Primes Within Primes

Today is a prime day, 24517, and it turns out to be an interesting prime in that it conforms to the membership requirements of OEIS A164077 which state that it must be:

• Prime p1 of the form a^b + c^d = p1, where a, b, c, d are primes
• a + b + c + d = p2, where p2 (A164078) is prime
• conc(abcd) = p3 (concatenation of a, b, c , d) is prime (A164079)

The first such number to qualify is 5527 because:

• 5^5 + 2^7 = 3253 is prime
• 5 + 5 + 2 + 7 = 19 is prime
• conc (abcd) = 5527 is prime

The next number to qualify is 24517 because:

• 29^3 + 2^7 = 24517 is prime
• 29 + 3 + 2 + 7 = 41 is prime
• conc (abcd) = 29327 is prime

After that comes 78157 because:

• 2^5 + 5^7 = 78157 is prime
• 2 + 5 + 5 + 7 = 19 is prime
• conc (abcd) = 2557 is prime

After that comes 366103 because:

• 2^13 + 71^3 = 366103 is prime
• 2 + 13 + 71 + 3 = 89 is prime
• conc (abcd) = 213713 is prime

... and so on and so on ...

Clearly primes satisfying such stringent prime number requirements are few and far between. Here is the list as shown on the OEIS site:

3253, 24517, 78157, 366103, 548677, 705097, 1030429, 1229257, 5735467, 6438391, 12221371, 17498881, 19618243, 74084347, 118370899, 263374849, 270840151, 286199371, 410180599, 418195621, 418719781, 529483321, 565609411, 698388391

I would imagine that there are an infinity of such primes but don't ask me to prove it.