Question

y + 1
is
a
*
factor of
- 29 +4K - 1​

Answer 1

Answer:

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Prove that any prime of the form 3k + 1 is of the form 6k +1.

Proof: The only even prime is 2 and it is not of the form 3k + 1. Thus any prime of the form 3k + 1 is odd. This means 3k must be even which implies that k must be even. Let k = 2m. Then a prime of the form 3k + 1 is of the form 3(2m) = 1 = 6m+1.

#10

Prove that any positive integer of the form 4k + 3 must have a prime factor of the same form.

Proof: Any number of the form 4k + 3 must be odd so it can not have any factors of the form 4k or 4k + 2 which are even. Thus all the factors of a 4k + 3 must be of the forms 4k + 1 and 4k + 3. Suppose that they were all of the form 4k + 1. Multiplying two such yields (4k+1)(4m+1) = 4(4km+k+m) +1, another 4k+1. Thus the product of any number of factors of the form 4k + 1 must be another 4k + 1. Thus a 4k + 3 must have a prime factor of the form 4k + 3.

Prove that any positive integer of the form 6k + 5 must have a prime factor of the same form.

Proof: Since a number of the form 6k + 5 is odd and not divisible by 3 it can’t have any factors of the form 6k, 6k + 2, 6k + 3, or 6k + 4. Thus all its prime factors are of the form 6k + 1 or 6k + 5. Multiplying any number of 6k + 1’s yields another 6k + 1. Thus it must have a prime factor of the form 6k + 5.

#11. If x and y are odd, prove that x2 + y2 can not be a perfect square.

Proof: If x and y are odd then x2 + y2 must be even. An even perfect square must be divisible by 4. (Why?) Let x = 2k + 1 and y = 2m + 1. Then

x2 + y2 = (2k + 1)2 + (2m + 1)2 = 4(k2 + k + m2 + m) +2 which is not a multiple of 4 and hence not a perfect square.

#12. If x and y are prime to 3, prove that x2 + y2 can not be a perfect square.

Step-by-step explanation:

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