Odd Integers whose Smaller Odd Coprimes are Prime

Theorem
$105$ is the largest integer such that all smaller odd integers greater than $1$ which are coprime to it are prime.

Proof
First it is demonstrated that $105$ itself satisfies this property.

Let $d \in \Z_{> 1}$ be odd and coprime to $105$.

Then $d$ does not have $3$, $5$ or $7$ as a prime factor.

Thus $d$ must have at least one odd prime as a divisor which is $11$ or greater.

The smallest such composite number is $11^2$.

But $11^2 = 121 > 105$.

Thus $d$ must be prime.

Thus it has been demonstrated that all odd integers greater than $1$ and smaller than $105$ which are coprime to $105$ are prime.