2003 Pan African MO Problems/Problem 3
Does there exists a base in which the numbers of the form: are all prime numbers?
Using the definition of base numbers, the number in base can be rewritten as , where .
The above expression can be factored, so In order for to be prime, either or (but not both) must equal . If , then or , but none of the values of are valid base numbers. If , then or . However, neither value are valid bases because is less than and the number has zeroes for digits (making base an invalid base).
Therefore, there is no base where are all prime numbers.
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