Proof of Infinite Positive Integers
Prove: There are infinitely many prime numbers p of the form 4n+3. In other words, show that there exist infinitely many positive integers, n, such that the number 4n+1 is prime.© SolutionLibrary Inc. solutionlibary.com 9836dcf9d7 https://solutionlibrary.com/math/algebra/proof-of-infinite-positive-integers-2uf
...us call them p_1, p_2, p_3, ... , p_r.
Now we consider a new number:
So this new number is constructed by multiplying together all of the primes in our list and adding 3. Note (this is very important): N itself has the form 4n+3.
This new number must have a prime factorization (by the fundamental theorem of arithmetic). But note that none of the primes p_1,p_2,p_3,...,p_r can appear in ...