Algebraic Roots

Algebraic  Roots
This refers to the algebraic factors of numbers b^n=1,  In decimal, this means numbers like 9, 99, 999, 9999, ...

There is a unique factor for each divisor of the number of digits. For example, 999999 has six digits, and the relevant factors are A(1)=9,  A(2) = 11, A(3) = 111, and A(6) = 91. A(1) = 9, A(1)A(2) = 9*11 = 99, A(1)A(3) = 9*111 = 999, all of these divide 999999. Note that A(4)=101 does not divide this, because 4 does not divide 6.

The algebraic roots defined in this way, are of the same order of the power of the base, as the Euler totient of the n in A(n). This means that there are four co-primes between 0 and 12, so A(12) has four digits. It's 9901. Likewise, the Euler totient of five is 4, its A(5) is 11111, where the first digit is the fourth power of the base.

The algebraic roots have the same form in every base, with the relation that the digits 6, 7, 8, 9 are replaced by the last four digits before 10. In base 7, A(12) becomes 6601, the last four digits being 3, 4, 5, 6. This particular relation happens because the numeral attached to any power of the algebraic roots does not exceed 4 (tested to numbers of 162 places.)

The factors of an algebraic root A(j) consists of primes of the form jx+1, that have a period j in base b. In decimal, A(7) is 1111111 = 239 * 4649, these two primes have a seven-place period. In base 12, it gives 1111111 = 659*4943. The Cunningham Project lists factors of algebraic roots for bases 1-10 and 12, as far as modern technology takes.

The other factors that algebraic roots might have are repeaters and sevenites.

The order given here follows the same as the Cunningham order, where the odd number is listed just before its double-value. This is because odd numbers have larger Euler totients than the surrounding even numbers, and this order arranges the values in approximately increasing size. The list here gives all roots up to 11 digits, using the Cunningham order. It includes all of the odd numbers to 11, the even numbers to 24, and the pair 15, 30.
 * 1:  9
 * 2:  11
 * 4:  101
 * 3:  111
 * 6:   91
 * 8:  10001
 * 5:  11111
 * 10:  9091
 * 12:  9901
 * 7:  1111111
 * 14:  909091
 * 16:  100000001
 * 9 :  1001001
 * 18:  999001
 * 20:  99009901
 * 11:  11111111111
 * 22  9090909091
 * 24  99990001
 * 15  90090991
 * 30  109889011

Repeaters
Where a prime p divides some A(j), then it will divide A(pj), A(p²j) etc. This reflects that prime-powers will eventually divide some rep-9'. In the list above, we see that 3 divides A(1), and also A(3) and A(9). Likewise 11 divides A(22),

Every prime will have a repeater in this manner, if it has an original period. In decimal, 2 and 5 do not have periods, so they do not have repeating instances.

Sevenites
On random occasions, a prime might divide its own period. This is extremely rare, but it does happen.

In base 32 (which is 2^5), we see that 1, 0, 1 is 5*5*41. This is because 32A(4) = 2A(4) * 2A(20) = 5 * (5*41). Writing the periods in full give 5 00110 01100 11001 10011 =  6, 12, 25, 19  25  00001 01000 11110 10111 =  1,  8, 30, 23 Although the period of 1/25 has increased in base 20 from 4 to 20 digits, in base 32, it has stayed at four digits.

The sevenites are named after the behaviour of 7 in base 18, where a third-order sevenite happens. 7 2 10  5  2 10  5 ...   49  0  6 11  0  6 11 ...  343  0  0 17  0  0 17 ... Sevenites are generally quite rare. When one serches for these into to 2^32, the list of primes that divide their own periods is a list less than 40 characters long. For example, just three are known for decimal, two for dozenal, and four for base 120. 6: 66161 534851  3152573                                                     10: 3  487  56598313                                                           12: 2693  123653                                                               14: 29  353                                                                    15: 29131                                                                      18: 5  7 # 37  331  33923  1284043                                             20: 281  46457  9377747  122959073                                             40: 11  17  307  66431                                                         60: 29                                                                         80: 3 # 7  13  6343                                                           120: 11  653  2074031  124148023         to 2*120**4 The # means that the preceeding prime goes back for a third shot.

Finding sevenites is something that chews up a lot of machine cycles, and extensive lists are only recently available. The values for primes under 1000 were found mostly in the nineteenth century, the larger values were discovered in the 20th century. In the case of base 120, I found each sevenite with increasing technology.