t-n-3-n-2-t-E-Give-a-general-form-solving-for-n-i-e-n-3-n-2-2k-n-

Question Number 8466 by FilupSmith last updated on 12/Oct/16

t = n^{3} - n^{2}

t \in E

Give a general form solving for n

i . e

n^{3} - n^{2} = 2 k

n = ?

Commented by Rasheed Soomro last updated on 12/Oct/16

n^{3} - n^{2} \in E \Rightarrow {\begin{cases} n^{3} \in E \land n^{2} \in E \Rightarrow n \in E \\ OR \\ n^{3} \in O \land n^{2} \in O \Rightarrow n \in O \end{cases}

n \in E, Let n = 2 k, n^{3} - n^{2} = {(2 k)}^{3} - {(2 k)}^{2} = {8 k}^{3} - {4 k}^{2}

= {4 k}^{2} (2 k - 1)

n \in O, Let n = 2 k + 1, n^{3} - n^{2} = {(2 k + 1)}^{3} - {(2 k + 1)}^{2}

= {(2 k + 1)}^{2} (2 k + 1 - 1) = 2 k {(2 k + 1)}^{2}

If n^{3} - n is of {4 k}^{2} (2 k - 1) type then n \in E

If n^{3} - n^{2} is of 2 k {(2 k + 1)}^{2} type then n \in O