Unprimeable numbers

Unprimeable numbers are special kind of numbers. No matter how you change one of its digits, you can never make it into a prime number. A prime number is a number that can only be divided evenly by 1 and itself. For example, 2, 3, 5, and 7 are prime numbers.

Examples of Unprimeable Numbers

Let's look at the number 14.

If we change the first digit to 0, 1, 2, 3, 4, 5, 6, 7, 8, or 9, we get numbers like 04, 24, 34, etc. None of these are prime numbers.
If we change the second digit to 0, 1, 2, 3, 4, 5, 6, 7, 8, or 9, we get numbers like 10, 11, 12, etc. But 11 is a prime number!

So, 14 is not an unprimeable number because we can change it into a prime number by changing one digit.

Now let's look at the number 200.

If we change the first digit to any other digit, we get numbers like 100, 300, 400, and so on. None of these are prime numbers.
If we change the second digit to any other digit, we get numbers like 210, 220, 230, and so on. None of these are prime numbers.
If we change the third digit to any other digit, we get numbers like 201, 202, 203, 204, 205, 206, 207, 208, and 209. Yet again, none of these are prime numbers, especially 203 and 209, since the factors for both of these numbers are 7 and 29 and 11 and 19, respectively.

So, 200 is an unprimeable number because no matter how you change a digit, you cannot make it into a prime number. Notably, it is the smallest unprimeable number. The first 28 unprimeable numbers are 200, 204, 206, 208, 320, 322, 324, 325, 326, 328, 510, 512, 514, 515, 516, 518, 530, 532, 534, 535, 536, 538, 620, 622, 624, 625, 626, 628, etc...

The smallest unprimeable numbers to end in 1, 3, 7, and 9 are 595631, 1203623, 872897, and 212159, respectively.

Why Are Unprimeable Numbers Interesting?

Unprimeable numbers are interesting to mathematicians because they are rare and have unique properties. Finding them can be challenging, and they help us learn more about how numbers work.

Conclusion

Unprimeable numbers are numbers that cannot become prime numbers, no matter how you change one of their digits. They are a fun and interesting part of mathematics, showing us that numbers can be more complex than they seem.