4 ms·

Haven't automated this yet but I've noticed, playing around with primes, that e.g. 1/17 is a repeating decimal (that means base 10) of length n-1 or 16. Others

by m3047 3y ago

Haven't automated this yet but I've noticed, playing around with primes, that e.g. 1/17 is a repeating decimal (that means base 10) of length n-1 or 16. Others such as 7, 11, 13, 31 are shorter and I have a working theory (not a proof) for this, that's how I backed into the first observation. Haven't automated it (yet), I've been toying with it one step more modern than pencil and paper. Not really related to the article except that it is apparent to me at this point that generating the sequences can be automated. It started as a hobby to see if e.g. telephone numbers could be represented as fractions yielding repeating decimals.

o11c 3y ago
You might want to look into "repunits" (though we really do want the 9s, actually) and the Cunningham Project. Mersenne primes, the largest known, are the base-2 case of these (in which case there is no "9"s distinction).
pretzellogician 3y ago
Yes! This comes from the multiplicative order of 10 modulo the number: https://mathworld.wolfram.com/DecimalPeriod.html https://mathworld.wolfram.com/DecimalPeriod.html IIRC, Gauss was the first to formalize this.
- m3047 3y ago
  Thank you! I have a weird story about Gauss. I was working on my motorcycle late at night with too little sleep and Gauss appeared in my head and was asking questions. Also gave me an answer to solve a "field repair" fab problem (damn, it worked!) and told me it was related to chain mail somehow, but I didn't understand.
Someone 3y ago
> Others such as 7, 11, 13, 31 are shorter ⅐ has maximum possible length of n-1 (142857) > to see if e.g. telephone numbers could be represented as fractions yielding repeating decimals. Every number can. For example, phone number 15224787 would yield n = 0.1522478715224787[15224787] ⇒ n × 10⁸ = 15224787.1522478715224787[15224787] = 15224787 + n ⇒ n × (10⁸ - 1) = 15224787 ⇒ n = 15224787 / 99999999 = 1691643 / 11111111 (I didn’t check whether further simplifications are possible)
- m3047 3y ago
  Yes: "the Nines". ;-) You didn't choose something which is prime. Primes end up being the number over the requisite number of Nines (and can't be simplified). The point of my game was to find simplifications. That's how I ended up asking myself "how can I predict how long a repeating decimal is?" and backed into the observation about 1/17 and its ilk. 3 is trivial because 9 is 3^2. 7 is a trick question because it happens to coincidentally be a factor of 999999. At my current rate of progress with this hobby, I might have something interesting to report in another decade or so.

by m3047 3y ago

o11c 3y ago

You might want to look into "repunits" (though we really do want the 9s, actually) and the Cunningham Project. Mersenne primes, the largest known, are the base-2 case of these (in which case there is no "9"s distinction).

pretzellogician 3y ago

Yes! This comes from the multiplicative order of 10 modulo the number: https://mathworld.wolfram.com/DecimalPeriod.html https://mathworld.wolfram.com/DecimalPeriod.html IIRC, Gauss was the first to formalize this.

m3047 3y ago
Thank you! I have a weird story about Gauss. I was working on my motorcycle late at night with too little sleep and Gauss appeared in my head and was asking questions. Also gave me an answer to solve a "field repair" fab problem (damn, it worked!) and told me it was related to chain mail somehow, but I didn't understand.

Someone 3y ago

> Others such as 7, 11, 13, 31 are shorter ⅐ has maximum possible length of n-1 (142857) > to see if e.g. telephone numbers could be represented as fractions yielding repeating decimals. Every number can. For example, phone number 15224787 would yield n = 0.1522478715224787[15224787] ⇒ n × 10⁸ = 15224787.1522478715224787[15224787] = 15224787 + n ⇒ n × (10⁸ - 1) = 15224787 ⇒ n = 15224787 / 99999999 = 1691643 / 11111111 (I didn’t check whether further simplifications are possible)

m3047 3y ago
Yes: "the Nines". ;-) You didn't choose something which is prime. Primes end up being the number over the requisite number of Nines (and can't be simplified). The point of my game was to find simplifications. That's how I ended up asking myself "how can I predict how long a repeating decimal is?" and backed into the observation about 1/17 and its ilk. 3 is trivial because 9 is 3^2. 7 is a trick question because it happens to coincidentally be a factor of 999999. At my current rate of progress with this hobby, I might have something interesting to report in another decade or so.