Numbers in Other Bases are Belong to Us

[sweety439]

Originally Posted by Stargate38

As shown on the following website, A-Z represent 10-35, a-z represent 36-61, and 62-93 are represented by punctuation (Case Sensitive):
http://home.ccil.org/~remlaps/DispConWeb/index.html

You can use ASCII character n+32 to represent n for base 95, see http://www.icerealm.org/FTR/?s=docs&p=base95

[tuckerkao]

Originally Posted by science_man_88

I see no counter example

proof:

the addition in base 3 goes

20*1-1 = 12
20*2-2 = 12+20 =32

since 20 affects the second place up and beyond the last digit wouldn't change unless maybe negative.

[Trinary] 20 * 2 = 110, 110 - 2 = 101

You should never have the "3" in base 3 because there are only [0, 1, 2].

I used my color balls and I figured the answer out without using the base calculator.

[R2357]

Originally Posted by Christenson

Theorem: Numbers can be represented in any base...when you get beyond 16, you need to explicitly state your rules of representation.
Theorem: A number ending in zero in any base representation is composite.

Except for prime based numbers!

[R2357]

I think base 30 might be interesting, primes (other than 2, 3 and 5) would only end by :"1, 7, B, D, H, J, N, T"

[S485122]

.

[tuckerkao]

Originally Posted by LaurV

Still did not get the part about prime numbers in base 3 ending in 2. So what? Are they even?

In base 3,

12 is a prime
22 is a composite
102 is a prime
112 is a composite
122 is a prime

Add all the digits together as in the decimal base, it should reveal whether the numbers are even or odd in base 3.

Originally Posted by tuckerkao

Maybe this is the wrong site for me to be on, I already found another site that specifically calculate math in other bases.

http://www.dozenalsociety.org.uk/doz...siteslist.html

I'd like to know whether it's possible to have my comments and stuffs removed from this site and delete my account as I finally realize most members don't engage and/or fully understand the calculations in other math bases here.

OF(Octal Fahrenheit), OC(Octal Celsius) -> both different from the decimal scales

32°OF = 0°OC -> Freezing
212°OF = 100°OC -> Boiling

Since 212° is the °F boiling point in decimal and 32° is the °F freezing point too. When switch the base, the numbers are re-located to the 32 and 212 of that specific base, thus become a different scale.


When breaking down, the octal scales have the different intervals compare to the decimal scales.

16°OF(Octal Fahrenheit) for every 10°OC(Octal Celsius)

32°OF = 0°OC
50°OF = 10°OC
66°OF = 20°OC
104°OF = 30°OC
122°OF = 40°OC
140°OF = 50°OC
156°OF = 60°OC
174°OF = 70°OC
212°OF = 100°OC

[sweety439]

Originally Posted by tuckerkao

In base 3,

12 is a prime
22 is a composite
102 is a prime
112 is a composite
122 is a prime

There is a problem, the minimal set of the strings for primes with at least two digits in base b:

https://primes.utm.edu/glossary/page...t=MinimalPrime

In 1996, Jeffrey Shallit [Shallit96] suggested that we view prime numbers as strings of digits. He then used concepts from formal language theory to define an interesting set of primes called the minimal primes:

A string a is a subsequence of another string b, if a can be obtained from b by deleting zero or more of the characters in b. For example, 514 is a substring of 251664. The empty string is a subsequence of every string.
Two strings a and b are comparable if either a is a substring of b, or b is a substring of a.
A surprising result from formal language theory is that every set of pairwise incomparable strings is finite [Lothaire83]. This means that from any set of strings we can find its minimal elements.
A string a in a set of strings S is minimal if whenever b (an element of S) is a substring of a, we have b = a.
This set must be finite!

For example, if our set is the set of prime numbers (written in radix 10), then we get the set {2, 3, 5, 7, 11, 19, 41, 61, 89, 409, 449, 499, 881, 991, 6469, 6949, 9001, 9049, 9649, 9949, 60649, 666649, 946669, 60000049, 66000049, 66600049}, and if our set is the set of composite numbers (written in radix 10), then we get the set {4, 6, 8, 9, 10, 12, 15, 20, 21, 22, 25, 27, 30, 32, 33, 35, 50, 51, 52, 55, 57, 70, 72, 75, 77, 111, 117, 171, 371, 711, 713, 731}

Besides, if our set is the set of prime numbers written in radix b, then we get these sets:

b, we get the set
2: {10, 11}
3: {2, 10, 111}
4: {2, 3, 11}
5: {2, 3, 10, 111, 401, 414, 14444, 44441}
6: {2, 3, 5, 11, 4401, 4441, 40041}

these are already researched in https://cs.uwaterloo.ca/~cbright/reports/mepn.pdf.

Now, let's consider: if our set is the set of prime numbers >= b written in radix b (i.e. the prime numbers with at least two digits in radix b), then we get the sets:

b, we get the set
2: {10, 11}
3: {10, 12, 21, 111}
4: {11, 13, 23, 31, 221}
5: {10, 12, 21, 23, 32, 34, 43, 111, 131, 133, 313, 401, 414, 14444, 30301, 33001, 33331, 44441, 300031}
6: {11, 15, 21, 25, 31, 35, 45, 51, 4401, 4441, 40041}
7: {10, 14, 16, 23, 25, 32, 41, 43, 52, 56, 61, 65, 113, 115, 131, 133, 155, 212, 221, 304, 313, 335, 344, 346, 364, 445, 515, 533, 535, 544, 551, 553, 1112, 1211, 1222, 2111, 3031, 3055, 3334, 3503, 3505, 3545, 4504, 4555, 5011, 5455, 5545, 5554, 6034, 6634, 11111, 30011, 31111, 33001, 33311, 35555, 40054, 300053, 33333301}
8: {13, 15, 21, 23, 27, 35, 37, 45, 51, 53, 57, 65, 73, 75, 107, 111, 117, 141, 147, 161, 177, 225, 255, 301, 343, 361, 401, 407, 417, 431, 433, 463, 467, 471, 631, 643, 661, 667, 701, 711, 717, 747, 767, 3331, 3411, 4043, 4443, 4611, 5205, 6007, 6101, 6441, 6477, 6707, 6777, 7461, 7641, 47777, 60171, 60411, 60741, 444641, 500025, 505525, 3344441, 4444477, 5500525, 5550525, 55555025, 444444441, 744444441}

However, I do not think that my base 7 and 8 sets are complete (I use PARI program to find these primes (all written in base b), but I only searched the primes with <= 8 digits, so there may be missing primes), I proved that my base 2, 3, 4, 5 and 6 sets are complete.

Can someone complete my base 7 and 8 set? Also find the sets of bases 9 to 36.

[tuckerkao]

Originally Posted by sweety439

However, I do not think that my base 7 and 8 sets are complete (I use PARI program to find these primes (all written in base b), but I only searched the primes with <= 8 digits, so there may be missing primes), I proved that my base 2, 3, 4, 5 and 6 sets are complete.

Can someone complete my base 7 and 8 set? Also find the sets of bases 9 to 36.

In base 7, the "14" is a prime as it's called 1 Hepta 4.
In base 8, the "13" is a prime as it's called 1 Octa 3.

It's quiet easy when I use my color balls to figure out the primes in the alternative bases up to several hundreds. I highlight the primes with the black stones in the decimal base first, then change the width of the arrays to 7 or 8, then I get the primes for base 7 or base 8.

There are also different scales derived from the alternative bases from length to weight to volume and so on -
http://www.dozenal.org/drupal/sites_...al_arith_0.pdf

Maybe there needs to be a subforum for the numbers and calculations in the different bases as most members don't understand what we are talking about.

[LaurV]

Originally Posted by tuckerkao

as most members don't understand what we are talking about.

As somebody pointed to me on PM, this is insulting for the most members of this forum, who are better qualified than you, to talk about the subject. Of course we understand numeration bases. But what you are doing with them, is called "mess". I think a week holiday may make you re-evaluate your priorities...

[R2357]

I noticed that if we classify the bases according to how efficient they are to identify at once (by the last digit) composites, base 10 is "quite" good, but not the best!
Not even up to 10, as base 6 is better.

I saw above that some wish a shifting to base 12, which is better than 10, but it's on the same rank as 6.

If we want to change our number system, as we already have existing bases, why not change for an even more efficient, how about 30, (I don't know if we have enough symbols for base 210).

[axn]

Originally Posted by R2357

I noticed that if we classify the bases according to how efficient they are to identify at once (by the last digit) composites

You're thinking like a human. For a computer, what you just said is a nanosecond of computation (to divide by the base and check the remainder). This brings no efficiency to prime finding.