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:
- $200, 202, 204, 205, 206, 208, \ldots$
This sequence is A192545 in the On-Line Encyclopedia of Integer Sequences (N. J. A. Sloane (Ed.), 2008).
Proof
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.
$\blacksquare$
Sources
- 1970: Wacław Sierpiński: 250 Problems in Elementary Number Theory: $101$
- 1997: David Wells: Curious and Interesting Numbers (2nd ed.) ... (previous) ... (next): $200$