9. (Primes of the form 4k +3) (a) Show that odd numbers have remainder either 1

Question

9. (Primes of the form 4k +3) (a) Show that odd numbers have remainder either 1 or 3 when divided by 4. (b) Let a,b have remainder 1 when divided by 4. Show that ab has the same reminder. c) Suppose a leaves reminder 3 when divided by 4. Show that a is divisible by a prime with (d) Let P be a non-empty set of primes, and let n-???? be their product. Show that no (e) Show that there are infinitely many primes of the form p 4k+3. Hint: Write a -4k+1, b 4l+1 and multiply. the same property. p P divides 4n-1

Explanation / Answer

c)

Let a be only divisible by primes which leave remainder 1 when divided by 4

Now from part b. the product of primes which leave remainder 1 when divided by 4 also leaves remainder 1 when divided by 4 but a leaves remainder 3 when divided by 4 so we have a contradiction

Hence, there is at least one prime of the form 4n+3 which divides a

d)

Let there be some prime p in P so that p divides 4n-1

Now,

n=Kp , K is some integer, product of all primes in P other than p to be specific.

4n=4Kp

m=4n-1=4Kp-1

p divides 4n-1 ie m hence,

m=Qp=4Kp-1

(4K-Q)p=1

Hence, p divides 1 which is not possible as p is a prime and hence >=2

Hence, no prime p in P divides 4n-1

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