September 19th, 2018, 08:11 AM  #11 
Newbie Joined: Sep 2018 From: tunis Posts: 27 Thanks: 0 
(1  2m)^3 = (8) m^3 + 12 m^2  6 m + 1 = 3 n^2 + 3 n + 1 how to find n? Last edited by skipjack; September 19th, 2018 at 10:31 AM. 
September 19th, 2018, 10:32 AM  #12 
Global Moderator Joined: Dec 2006 Posts: 21,038 Thanks: 2274 
The righthand side is the difference between the cubes of consecutive integers; as this difference is a perfect cube, it must be 1, so n is 1 or 0, and m = 0.

September 19th, 2018, 03:01 PM  #13 
Newbie Joined: Sep 2018 From: tunis Posts: 27 Thanks: 0 
( b +1)^3b^3=The righthand side but the difference between consecutive cubes is not a cube. 1^3+(3b^2+3b+1)=3b^2+3b+2 1+ 3*1+3.1+1 = 3 +3 +2 1+ 7 = 8 =2^3 n=1 et 0 m=0 is it a complete study on this subject? Last edited by skipjack; September 19th, 2018 at 08:15 PM. 
September 19th, 2018, 08:26 PM  #14 
Global Moderator Joined: Dec 2006 Posts: 21,038 Thanks: 2274  The equation states that its righthand side, 3 n^2 + 3 n + 1, which is the difference between n³ and (n + 1)³, equals its lefthand side, which is (1  2m)³. Consecutive cubes differ by a perfect cube only when they are 1 and 0 or 0 and 1.

September 20th, 2018, 12:56 AM  #15 
Newbie Joined: Sep 2018 From: tunis Posts: 27 Thanks: 0 
thank you very much I bring here the support of 2n + 1 = x ^ 2? if x is odd we can write it 12m x ^ 2 = 4m ^ 2  4m + 1 = 2 (2m ^ 2  2m) + 1 = 2n + 1 and 4m ^ 2  4m + 1 = 2n + 1 (4m ^ 2  4m) = 2n (4m ^ 2  4m) / 2 = n 2m ^ 2  2m = n 2 (m ^ 2  m) = n From what you showed me. Do we get this with (12m)^3 if not show how to do it? Last edited by skipjack; September 20th, 2018 at 03:05 AM. Reason: to correct typos and sign error 
September 20th, 2018, 02:34 AM  #16 
Newbie Joined: Sep 2018 From: tunis Posts: 27 Thanks: 0 
If x^2 = 2n + 1 = (1  2m)^2 sqrt(2n + 1) = 1  2m n = ((2m + 1)^2  1)/2 for any m $\in$ N Last edited by skipjack; September 20th, 2018 at 02:54 AM. Reason: to correct 
September 20th, 2018, 03:00 AM  #17 
Global Moderator Joined: Dec 2006 Posts: 21,038 Thanks: 2274 
(1  2m)³ = (1  2m)(1  2m)² = (1  2m)(1  4m + 4m²) = 1  6m + 12m²  8m³ In your last post, I made some corrections. It would make more sense to say "for any m $\small\in$ ℤ" than "for any m $\small\in$ N". 
September 20th, 2018, 03:43 AM  #18 
Newbie Joined: Sep 2018 From: tunis Posts: 27 Thanks: 0 
12m=(3n^2+3n+1)^1/3 there are no solutions because there is the negative root 
September 20th, 2018, 03:53 AM  #19  
Newbie Joined: Sep 2018 From: tunis Posts: 27 Thanks: 0  Quote:
 
September 20th, 2018, 04:34 AM  #20 
Global Moderator Joined: Dec 2006 Posts: 21,038 Thanks: 2274  

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