How to Convert to Base6
The base6 numbering system is also known as heximal or senary, and uses only the six digits 0, 1, 2, 3, 4 and 5 to represent numbers. Heximal has many interesting properties that are analogous to features of our usual base10 numbering system. For example, like 10, the number 6 is also the product of two distinct primes. In base6, any fraction whose denominator is of the form 2^{n}3^{m} can be expressed as a terminating decimal. This means 1/2, 1/3, 1/4, 1/6, 1/8, 1/9, 1/12, 1/16, 1/18, 1/24, 1/27, 1/32, etc., are all terminating decimals. The same is true of base12, since 12 = 2*2*3. The parallel situation in base10 is that all fractions whose denominators are of the form 2^{n}5^{m} have terminating decimal representations since 10 = 2*5.
In senary, all prime numbers end in 1 or 5, and all perfect numbers end in 44. (A perfect number is one whose proper factors add up to the number itself.) Square numbers never end in 2 or 5.
An easy way to generate random base6 numbers with standard cubic dice, where the face numbered 6 represents 0. For example, if you roll three dice in succession and obtain 4, 6, 1, this represents the senary number 401, which is equivalent to the base10 number 145.
To convert a number from base10 to base6, you can use the remainder algorithm. You can also convert the long way by expressing the number as the sum of multiples of powers of 6.
Converting to Base6 with the Remainder Method
To apply the remainder method to a regular base10 number, you begin by dividing it by 6, and noting the quotient and remainder. Next, you divide that quotient by 6 and again note the new quotient and new remainder. Proceed this way until you get a quotient of 0. Now take the list of remainders you generated and concatenate them from last to first. This is the base6 representation of the number.
As an example, let's convert the base10 number 79148 into a senary number. We start with simple division
79148÷6 = 13191 remainder 2
And now we repeat the process, this time dividing 13191 by 6 and noting the new quotients and remainders.
13191÷6 = 2198 remainder 3
2198÷6 = 366 remainder 2
366÷6 = 61 remainder 0
61÷6 = 10 remainder 1
10÷6 = 1 remainder 4
1÷6 = 0 remainder 1
Stringing together the remainders from last to first gives us the number 1410232. This is the base6 representation of the base10 number 79148.
How to Convert to a Sum of Multiples of 6^n
Another way to convert a base10 number to a heximal number is to rewrite the number as the sum of multiples of powers of 6, where the multiples range from 0 to 5. The lowest power of 6 is 6^0 = 1 and the highest power is floor{log_{6}(n)}. For example,
floor{ log_{6}(79148) } = floor{ 6.295 } = 6, and
79148 = 1(6^6) + 4(6^5) + 1(6^4) + 0(6^3) + 2(6^2) + 3(6^1) + 2(6^0)
The coefficients or multiples are precisely the digits of the base6 representation. Restricting the values of the multiples to {0, 1, 2, 3, 4, 5} ensures that the representation is unique. Thus, converting from one integer base to another is a onetoone operation.
Fractions to Decimals in Base6
Since 6 has a different prime factorization than 10, the representation of rational numbers by decimals (positional representation) is different. In base10, fractions over 2, 4, 5, 8, 10, 16, 20, 25, etc., all terminate, but any fraction over a multiple of 3 turns into a repeating decimal. In base6, fractions over multiples of 5 turn into repeating decimals, whereas denominators that are solely divisible by powers of 2 and/or 3 all terminate. Here is a sample of some rational numbers expressed as decimals in base6.
Base10 Fraction
 Base10 Decimal, with Repeating Part in {Braces}
 Base6 Decimal, with Repeating Part in {Braces}


1/2
 0.5
 0.3

1/3
 0.{3}
 0.2

2/3
 0.{6}
 0.4

1/4
 0.25
 0.13

3/4
 0.75
 0.43

1/5
 0.2
 0.{1}

2/5
 0.4
 0.{2}

3/5
 0.6
 0.{3}

4/5
 0.8
 0.{4}

1/6
 0.1{6}
 0.1

5/6
 0.8{3}
 0.5

1/7
 0.{142857}
 0.{05}

1/8
 0.125
 0.043

1/9
 0.{1}
 0.04

1/10
 0.1
 0.0{3}

1/11
 0.{09}
 0.{0313452421}

1/12
 0.08{3}
 0.03

1/13
 0.{076923}
 0.{024340531215}

1/14
 0.0{714285}
 0.0{23}

1/15
 0.0{6}
 0.0{2}

1/16
 0.0625
 0.0213

Number Theory in Base6
Why do all prime numbers (except 2 and 3) have to end in either 1 or 5 in base6? If a number in base6 ends in 0, it is divisible by 6. If it ends in 2 or 4, it is divisible by 2. And if it ends in 3 it is divisible by 6. That leaves only 1 and 5 as the possible last digits of a prime number.
Why do all perfect numbers (except 6) end in "44" in base6? If you recall, a perfect number is one whose proper factors add up to the number itself. All even perfect numbers P are of the form
P = [2^n]*[2^(n+1)  1]
where 2^(n+1)  1 is a prime number. It is not known if there exist any odd perfect numbers, as none have ever been found, and no proof of their nonexistence has ever been devised. The base10 representations of the first few perfect numbers are 6, 28, 496, 8128, and 33550336. All even perfect numbers except for 6 are of the form 36k + 28, meaning they are 28 plus a multiple of 36.
The base6 representations of these numbers are 6, 44, 2144, 101344, and 3155033344. The fact that they all end in the digits 44 is a consequence of the fact that they are of the form 36k + 28, because if you reformulate that linear expression in base6, it's 100k + 44.