My Math Forum how to show that 2n + 1 = x ^ 2 ?

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 September 18th, 2018, 01:54 PM #1 Newbie   Joined: Sep 2018 From: tunis Posts: 27 Thanks: 0 how to show that 2n + 1 = x ^ 2 ? how to show that 2n + 1 = x ^ 2 ? the solution is: n = 2 * (m ^ 2 - m) x = 1 - 2 * m and 1 + 3b + 3b ^ 2 = x ^ 3 the solution is: b = -1, x = 1 step by step I calculated that with a software it gives only the solution. Thank you. Last edited by skipjack; September 19th, 2018 at 10:27 AM.
 September 18th, 2018, 02:54 PM #2 Global Moderator   Joined: Dec 2006 Posts: 20,926 Thanks: 2205 For the second problem, the software didn't find the solution b = 0, x = 1. Last edited by skipjack; September 19th, 2018 at 10:27 AM.
September 18th, 2018, 07:24 PM   #3
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Quote:
 Originally Posted by Ak23 how to show that 2n + 1 = x ^ 2 ? the solution is: n = 2 * (2 ^ m-m) x = 1-2 * m and 1 + 3b + 3b ^ 2 = x ^ 3 the solution is: b = -1, x = 1 step by step I calculated that with a software it gives only the solution. thank you.
I think we can simplify this a bit and get rid of the "x", computer or not. If we have $\displaystyle n^2$ then we can get the next square (integer) by adding 2n + 1. ie. $\displaystyle (n + 1)^2 = (n^2) + (2n + 1)$

Similar for the cube:
$\displaystyle (n + 1)^3 = (n^3) + (3n^2 + 3n + 1)$

-Dan

 September 19th, 2018, 12:15 AM #4 Newbie   Joined: Sep 2018 From: tunis Posts: 27 Thanks: 0 @topsquark 2n+1=4n+2 2n-4n=-1 2n=-1 n=-1/2 2.(-1/2)+1 =0 it does not work pair = odd Last edited by Ak23; September 19th, 2018 at 12:17 AM.
 September 19th, 2018, 03:28 AM #5 Newbie   Joined: Sep 2018 From: tunis Posts: 27 Thanks: 0 n=2*(m^2-m) x=1-2*m is solution when z^2=(n+1)^2 y^2=n^2 x^2=2n+1 It's not all numbers only with the squares that is on the odd numbers 1 3 5 7 9 11, .... 2N-1 9 25 49 .... ETC
 September 19th, 2018, 03:56 AM #6 Newbie   Joined: Sep 2018 From: tunis Posts: 27 Thanks: 0 m n x x^2 0 0 1 1 1 0 -1 1 2 4 -3 9 3 12 -5 25 4 24 -7 49 Last edited by Ak23; September 19th, 2018 at 03:59 AM.
 September 19th, 2018, 05:17 AM #7 Global Moderator   Joined: Dec 2006 Posts: 20,926 Thanks: 2205 If x is odd, it can be written as 1 - 2m. That means that x² = 4m² - 4m + 1 = 2(2m² - 2m) + 1 = 2n + 1, where n = 2(m² - m). If you have no objection, I'll correct your error in the original post, so that the thread can be shortened. Thanks from Ak23
 September 19th, 2018, 05:59 AM #8 Newbie   Joined: Sep 2018 From: tunis Posts: 27 Thanks: 0 how to find solutions? for x ^ 2 and x ^ 3
 September 19th, 2018, 06:34 AM #9 Global Moderator   Joined: Dec 2006 Posts: 20,926 Thanks: 2205 Doesn't my previous post show how to find the solutions for x²? Thanks from Ak23
 September 19th, 2018, 07:33 AM #10 Newbie   Joined: Sep 2018 From: tunis Posts: 27 Thanks: 0 yes I understood thank you very much

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