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There are infinitely many prime numbers that can be expressed as 4K + 1. There are infinitely many prime numbers that can be expressed as 3K + 1. Question: what is k?
4K + 1-k = 3K + 1, 3K + 1 is prime, 4K + 1 is prime, k = 4
It is proved that there are infinitely many prime numbers of 4N-1
Please help me! ~ ~ ~ thank you
There are finite prime numbers such as 4N-1, P1 = 4n1-1, P2 = 4n2-1... PM = 4nm-1, let n = 4 * P1 * P2 *... * PM-1. If n has a prime factor of 4N-1, then let it be p. so p must be in P1, P2... PM, let P = Pi, so pi|n, and pi|n + 1
Euclid proved that the number of primes is infinite by the method of disproportion
Suppose all prime numbers are 2, 3, 5... P in turn
Let m = 2 * 3 * 5 *... * P + 1
Because 2,3,5... P can't divide m, then M is either a prime number or a prime number larger than P can divide M. in both cases, it means that there are new and larger primes, which contradicts the hypothesis that all primes are infinite
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