SOLUTION: How many bases $b \ge 2$ are there such that $100_b + 1_b$ is prime?

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Question 1207586: How many bases $b \ge 2$ are there such that $100_b + 1_b$ is prime?
Answer by ikleyn(52879) About Me  (Show Source):
You can put this solution on YOUR website!
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How many bases $b \ge 2$ are there such that $100_b + 1_b$ is prime?
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If I understand this question correctly, it asks (in hidden form)


    +---------------------------------------------+
    |        For how many whole numbers n         |
    |   the expression n^2+1 is a prime number?   |
    +---------------------------------------------+


Infinitude of primes of the form 𝑛^2+1 is an open question in theory of numbers

(see web-site https://oeis.org/wiki/List_of_prime_conjectures).


In other words, till today  (which is June 2024),  nobody knows a proven answer to this question 
and nobody has a commonly accepted proof.