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Prove that there are an infinite number of primes of the form $4n - 1$. I am having trouble solving this problem, so any help would be appreciated.

GEdgar
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user34304
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  • @Arthur it is not really a duplicate as it asks for the explication of one particular proof (other than the two given below). – quid Feb 18 '15 at 15:06

2 Answers2

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It is essentially as for the usual infinitude of primes proof.

  • Consider any finite collection of such primes and form their product $P$.

  • Consider $P+2$ or $P+4$ depending on which of the two is of the form $4n-1$.

  • Argue that this number must be divisible by a prime of the form $4n-1$ yet cannot be divisible by any one used in the product.

You get that no finite list can be complete.

quid
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Assume that there are finitely many primes of this form. Then we can list them:

$$\{P_1,\cdots,P_n\}$$

Now $P_1*\cdots*P_n$ is either $1$ mod $4$ or $-1$ mod $4$. If it is $1$ mod $4$, we multiply by $P_1$ again. Now add $4$ to the result. This is again $-1$ mod $4$, so it will have a prime divisor other than $P_i$ that is $-1$ mod $4$, which is a contradiction.

Uncountable
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