Discrete Math 2 Prove that if n 2 then n31 is composite by c

Discrete Math

2) Prove that if n >2 then n3-1 is composite by contrapositive

3) Show that if x and y are odd numbers, then there is no integer z so that x2+ y2 = z2

Solution

2)

Let, n^3-1 be contrapositive

n^3-1=(n-1)(n^2+n+1)

n=1 give, n^3-1=0 which is not composite

n=2 gives n^3-1=7 which is prime

For n>2

n-1>1

Hence n-1 is a factor of n^3-1

HEnce n^3-1 is composite

Hence n>2

2)

Assume such a z exists

x,y are odd

Hence, x^2=y^2=1 modulo 4

x^2+y^2=2 modulo 4

HEnce, z^2=even

Hence z is even

So,z=0 modulo 4

Hence a contradiction

So no such integer z exists