About the maximum prime number

These days are very boring to understand a few math problems by the king of the 40 lines of code triggered, first studied the tail recursion, and then because of Wang Yin's article "Talk about P=NP." "Understanding of the seven problems of today's mathematics, and then to check one of the Poincaré conjecture things (Poincaré conjecture has been resolved, after the Shing event), and Goldbach conjecture related things (Chen Jingrun 1+2, not seven problems), and finally back to the P/NP problem (one of the seven problems), The result is careless and bored to check the maximum prime number problem, more boring is also followed to prove a bit ... It has nothing to do with my programming work. I found my mind too scattered ...
On the maximum prime number question whether there is the maximum prime number, the following is the Baidu encyclopedia to give the certification process:
There is no maximum prime number!In primary school, we know that all natural numbers can be divided into prime numbers (primes) and composite two categories, of course, also specifically stipulates that "1 is neither prime, nor composite." Prime numbers within 100, from small to large, are: 2, 3, 5, 7, 11, 13, 17, 、......、 83, 89, 97. Needless to say, you'll be sure to memorize it. So the number of prime numbers is not a finite number of it. Before we solve this problem, let's look at another question: How to tell if a known natural number is prime. For example, 143 is not prime. You will certainly follow this step to judge: first with the smallest prime number 2 to remove 143, not divisible, then 3 to try, or not, and then in turn with 5, 7 try, or not, 11. Yes. 143=11x13, so 143 is not prime, but composite. So, to judge whether a number is a prime, just use all the prime numbers smaller than this number, then remove it in turn, if not divisible, the number must be prime; on the contrary, as long as the number can be divisible by one prime, the number must be composite. This method is based on the principle that each composite can be expressed as a product of several prime numbers. Needless to say, this is called "decomposition factorization", but also the elementary school mathematics knowledge. We first assume that the number of prime numbers is limited, then there must be a "maximum prime", set this "maximum prime" for N. Here we find all the prime numbers from 1 to N, and multiply them, that is: 2x3x5x7x11x13x......xn the multiplication of the product plus 1, to get a fairly large number m:m=2x3x5x7x11x13x......xn+1 then this M is prime or composite. At first thought, it is not difficult to judge, since N is the largest prime number, and m>n, then M should be composite. Since M is composite, I can decompose factorization on M. But try to find out, we use any prime number from 1 to N to remove M, always remaining 1. This reality, also shows that M must be prime. This paradoxical result is nothing more than a description: The largest prime number is nonexistent. If you have a prime number that is large enough, n, you can find a prime number m larger than N, as above. Since there is no maximum prime number, it can be inferred that prime numbers in natural numbers should be infinitely numerous.
At the same time Baidu encyclopedia someone gave another counter example: M=2x3x5x7x11x13x......xn+1, with any one prime number from 1 to N to remove M, always remaining 1. This reality, also shows that M must be prime. This conclusion is a big mistake, for example, 2x3x5x7x11x13+1=30031=59x509,30031 is a composite.

Seeing this counter-example, I began to doubt the correctness of the above proofs, so I thought for a long time, finally want to understand that the proof is correct.     The wrong point of the inverse example is that he does not have all the primes except the 30031 small, except to 13, and to prove that a prime is not divisible by any number of primes other than his small to prove that it is prime. It is good proof that a number is quickly judged by the number of primes, divided by all the primes that are smaller than his. Each composite must be expressed as a product of several primes, as easily proved.     Don't dwell on it here. The reason why we question the above proof process is mainly from the conclusion that if N is the maximum prime number, then M is the prime number. So by counter example n=13, then m=30031, can 30031 is composite namely M is composite, with the above inference m is prime contradiction. What the hell is going on here? Does it mean that the inference is wrong? After thinking we found that the proof that M is the prime number of the premise is first we have to assume that N is the largest prime number. Before the introduction of M or prime, and then self-contradictory, can disprove n is not the largest prime.     And the precondition of the inverse example is error, n=13, obviously 13 is not the maximum prime number he can not become N. We do not consider the M and n who is larger, according to the proof that N is the largest prime number, then the introduction of M must be the prime number of this conclusion.     Finally found M>n, so N is not the largest prime. Then take a look at the inverse: "m=2x3x5x7x11x13x......xn+1, using any prime number from 1 to N to remove M, always remaining 1." This reality, also shows that M must be prime. This conclusion is a big mistake, for example, 2x3x5x7x11x13+1=30031=59x509,30031 is a composite. ", in fact m divided by 1 to N total 1 means m must be prime this conclusion is not wrong, because the precondition of this conclusion is that N is the premise that we assume the largest prime number, rather than the arbitrary prime number, the author confuses the concept, such as example will 13=n, the introduction of 30031 m, and M is not a prime number so overturn the conclusion This precondition is not correct.