### Post by on Sept 16, 2015 14:08:40 GMT -8

It sounds like Buffoonery requires a base number to contain both 3 and 4; the least common multiple of these is twelve, and no multiple of twelve below 60 has three prime factors. Oh well, that means sticking to only 2 and 3. He also complained if 1/8, 3/8, 5/8, and 7/8 needed two places after the radix point to terminate (in other words, the base wasn't divisible by 8). This would raise the minimum number to twenty-four (8*3), and he will probably write out fractions in this base and say "Oh no! Sixteenths still need two floating places to terminate!" LOL he could go on and on, adding factors of two, maybe even forty-eight (2^4 * 3) would not have enough of those for him. It's much more useful to add a third prime factor, but this can't be done, since sixty and perhaps forty-eight are too high. There are only three choices: twelve, twenty-four, and thirty-six, all of which terminate the same set of fractions. Thirty-six, hands down, does the best at not making the non-terminating fractions (recurring) ugly. It does, in total, have nine factors as well, and using 10(alphadecimal) as 6^2 = 2^2 * 3^2 is quite notable.

(custom random looking symbol set)

1/2 = 0.y

1/3 = 0.U

1/4 = 0.9

1/5 = 0.7777777777777...

1/6 = 0.6

1/7 = 0.5555555555555...

1/8 = 0.4y

1/9 = 0.4 // Only alphadecimal makes 1/9 have just one significant digit if other multiples of 9 don't count because they don't have 4.

1/t = 0.3aaaaaaaaaaaaaa...

1/r = 0.39m

1/U = 0.3

1/H = 0.2eEn8r2eEn8r2eEn8r2eEn8r2....

1/C = 0.2PPPPPPPPPPPPP...

1/A = 0.2CCCCCCCCCCCCC....

1/J = 0.29

1/N = 0.248JnFeG248JnFeG248JnFeG24....

1/y = 0.2

1/G = 0.1R7PdrH9N1R7PdrH9N1R7PdrH9N1....

1/P = 0.1ZZZZZZZZZZZZZZZZ....

1/a = 0.1TTTTTTTTTTTTTTTT.....

1/] = 0.1]R?6G]R?6G]R?6G]R?6G]R?6G]R?.....

1/j = 0.1PUyZ69C34T1PUyZ69C34T1PUyZ69C34T1PU....

Unfortunately, alphadecimal is no better of a sieve for prime numbers than senary or dozenal (unless one does more painstaking digit summation to check for multiples of 5 or 7). That little j, twenty-three, is a typical pain in the butt - for any base not adjacent to it, it repeats at least eleven digits. Wait, 23 is below 24, so one twenty-third would be 0.111111... in base twenty-four! Never mind. The primes 7, 11, 13, and 17 are all cyclic in that base, repeating 6, 10, 12, and 16 digits, respectively. Because thirty-six is an even square, not a single prime can be cyclic in alphadecimal. It's the only viable base with this property that terminates the common fractions. For only the use of 2s, 3s, and 4s (without other prime factors), alphadecimal is the best.

(custom random looking symbol set)

1/2 = 0.y

1/3 = 0.U

1/4 = 0.9

1/5 = 0.7777777777777...

1/6 = 0.6

1/7 = 0.5555555555555...

1/8 = 0.4y

1/9 = 0.4 // Only alphadecimal makes 1/9 have just one significant digit if other multiples of 9 don't count because they don't have 4.

1/t = 0.3aaaaaaaaaaaaaa...

1/r = 0.39m

1/U = 0.3

1/H = 0.2eEn8r2eEn8r2eEn8r2eEn8r2....

1/C = 0.2PPPPPPPPPPPPP...

1/A = 0.2CCCCCCCCCCCCC....

1/J = 0.29

1/N = 0.248JnFeG248JnFeG248JnFeG24....

1/y = 0.2

1/G = 0.1R7PdrH9N1R7PdrH9N1R7PdrH9N1....

1/P = 0.1ZZZZZZZZZZZZZZZZ....

1/a = 0.1TTTTTTTTTTTTTTTT.....

1/] = 0.1]R?6G]R?6G]R?6G]R?6G]R?6G]R?.....

1/j = 0.1PUyZ69C34T1PUyZ69C34T1PUyZ69C34T1PU....

Unfortunately, alphadecimal is no better of a sieve for prime numbers than senary or dozenal (unless one does more painstaking digit summation to check for multiples of 5 or 7). That little j, twenty-three, is a typical pain in the butt - for any base not adjacent to it, it repeats at least eleven digits. Wait, 23 is below 24, so one twenty-third would be 0.111111... in base twenty-four! Never mind. The primes 7, 11, 13, and 17 are all cyclic in that base, repeating 6, 10, 12, and 16 digits, respectively. Because thirty-six is an even square, not a single prime can be cyclic in alphadecimal. It's the only viable base with this property that terminates the common fractions. For only the use of 2s, 3s, and 4s (without other prime factors), alphadecimal is the best.