For a=10 it means that 10^(p-1)-1 is multiple of p. That's why, f. ex.: 999999 is multiple of 7, and 999999999999 is multiple of 13.
In addition, many of this primes (called: full reptend primes) has the property that they generate cyclic numbers. This means that when you divide [10^(p-1)-1] by p, the number you get "q" and its multiples from 1 to (p-1) [=q, 2*q, 3*q,..., (p-1)*q] have all the same digits.
The most famous example of a cyclic number is for p=7 where 999999/7=142857.
See that taking the extreme left digit of 142857 and adding it to the right extreme, the new number 428571 is 3*142857, and so, repeating the operation, is 285714 (=2*142857), etc. Rotation, digit by digit, result in six numbers that are multiples from 1 to 6 of 142857 (while not necessarily in order from 1 to 6).
It has been conjectured, but non proven, that infinite cyclic numbers, with origin in primes, exists. In fact, the fraction of cyclic numbers out of all primes has been conjectured to be the Artin's constant: C=0,3739558126...
It means, if we accept the conjecture, that more than 1/3 of the primes are "full reptend primes" (base for cyclic numbers).
Knowing this all our question is straight:
Given a positive integer m the conjecture implies that there is a positive full reptend prime q such that:
q>2m.
[10^q-1]/q is a cyclic number k.
Then m*k and m*2k have the same digits, and 2km is a digit anagram of km.
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Ex:
m=53
q>106
From http://oeis.org/A086018 you see that 109 is a full reptend prime.
k=[(10^108)-1]/109=
(00)9174311926605504587155963302752293577981651376146788990825688073394495412844036697247706422018348623853211
k, 2*k, 3*k, 4*k, ..., 108*k all will have the same digits. Each one-digit rotation is in corrispondence with one of this multiples, while non necessarily in order from 1 to 108.
m*k=486238532110091743119266055045871559633027522935779816513761467889908256880733944954128440366972477064220183
m*2k=972477064220183486238532110091743119266055045871559633027522935779816513761467889908256880733944954128440366
Solution "if"
