2
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Pan Digital Number is the smallest with all the digits 1 to 9 with no repeats.

When looking forwards or backwards, eleven Primes can be extracted from that number preserving the same sequence.

Among the eleven primes, one of each 4 digit, 5 digit, 8 digit ones exist. Rest are single and double digit primes.

Just using deductive reasoning and no brute force, deduce all the primes.

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  • $\begingroup$ I may need a computer to determine if an 8-digit number is prime. Is that allowed? $\endgroup$ – hexomino Jun 27 at 10:32
  • $\begingroup$ For just one number..you may need it. All others should follow from logic. $\endgroup$ – Uvc Jun 27 at 10:34
4
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Using:

$123456789$

we find:

$2,3,5,7$
$23,67,89$ and $43$
4-digit - either $4567$ or $4321$ (no even numbers, multiples of $3$ or $5$)
5-digit - $34567$, $56789$ or $76543$
$23456789$, because the other option is a multiple of $3$, and this one's reverse is a multiple of $2$.

I can't yet see how to find the $4$ and $5$ digit primes without using a computer/calculator.

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  • $\begingroup$ I am also stuck at the same point. There are still 3 numbers to rule out, but I don't see any way to do so by logic. I know which numbers they are and their factors are absolutely not trivial to find. $\endgroup$ – jarnbjo Jun 27 at 11:22
  • $\begingroup$ Small calculator will narrow down 4 digit....almost 5 digit..like I commented on hexomino’s Q, you need 1 computer calculation $\endgroup$ – Uvc Jun 27 at 11:35
  • $\begingroup$ @Uvc So you don't consider checking with a calculator if a 4 digit number is prime to be 'brute force'? I would very much do so. There is not much 'deductive reasoning' in using a calcuator. $\endgroup$ – jarnbjo Jun 27 at 12:22
  • $\begingroup$ You can do by hand calculations when. Doing small number check....considering the last digit options can be narrowed....I just said calculator as most people use phones to check.. $\endgroup$ – Uvc Jun 27 at 12:28
  • $\begingroup$ We are using the calculator sparingly to narrow down final options after initial reasoning...you $\endgroup$ – Uvc Jun 27 at 12:35

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