Have infinitely many prime numbers in the forms of 8k+1,8k-1,8k+3and 8k-3(k∈Z)
This article attempts to use the proof methods of reduction to absurdity and classification discussion to provide the strict proofs for"there are infinite primes in the form of 8k+1(k∈Z)","there are infinite primes in the form of 8k-1(k∈Z)","there are infinite primes in the form of 8k+3(k∈Z)",and"there are infinite primes in the form of 8k-3(k∈Z)".The used knowledge is the fundamental knowledge in elementary number theory,limited to some basic properties of prime number,integer division,congruence,and Legendre symbol.To prove the main conclusions,this article first derives two very useful lemmas.
reduction to absurdityprime numberexact divisioncongruenceLegendre symbol
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