Unprimeable Composites and Digitally Delicate Primes

I consult Numbers Aplenty on a daily basis and one of the categories always mentioned is that unprimeable numbers. It's one that I have always ignored but a recent mathematical article kindled my interest:

Mathematicians Discovered a New Kind of Prime Number

In new research, mathematicians have revealed a new category of “digitally delicate” prime numbers. These infinitely long primes turn back to composites faster than Cinderella at midnight with a change of any individual digit.

Digitally delicate primes have infinite digits, and changing any digit to any other value bears a composite number outcome instead. To use a more bite-size example, consider 101, which is a prime. Change the digits to 201, 102, or 111, and you have values that are divisible by 3 and therefore compound numbers.

This idea is decades old, so what’s new? Now, mathematicians from the University of South Carolina have established an even more specific niche of the digitally delicate primes: widely digitally delicate primes. These are primes with added, infinite “leading zeros,” which don’t change the original prime, but make a difference as you change the 0s into other digits to test for delicacy.

So instead of 101, consider 000101. That value is prime, and the zeros are just there for show, basically. But if you change the zeros, like 000101 to 100101, now you have a composite number that’s divisible by 3. The mathematicians believe there are infinite widely digitally delicate primes, but so far, they can’t come up with a single real example. They’ve tested all the primes up to 1,000,000,000 by adding leading zeros and doing the math. 

I won't go into the topic of widely digitally delicate primes in this post but will look at digitally delicate primes that form OEIS A050249:

A050249

Weakly prime numbers (changing any one decimal digit always produces a composite number). Also called digitally delicate primes.

The first few of these sorts of primes are:

294001, 505447, 584141, 604171, 971767, 1062599, 1282529, 1524181, 2017963, 2474431, 2690201, 3085553, 3326489, 4393139, 5152507, 5564453, 5575259, 6173731, 6191371, 6236179, 6463267, 6712591, 7204777, 7469789, 7469797

Clearly these sorts of primes are quite rare. Terence Tao however, has proved that this sequence is infinite. For values 6, 7, 8, 9, 10 of \(k\), the number of terms \(< 10^k \) in this sequence is 5, 35, 334, 3167, 32323. It should be noted that the digits include 0 and so the first number in the sequence, 294001, can become 094001 or simply 94001 which of course is composite.

200 is the first unprimeable number

Unprimeable numbers turn out to be far more numerous and are defined as follows:

A composite number \(n\) is called unprimeable if it cannot be turned into a prime by changing a single digit. 

For example, 144 is not unprimeable, because changing the last digit into a nine we obtain 149, a prime. The number 200 is instead unprimeable (the smallest one), since none of the numbers 201, 203, 207, 205, and 209 are prime and all the other numbers which can be obtained from 200 (say, 300, or 270, or 208) are even, so they are not prime. 

It is easy to prove that unprimeable numbers are infinite, since, for example, all the numbers of the form  \(510+k\cdot 2310\) are unprimeable. Source.

These numbers form OEIS A118118:

A118118

Composite numbers that always remain composite when a single decimal digit of the number is changed.

 The initial members of the sequence 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, 840, 842, 844, 845, 846, 848, 890, 892, 894, 895, 896, 898, 1070, 1072, 1074, 1075, 1076, 1078

Notice that 202 is not in the sequence because it can be changed to 002 = 2 which is prime. Otherwise, within a suitable decad, all numbers ending in 0, 2, 4, 5, 6 or 8 will be in the sequence e.g. 320, 322, 324, 325, 326 and 328. Thus the numbers comes in batches within decads that often have wide gaps between them:

• 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
• 840, 842, 844, 845, 846, 848
• 890, 892, 894, 895, 896, 898
• 1070, 1072, 1074, 1075, 1076, 1078

So this post brings together two different but related categories of numbers: 

• the digitally delicate prime that always changes into a composite number with the alteration of the single digit AND 
• the unprimeable composite that can never change into a prime by the alteration of a single digit