# Primes from the Pan Digital

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.

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

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.

• 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. – jarnbjo Jun 27 at 11:22
• Small calculator will narrow down 4 digit....almost 5 digit..like I commented on hexomino’s Q, you need 1 computer calculation – Uvc Jun 27 at 11:35
• @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. – jarnbjo Jun 27 at 12:22
• 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.. – Uvc Jun 27 at 12:28
• We are using the calculator sparingly to narrow down final options after initial reasoning...you – Uvc Jun 27 at 12:35