Number Theory Test 52

A complex number is said to be purely imaginary if $$z+\overline { z } =0$$

If $$z=\dfrac { 3+2isinθ }{ 1−2isinθ } $$

then $$\overline { z } =\overline { \dfrac { 3+2isinθ }{ 1−2isinθ }  } =\dfrac { 3-2isinθ }{ 1+2isinθ } $$

$$z+\overline { z } =\dfrac { 3+2isinθ }{ 1−2isinθ } +\dfrac { 3-2isinθ }{ 1+2isinθ } $$

So,$$\dfrac { 3+2isinθ }{ 1−2isinθ } +\dfrac { 3−2isinθ }{ 1+2isinθ } =0$$

$$\dfrac { (3+2isinθ)(1+2isinθ)+(3−2isinθ)(1−2isinθ) }{ (1−2isinθ)(1+2isinθ) } =0$$

$$3+6isinθ+2isinθ−4{ sin }^{ 2 }θ+3−6isinθ−2isinθ−4{ sin }^{ 2 }θ=0$$

$$6−{ 8sin }^{ 2 }θ=0$$

$${ sin }^{ 2 }θ=\dfrac { 3 }{ 4 } $$

$$sinθ=\dfrac { \sqrt { 3 }  }{ 2 } =sin\dfrac { \pi  }{ 3 } $$

$$θ=nπ+{ (−1) }^{ n }\left( \dfrac { \pi  }{ 3 }  \right) $$

$$sinθ=\dfrac { \sqrt { 3 }  }{ 2 } =sin−\dfrac { \pi  }{ 3 } $$

$$θ=nπ+{ (−1) }^{ n }\left( \dfrac { -\pi  }{ 3 }  \right) =nπ+{ (−1) }^{ n+1 }\left( \dfrac { \pi  }{ 3 }  \right) $$

Twin primes are prime numbers that differ by$$2$$.

For example 11 and 13 are twin primes.

In a sense they are “right next to each other” in the sense that(with the exception of the prime number $$2)$$, 

two primes can’t be closer together.

$$21$$ is not prime though. so $$19$$ and $$21$$ are not twin primes.

$$19$$ and $$17$$ are twin primes, but the next prime after $$19$$ is $$23$$ which is more than $$2$$ greater.