We can define Euclid numbers, e(n), by the following inductive definition:

e(1) = 2

e(n) = e(1) *...*e(n-1) + 1, n > 1

Calculating, we get

e(1) = 2

e(2) = e(1) + 1 = 2+1 =3

e(3) = e(1) * e(2) + 1 = 2*3+1 =7

e(4) = e(1) * e(2) * e(3) + 1 = 2*3*7+1 =43

Show that,

(i) when j /= k ( '/=' is supposed to mean 'NOT equal to!), e(j) and e(k) are

relatively prime, i.e. gcd(e(j), e(k)) = 1

(ii) for n > 0,

e(n+1) = (e(n))^{2 }- e(n) + 1

No idea how to do (i), tried (ii) using induction but didn't get very far. Please help, would

be ever so grateful!!!