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A number $ A_{1}A_{2}...A_{n} $ is called a (decimal) pure prime number iff $ A_{1},A_{1}A_{2} $ ... and $ A_{1}A_{2}...A_{n} $ are all prime. Decimal pure prime numbers are finite in number; here is full list of those $$ \{53, 317, 599, 797, 2393, 2399, 3793, 3797, 7331, 23333, 23339, 31193, 31379, 37339, 37397, 71933, 73331, 373393, 593993, 719333, 739397,739399, 2399333, 7393931,7393933, 23399339, 29399999, 37337999, 59393339, 73939133\} $$ (+ subparts, i.e. $5, 31 $ etc., $ 83 $ total)

So, this might be proved by brute-force. But is true that for every basis $b$, $b$ -ary pure prime numbers are finite in number?

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    $23$, $37$ and others seem to be missing from your list of "decimal pure prime numbers". Where did you get your list from and why do you think the list is finite? – 2017-01-04
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    Yes. I wrote only a "maximum", i.e 23333 or 23339 instead of {23, 233, 239, 23333, 23339 } Absolutely full version is $$ \{23,29,31,37,53,59,71,73,79,233,239,293,311,313,317,373,379,593,599,719,733,739,797,2333,2339,2393,2399,2939,3119,3137,3733,3739,3793,3797,5939,7193,7331,7333,7393,23333,23339,23399,23993,29399,31193,31379,37337,37339,37397,59393,59399,71933,73331,73939,233993,239933,293999,373379,373393,593933,593993,719333,739391,739393,739397,739399,2339933,2399333,2939999,3733799,5939333,7393913,7393931,7393933,23399339,29399999,37337999,59393339,73939133\} $$ – 2017-01-04
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    But where did you get the list from and why do think it is finite? – 2017-01-04
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    A term you may or may not know is (right) [truncatable prime](https://en.wikipedia.org/wiki/Truncatable_prime). – 2017-01-04
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    Because, for example, 739391330, 739391331, ... 739391339 aren´t prime. And the same is true for any "maximal" pure prime – 2017-01-04
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    term "pure prime" is from Hazrat (Mathematica - A Problem Centered Approach) – 2017-01-04
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    @DanielFischer: thanks for the reference. – 2017-01-04
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    Your so called "Absolutely full version" is missing at least $2$, $3$, $5$ and. $7$. – 2017-01-04
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    @RobArthan For a specific b, may be demonstrated it via hard power, human or computational. For example: for b=10, one-digits primes are 2,3,5,7. Two-digits candidates are 20-29, 30-39, 50-59, 70-79, but primes are only 23, 29, 31,37, 53, 59, 71,73,79. For example, 53 have not any successor, 530-539 aren´t primes. So, we construct a tree, such that for leaves is not hard proved by computation, that they are really leaves. But the same is a problem for general b. – 2017-04-19

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