Prime, non-prime and pure burgers

Claes Oldenburg, Floor Burger, 1962

A prime burger (PB) is a grilled prime number served in a split bun of two nonnegative nonprimes. Examples are 870, 436 and 1239 [the latter remaining a PB whatever the way (among three possible) to partition it: 1.2.39, 1.23.9 or 12.3.9].

Similarly, a nonnegative nonprime burger (NNB) is a fried nonnegative nonprime number served in a split roll of two prime numbers. Examples are 282, 507 and 2913 [the latter remaining a NNB whatever the way (among three possible) to partition it: 2.9.13, 2.91.3 and 29.1.3].

All the above burgers are pure, as they are not a mix of PBs, NNBs and junk food [what we call junk food is a 3-integer succession (like 1.2.3 or 7.8.9) that doesn’t form a PB or a NNB].

A good example of impure burger would be 1782 — which indeed can be considered as a prime burger (according to the partition 1.7.82), but also as a nonnegative nonprime burger (17.8.2) or plain junk food (1.78.2).

[Note also that numbers having one or more partitions starting with a leading 0 (like 01 or 003) cannot have access to the envied status of pure burgers.]

Question

Say we give the name P to the sequence containing only pure burgers: is P finite?

[As we couldn’t find any 6-digit pure burger, we conjecture that yes, P is finite. The best we could find (by hand) was six 5-digit pure burgers. Two of them are NNBs (59947 and 59997), four of them are PBs (63730, 63734, 63736 and 63738)]. 

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Scott Shannon has just sent (in private, July 25th 2023):

> I think you are correct in saying there are no more prime or nonprime with more than 5 digits.

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Note that a prime burger is not a prime sandwich