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Cube of a Number

if 12p + 1 is...

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if 12p + 1 is the cube of a positive intiger , where p is a positive odd intiger , then find cube root of 12p + 1

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In Euclids division lemma,to show thatthe cube of any positive intiger is of the form of 9m,9m+1,9m+8,why we r taking b=3

{(1 + ω^{2})}^{m} = {(1 + ω^{4})}^{m}

ω

is an imaginary cube root of unity, then least positive integral value of

m

\sec θ + \tan θ = p

(i) \sec θ = \frac{1}{2} (p + \frac{1}{p}) (ii) \tan θ = \frac{1}{2} (p - \frac{1}{p}) (iii) \sin θ = \frac{p^{2} - 1}{p^{2} + 1}

ω (\neq 1)

is a cube root of unity and

(1 + ω^{2})^{n} = (1 + ω^{4})^{n}

, then the least positive value of n is

2 x^{2} - 3 x - (p + 1) (2 p - 1)

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