is it true?
what is the proof?
I prefer a hint if it ain't difficult to prove.
tnx
Hint(?):Originally Posted by srulikbd
Let $\displaystyle p>3$ be a prime. Consider the possible values of the remainder
when $\displaystyle p$ is divided by $\displaystyle 6$. As $\displaystyle p>3$ is prime the remainder cannot
be divisible by any factor of $\displaystyle 6$.
RonL
Or to say it another way.
Any number must have form,
$\displaystyle \begin{array}{c}6k\\6k+1\\6k+2\\6k+3\\6k+4\\6k+5$
Note that,
$\displaystyle \begin{array}{c}6k\\6k+2\\6k+4$
are all divisible by 2 and $\displaystyle p>2$ thus not a prime.
And that,
$\displaystyle 6k+3$ is divisble by 3 and $\displaystyle p>3$ thus not a prime.
Hence the only possibilites are,
$\displaystyle 6k+1$
or,
$\displaystyle 6k+5=6k+6-1=6(k+1)-1$
Thus,
$\displaystyle 6k\pm 1$ for some integer $\displaystyle k$.
Whoa!!!Originally Posted by srulikbd
That is too complicated I believe the proof use's funtional analysis. It was first proven by Dirichelt (my avatar ).
In fact in the book "Introduction to Theory of Numbers" by Hardy and Wright. Which contains elementary,algebraic and analytic number theory; proves all theorems in the book (even the prime number theorem) except this theorem! That is how complicated it is.
Lemma: The product of integers of the form $\displaystyle 4k+1$ is again an integer of the form $\displaystyle 4k+1$.Originally Posted by srulikbd
Proof:: If $\displaystyle a=4k+1$ and $\displaystyle b=4j+1$ then, $\displaystyle ab=(4k+1)(4j+1)=4(4kj+k+j)+1$. Q.E.D.
Theorem: There are infinitely many primes of form $\displaystyle 4k+3$.
Proof: Assume there are finitely many primes of form $\displaystyle 4k+3$ call them $\displaystyle \{ p_1,...,p_s} \}$. Then form an integer,
$\displaystyle n=4p_1p_2\cdot ...\cdot p_s-1$
Prime factorize $\displaystyle n$ as,
$\displaystyle n=q_1q_2\cdot ...\cdot q_r$
Note that $\displaystyle 2\not | n$ because it is odd. Thus, $\displaystyle q_i\not =2$ for any $\displaystyle 1\leq i\leq r$.
We cannot have that all the primes factors of $\displaystyle n$ are of the form $\displaystyle 4k+1$ for that would imply that $\displaystyle n$ has form $\displaystyle 4k+1$, which is does not because $\displaystyle 4k-1=4(k-1)+3$ by the lemma. Thus one of its factors $\displaystyle r_i$ has form $\displaystyle 4k+3$. Because an odd prime has one of two forms $\displaystyle 4k+1,4k+3$. But then $\displaystyle r_i|n$ thus, $\displaystyle r_i|(4p_1p_2\cdot ...\cdot p_s-1)$ thus, $\displaystyle r_i|1$, because $\displaystyle r_i$ is found among $\displaystyle p_1,...,p_s$ thus an impossibility. Q.E.D.