Numbers that cannot be made Prime by changing 1 Digit

Theorem

The following positive integers cannot be made into prime numbers by changing just one digit:

$200, 202, 204, 205, 206, 208, \ldots$

In order to make any one of these positive integers into a prime number one would have to change the last digit.

Otherwise the number it was changed into would be either even, or divisible by $5$, and so not prime.

But all the other integers between $200$ and $209$ are composite:

$\ds 201$

$=$

$\ds 3 \times 67$

$\ds 203$

$=$

$\ds 7 \times 29$

$\ds 207$

$=$

$\ds 3^2 \times 23$

$\ds 209$

$=$

$\ds 11 \times 19$

In order for there to be a smaller number with this property, a prime gap would need to be found which spans an entire decade.

There are none such.

On the other hand, each of $201, 203, 207, 209$ can themselves be turned into a prime number by changing the initial $2$ into a $1$:

$101, 103, 107,109$

are all prime.

$\blacksquare$