I have to proof if true or wrong: There are infinite prime numbers of the form 4k+3.  I want to proof: Yes, this is true. My ideas:  1) Assume - as a contradiction - that there are only infinite prime numbers. Let $p_k$ be the highest of them of the form 4k+3. Let $p_1 = 3, p_2 = 5, ...$ the sequence of prime numbers.  So define $N := 4*(3*5*...*p_k) - 1$. Then we have  a) N divided by 4 has remainder 3 and 3 divided by 4 has remainder 3, because: $ N \equiv -1 \equiv 3 (mod 4)$.  b) Then I'd like to show that N does have a prime-factor of the form 4k+3 which is higher than $p_k$  &lt;&lt; here I don't know how to show this.  I'd appreciate any comments on whether my ideas are correct or not.
This is the proof from the book:     Theorem. There are infinitely many primes of the form $4n+3$.    Lemma. If $a$ and $b$ are integers, both of the form $4n + 1$, then the product $ab$ is also in this form.  Proof of Theorem: Let assume that there are only a finite number of primes of the form $4n + 3$, say  $$p_0, p_1, p_2, \ldots, p_r.$$ Let $$Q = 4p_1p_2p_3\cdots p_r + 3.$$ Then there is at least one prime in the factorization of $Q$ of the form $4n + 3$. Otherwise, all of these primes would be of the form $4n + 1$, and by the Lemma above, this would imply that $Q$ would also be of this form, which is a contradiction. However, none of the prime $p_0, p_1,\ldots, p_n$ divides $Q$. The prime $3$ does not divide $Q$, for if $3|Q$  then $$3|(Q-3) = 4p_1p_2p_3\cdots p_r,$$ which is a contradiction. Likewise, none of the primes $p_j$ can divides $Q$, because $p_j | Q$ implies $p_j | ( Q - 4p_1p_2\cdots p_r ) = 3$, which is absurd. Hence, there are infinitely many primes of the form $4n +3$.   END  From "however, none of the prime ...." to the end, I totally lost!    My questions:   Is the author assuming $Q$ is prime or is not? Why none of the primes $p_0, p_1,\ldots, p_r$ divide $Q$? Based on what argument?     Can anyone share me a better proof?     Thanks.
What is a good introduction to statistics for a mathematician who is already well-versed in probability?  I have two distinct motivations for asking, which may well lead to different suggestions:   I'd like to better understand the statistics motivation behind many problems considered by probabilists. I'd like to know how to better interpret the results of Monte Carlo simulations which I sometimes do to form mathematical conjectures.   I'm open to the possibility that the best way to go is not to look for something like "Statistics for Probabilists" and just go to a more introductory source. Why can't I join a friend's world on Minecraft PE? I'm on my tablet playing Minecraft PE. My boyfriend rang me and asked me to join his Minecraft PE world. However, I can't do it because the game doesn't show his world. What's wrong and how can I fix it? Can we insert or update FieldSets using apex code? 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However, while restoring I ran into no space available issue.   Is it possible to change the directory where the databases are stored? Determine if Type is a pointer in a template function Inspired by question :  When do I need to invoke \phantomsection?  In what condition is invoking \phantomsection necessary? NSString question - rangeOfString method Why is it so hard to find the roots of polynomial equations? The question that follows was inspired by  question:  When trying to solve for the roots of a polynomial equation, the quadratic formula is much more simple than the cubic formula and the cubic formula is much more simple than the quartic formula.    That the general solutions to various polynomial equations are so complex and difficult to derive seems to suggest a fundamental limitation in the problem solving capabilities of the mathematical machinery. Does this intuition of mine make any sense? What should I make of it? 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