A Number Theory Observation

Six sided dice can result in random numbers, when the face value is converted to Base 10, DESPITE displaying no "0."

I suspect that Number Theory, as a class, would have taught me more than some finite number of Arithmetic tricks. It is my understanding that Number Theory investigates how these apparent coincidences are true using Algebra. It has been brought to my attention that, IF Erasthmus's Sieve is correct, THEN the modified Sieve in the previous post is also correct, as follows:

Erasthmus used a positive integer number line, and stepped by the relevant n. In the sieve I uncovered (seeking a way to improve compactness in computer RAM) I start with odd integers, as a number line, and as a practical matter strike by 2*n. Explicitly, if we start with 3 and strike every third value, for example, we step over an even number in between each odd number skipped, resulting (algebraically) in striking by 2*3, or 6. Under this constraint, 3 being odd, we strike all ODD multiples of three, and the omissions from our number line omit (instead of striking) all EVEN multiples of 3. The same holds true for 5, 7, and all subsequent unstruck numbers.

A better known Number Theory theorem is that the total of a column of numbers with exactly ONE transposition error, when operated on correctly, results in a difference of EXACTLY nine from the same column of numbers without the error.

To these, I would like to add the observation that six-sided dice result in an unbroken series of integers when their face value is converted to Base 10.

I was entertaining myself, converting 5 digit dice casts to Base 10, when it occurred to me that, in Base 6, there are NO SIXES. Upon consideration, I finally observed that there were also no "0s." This led me to modify my entertainment by decrementing each face value by 1, resulting in a true Base 6 numeral, and converting THESE to Base 10. Now that "11111," converted to 0, and "66666," converted to "55555," or 7775 Base 10, I was able to document a range for the random numbers so derived, as "0 to 7775."

This inevitably led me to investigate if the previous experiment resulted in a pattern. While Statistics was open to me as an avenue, I chose to write a small Python3 script, to roll every possible roll from 11111 to 66666, convert the resulting list to decimal, and tabulate it as an output file. To my surprise (and making me grateful I was not beholden to Hypothesis Testing,) I found that the list from 1555 to 9330 is contiguous and unbroken. This is NOT intuitive, since 11116 increments to 11121, NOT 11120. 

I write this post as a curiosity and entertainment, freely admitting that the most constructive thing I can observe about it, is that 6 + 1 results in 7, decimal, which is coincidentally also "10" Base 6. 

I wish I had been able to capitalize on a course in Number Theory.

******* Amended Mar 11 ******* (Day of publishing.) *********

My final paragraph contains an error, and the correction addresses the question. 

7 is 11 Base 6... 6 is 10 Base 6

...so IN Base 6, the count goes 1, 2, 3, 4, 5, 10, 1, carry 1. Meanwhile, the 6s column continues to hold a single 6, or 1 Base 6, and the 1s column runs 1,2,3,4,5, ??? 10 in the 1's column despite being a full six, then the 6s column updates to 2 6s, but the 1s column shows 1 again, having accounted for 10, as "16" in a kind of hexadecimal way, for Base 6.

Please feel free to chastise me Algebraically.

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