nicole! I actually looked into this. One of the things we wanted was


































, this becomes






which explains our pattern.





doesn't hold, so it won't be that easy to find repeats
DeleteIf
Unfortunately there are solutions to this equation where
Thank you so much for taking the time to do this!
DeleteHi Vijay! Once again, a little bit of this went over my head, but I think I understand most of it. The work you have done toward this project is very impressive. Keep it up!
ReplyDeleteHello Jackson! Sorry I was a little bit all over the place this week. Next week will definitely flow more smoothly.
DeleteHi Vijay! I'm glad you've gotten back at your research this week. Can't wait for next week and more calculators. :)
ReplyDeleteHello Carla! I'm also really excited. I'm finally getting into the flow of things, so I hopefully will get somewhere.
DeleteHi Vijay,
ReplyDeleteAmazing article! I really enjoyed the cartoons and vivid descrption! Great Job!
Sincerely, Bhavik Rajaboina
Hello Bhavik! That is exactly what I was hoping to hear. If I can educate my readers in an entertaining way, my job is done.
DeleteHi Vijay! While this stuff is really complex, you did a good job explaining it so I understand now. The work you have done on this project is very impressive as always. I'm looking forward to next week's blog. Thanks.
ReplyDeleteHello Jack! It's good to know that you understand what I'm explaining. Hopefully, I'll have something very impressive to share at the end.
DeleteHi Vijay! I had to go back and read week five's post to understand a lot of the things in this post, but I was still able to understand a lot of it because of your good description. As always, it seems like you are really making good progress on your project! How much closer are you to solving (or proving it cannot be solved) the magic square of squares? I look forward to next week's post and calculator.
ReplyDeleteHello Gokul! I'll try to avoid things like this for future posts. As far as progress goes, it's hard to tell; if one component of a 3x3 magic square of squares doesn't work and I examine it, the whole thing would be solved in an instant. Otherwise, it could take a long time to find components of a 3x3 magic square of square and make them work together.
DeleteHi Vijay! Love your description, I think I understood most of the material in this post. How much closer are you to solving this problem, and how specifically does this week's post contribute twards the end goal?
ReplyDeleteHello Abhinav! The last few posts have been dedicated to studying different families of triples because of their interesting properties in the context of my problem. Hopefully, I'll be able to use these properties to find parts of the 3x3 magic square of squares. For progress I would say that I could find part of the magic square that doesn't work and instantly end it (unlikely) or I could slowly build the magic square of squares piece by piece.
DeleteHey Vijay! Loved your explanation of the Plato triples! What has been your favorite part of researching magic squares so far?
ReplyDeleteHello Nathan! It's so hard to choose what my favorite part is! I really liked learning about Pellian equations (








where 
and 
are integers). "Diophantine Equations" always sounded scary, but this experience showed me how awesome they are. Another part I really liked was creating all these different cool calculators.
Delete