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 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 right-hand 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. Thanks from Ak23 September 19th, 2018, 03:01 PM #13 Newbie   Joined: Sep 2018 From: tunis Posts: 27 Thanks: 0 ( b +1)^3-b^3=The right-hand 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
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Quote:
 Originally Posted by Ak23 . . . but the difference between consecutive cubes is not a cube.
The equation states that its right-hand side, 3 n^2 + 3 n + 1, which is the difference between n³ and (n + 1)³, equals its left-hand 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 1-2m 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 (1-2m)^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". Thanks from Ak23 September 20th, 2018, 03:43 AM #18 Newbie   Joined: Sep 2018 From: tunis Posts: 27 Thanks: 0 1-2m=(3n^2+3n+1)^1/3 there are no solutions because there is the negative root September 20th, 2018, 03:53 AM   #19
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Quote:
 It would make more sense to say "for any m $\small\in$ ℤ" than "for any m $\small\in$ N".
I did not understand September 20th, 2018, 04:34 AM   #20
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There's no reason to exclude negative values of m.

Quote:
 Originally Posted by Ak23 . . . there are no solutions because there is the negative root
There is no negative cube root of 3n² + 3n + 1. Tags show Thread Tools Show Printable Version Email this Page Display Modes Linear Mode Switch to Hybrid Mode Switch to Threaded Mode Similar Threads Thread Thread Starter Forum Replies Last Post mared Geometry 7 May 16th, 2015 09:50 PM Few_But_Ripe Complex Analysis 1 November 11th, 2011 09:08 AM notnaeem Real Analysis 4 August 16th, 2010 12:32 PM naserellid Algebra 2 August 15th, 2010 02:20 AM

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