Answer only 912 please Answer only 912 please Find all prime

Find all primes between 1025 and 1050. In the nest problems, pretend the number P used in the first proof of Theorem 2.9 really exists. Find (P, n) for n = 11, 12, ..., 20 Find (P, n) for n = 21, 22, ..., 30. Define a legitimate function P(n) to take the place of the fictional (P, n). Give a detailed proof of why the number p_1p_2 ... p_k + 1 of Euclid\'s proof of Theorem 2 9 is not divisible by any of the p_s. How many primes are congruent to 0 (mod 4)? To 2 (mod 4)? Show that if (a, b) > 1. then there exists at most one prime congruent to a (mod 6). Show that if p and q are twin primes and p + q > 8, then 12 divides p + q. Show that if p, p + 2. and p + 4 are all prime, then p = 3. Show that if ab = c^2. with a and b relatively prime positive integers, then a = r^2 and b = s^2 for some integers r and s.

Solution

9. If a prime is congruent to 0 or 2 mod 4 then the prime must be even

Only even prime number is 2

So, there are no primes congruent so 0 mod 4

2 is only prime which is 2 mod 4

12.

p must be odd. Else,p+2,p+4 are also even and hence not prime

Case 1. p=3

There is nothing to prove

Case 2. p=3k+1

Then,p+2=3k+3 hence not prime

Case 3. p=3k+2

Then,p+4=3k+6 hence not prime

Hence, p=3