he integers in the puzzle are all Reaman Numerals with the common multiplier digit of 4.

Each has a distinct do-si-do digit as indicated.

 102,564 times 4 equals 410,256 128,205 times 4 equals 512,820 153,846 times 4 equals 615,384 179,487 times 4 equals 717,948 205,128 times 4 equals 820,512 230,769 times 4 equals 923,076

s for that 57-digit number in the hint -- why, it is merely another Reaman Numeral.

 1,016,949,152,542,372,881,355,932,203,389,830,508,474,576,271,186,440,677,966 times 6 equals 6,101,694,915,254,237,288,135,593,220,338,983,050,847,457,627,118,644,067,796
It has one property in common with the first number in the puzzle; namely, the multiplier digit and the do-si-do digit are the same.

he following equation is a generalized version of the one we derived elsewhere.

m x = (x - d) / 10 + 10n-1 d , where:
d = the do-si-do digit (constrained)
m = the multiplier digit (given)
n = the number of digits in x (unknown)
x = the Reaman Numeral (unknown)
Using a few algebraic steps we are able to rearrange our equation so that x winds up all by itself over on the left hand side.
x = (10n d - d) / (10m - 1)
We will always know the value of the denominator of this fraction, inasmuch as, in finding a Reaman Numeral, m must be given. Here is a handy table of denominators.

 m = 9 8 7 6 5 4 3 2 10m -1= 89 79 69 59 49 39 29 19

The numerator has the same form for any value of digit d. Thus, the fraction may be expressed as follows:

 x = d-1 9 9 9 ... 9 10-d / (10m - 1)

The algorithm applies four steps...

Step 1. Pick a value for digit d,
Step 2. Put digit d - 1 at the leftmost end of a string of nines,
Step 3. Put digit 10 - d at the rightmost end of that string of nines, and
Step 4. Carry out the indicated long division by 10m - 1.
Here is a handy table of numerators.

 d = 9 8 7 6 5 4 3 2 Numerator 89~91 79~92 69~93 59~94 49~95 39~96 29~97 19~98
During the long division in Step 4, we have to keep inserting nines as we go along until the quotient comes out even. As we have seen, that can take quite a long time.

Finally, one correspondent, Don Lauria, has suggested that the Reaman Numerals should be given a symbolic representation as R(d,m), such that, for example, R(4,4) = 102,564, R(5,4) = 128,205, and R(5,5) = 102,040,816,326,530,612,244,897,959,183,673,469,387,755.  So, then, how many Reaman Numerals are there?

Epilog:

Few activities given to mankind carry more solemn responsibilities than those that arise in creating puzzles.  As a courtesy to solvers, both the statement and the solution must be precisely correct.

More than six years went by after the publication of Reaman Numerals Plural, and with no challenges received, one might reasonably assume that the material is error free.  So then several Reaman Numerals were appropriated for what was purported to be the world's first Cross-Number Puzzle, and zowie: a typographical errer was reported and reported and reported...  The mistrake has been corected now, thanks to the allertness of two many solvers to acknowlage.

 Home Page Puzzle Page Math and Models The Puzzle as a Literary Genre