mersenneforum.org  

Go Back   mersenneforum.org > Extra Stuff > Blogorrhea > enzocreti

Reply
 
Thread Tools
Old 2020-03-20, 14:35   #1
enzocreti
 
Mar 2018

10178 Posts
Default 216 is the only cube that can be written in this way?

216=3^2*(182^2-331*10^2)


is there any other positive cube that can be written as


a^2*(b^2-r*10^2)?


with a and b positive and coprime, r prime?
enzocreti is offline   Reply With Quote
Old 2020-03-20, 15:16   #2
xilman
Bamboozled!
 
xilman's Avatar
 
"π’‰Ίπ’ŒŒπ’‡·π’†·π’€­"
May 2003
Down not across

1039710 Posts
Default

Quote:
Originally Posted by enzocreti View Post
216=3^2*(182^2-331*10^2)


is there any other positive cube that can be written as


a^2*(b^2-r*10^2)?


with a and b positive and coprime, r prime?
Since d^3 = d^2 * d, all you are asking is whether d = (b^2 - 100r) is positive. In other words, for what values of b and r is b^2 > 100r and b is co-prime to d..

It should now be obvious that there are an infinite number of solutions. Picking just one pretty much at random, let r =1. In that case b^2 > 100, or b> 10. One such value is 11. Plugging in the numbers, d=121-100 = 21.

Indeed 21^3 = 9261 = 441 * (11^2 - 10 * 1^2).

Why do you post questions which are so trivial to answer with even the slightest amount of thought?

Last fiddled with by xilman on 2020-03-20 at 15:20
xilman is offline   Reply With Quote
Old 2020-03-22, 11:13   #3
LaurV
Romulan Interpreter
 
LaurV's Avatar
 
Jun 2011
Thailand

8,963 Posts
Default

Quote:
Originally Posted by xilman View Post
r =1
Neee.. gotcha! (so long I waited for that! )
r must be prime, I guess you didn't know that 1 is not considered a prime
LaurV is offline   Reply With Quote
Old 2020-03-22, 15:30   #4
xilman
Bamboozled!
 
xilman's Avatar
 
"π’‰Ίπ’ŒŒπ’‡·π’†·π’€­"
May 2003
Down not across

37·281 Posts
Default

Quote:
Originally Posted by LaurV View Post
Neee.. gotcha! (so long I waited for that! )
r must be prime, I guess you didn't know that 1 is not considered a prime
Fairy Nuff: I over-looked that condition.

OK, choose r=2. Prime, right?

Let b=17, so B^2-100r = 289-200 = 89 = a. All are positive and co-prime.

Let b=19, so b^2-100r = 361-200 = 161 = 7 * 23 = a. All are positive and co-prime.

Let b=21, so a=241 which is a prime.

And so on.

It is still a trivial problem.
xilman is offline   Reply With Quote
Reply

Thread Tools


Similar Threads
Thread Thread Starter Forum Replies Last Post
Numbers that can be written in two different ways enzocreti enzocreti 3 2020-02-24 17:59
Prime numbers test primality - with proof written in invisible ink Godzilla Miscellaneous Math 40 2018-10-17 00:11
any suitable sieve written in C or Python? Shen Information & Answers 4 2017-01-05 04:49
Looking for British written American history on Kindle jasong Homework Help 3 2015-01-30 21:36
Math of LLR(written by Jean Penne) TTn 15k Search 3 2006-01-06 00:08

All times are UTC. The time now is 15:35.

Sat Dec 5 15:35:51 UTC 2020 up 2 days, 11:47, 0 users, load averages: 1.44, 2.14, 2.27

Powered by vBulletin® Version 3.8.11
Copyright ©2000 - 2020, Jelsoft Enterprises Ltd.

This forum has received and complied with 0 (zero) government requests for information.

Permission is granted to copy, distribute and/or modify this document under the terms of the GNU Free Documentation License, Version 1.2 or any later version published by the Free Software Foundation.
A copy of the license is included in the FAQ.