Magnanimous Primes

On December 27th 2020, almost two years ago, I created a post titled Magnanimous Numbers which are defined as follows:

A magnanimous number is a number (which we assume to be of at least 2 digits) such that the sum obtained inserting a "+" among its digit in any position gives a prime.

Such numbers are quite rare. At the time of that post, I had just turned 26201 days old and this number is magnanimous. Since then my diurnal age has corresponded to only one other such number: 26285. That is, until today, when I turned 26881 days old, a number that is also magnanimous. Checking we find that:

$$\begin{align} 2+6881&= 6883 \text{ which is prime}\\26+881&=907 \text{ which is prime}\\268+81&=349 \text{ which is prime}\\2688+1&=2689  \text{ which is prime}\end{align}$$ What differentiates 26881 from these other numbers is that it is prime itself and is thus a member of a special class of numbers known as magnanimous primes. Though magnanimous numbers are rare, magnanimous primes are naturally even rarer. Such primes belong to OEIS  A089392 with the single digit primes (2, 3, 5 and 7) being included for completeness.

A089392

Magnanimous primes: primes with the property that inserting a "+" in any place between two digits yields a sum which is prime.

 Up to one million, there are only 71 such primes (see permalink). They are:

2, 3, 5, 7, 11, 23, 29, 41, 43, 47, 61, 67, 83, 89, 101, 227, 229, 281, 401, 443, 449, 467, 601, 607, 647, 661, 683, 809, 821, 863, 881, 2221, 2267, 2281, 2447, 4001, 4027, 4229, 4463, 4643, 6007, 6067, 6803, 8009, 8221, 8821, 20261, 24407, 26881, 28429, 40427, 40483, 42209, 60443, 68683, 80849, 220021, 224027, 228601, 282809, 282881, 282889, 404249, 408049, 464447, 600881, 602081, 626261, 800483, 806483, 864883

I'll see only one more (28429) during my lifetime. This site has a useful table that shows the number of magnanimous primes and their range for number of digits ranging from 2 to 9. It is reproduced in Figure 1.

Figure 1: source

It is most likely that there are only a finite number of magnanimous primes and it could well be that 608,844,043 is the largest.