Using your Head is Permitted

Define a "Self-Assembled Number" (SAN) to be a positive integer that is the result of a calculation involving its own digits plus the four basic arithmetic operations and exponentiation, where each digit is used exactly once. Notably, the definition of SAN is dependent on the base in which the integer is represented. Here are some examples of SANs:

In base ten:
(i) 25 = 5²
(ii) 625 = 5^6-2
(iii) 1024 = (4/2)¹⁰

In binary:
(iv) 11001_TWO = 101_TWO^10_TWO

(Interestingly, examples (i) and (iv) are really the same calculation in a different representation base.)

Note that, in the calculation, digits can also be concatenated into integers, such as in the case of the "10" in examples (iii) and (iv). We do not allow, however, extraneous leading zeros, decimal points and the like.

This month's riddle has two parts. Answer either or both to be credited as a solver. There will be separate solvers' lists kept for each part.

In each of the parts, you are asked to determine about a specific subset of SANs in which bases there is an infinite number of them. For any base of representation where you claim an infinite number does not exist, prove this point. Where you claim it does exist, show a construction.

Part 1:

127 = -1 + 2⁷

In which bases are there an infinite number of these?

Part 2:

In order to be considered a solver for Part 2, you should present an answer for all bases that are composite numbers. The answer for primes is a bonus question.

(The published answer will cover both cases.)

As another bonus question, consider the numbers described in Part 1. In which bases are there an infinite number of these that are also prime numbers?

The solution to this month's riddle will include references to further reading. Until these references are published, readers are kindly asked to refrain from Internet searching on the topic if they want to qualify as original solvers.

One last word: "Using your Head is Permitted" and IBM's "Ponder This" decided to coordinate themes this month on their respective riddles. As a result, readers interested in more riddles on a similar vein can also refer to the Ponder This site.

List of solvers:

Part 1:

Part 2: