- #1

- 734

- 186

- Homework Statement
- By considering the number ## N=16p_{1}^2p_{2}^2\dotsb p_{2}^2-2 ##, where ## p_{1}, p_{2}, ..., p_{n} ## are primes, prove that there are infinitely many primes of the form ## 8k-1 ##.

- Relevant Equations
- If ## p ## is an odd prime, then ## (2|p)=1 ## if ## p\equiv \pm 1\pmod {8} ##.

Proof:

Suppose for the sake of contradiction that the only primes of the form ## 8k-1 ## are ## p_{1}, p_{2}, ..., p_{n} ##

where ## N=16p_{1}^2p_{2}^2\dotsb p_{n}^2-2 ##.

Then ## N=(4p_{1}p_{2}\dotsb p_{n})^2-2 ##.

Note that there exists at least one odd prime divisor ## p ## of ## N ## such that ## (4p_{1}p_{2}\dotsb p_{n})^2\equiv 2\pmod {p} ##.

This implies ## (2|p)=1 ##.

Thus ## p\equiv \pm 1\pmod {8} ## where ## p ## is one of the primes ## p_{i} ##.

Observe that if all the odd prime divisors of ## N ## were of the form ## 8k+1 ##, then ## N ## would be of the form ## 8a+1 ##.

This is impossible, because ## N ## is of the form ## 16a-2 ##.

Thus, ## N ## must have a prime divisor ## q ## of the form ## 8k-1 ##.

But ## q\mid N ## and ## q\mid (4p_{1}p_{2}\dotsb p_{n})^2 ## leads to the contradiction that ## q\mid 2 ##.

Therefore, there are infinitely many primes of the form ## 8k-1 ##.

Suppose for the sake of contradiction that the only primes of the form ## 8k-1 ## are ## p_{1}, p_{2}, ..., p_{n} ##

where ## N=16p_{1}^2p_{2}^2\dotsb p_{n}^2-2 ##.

Then ## N=(4p_{1}p_{2}\dotsb p_{n})^2-2 ##.

Note that there exists at least one odd prime divisor ## p ## of ## N ## such that ## (4p_{1}p_{2}\dotsb p_{n})^2\equiv 2\pmod {p} ##.

This implies ## (2|p)=1 ##.

Thus ## p\equiv \pm 1\pmod {8} ## where ## p ## is one of the primes ## p_{i} ##.

Observe that if all the odd prime divisors of ## N ## were of the form ## 8k+1 ##, then ## N ## would be of the form ## 8a+1 ##.

This is impossible, because ## N ## is of the form ## 16a-2 ##.

Thus, ## N ## must have a prime divisor ## q ## of the form ## 8k-1 ##.

But ## q\mid N ## and ## q\mid (4p_{1}p_{2}\dotsb p_{n})^2 ## leads to the contradiction that ## q\mid 2 ##.

Therefore, there are infinitely many primes of the form ## 8k-1 ##.