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-   -   Numbers in Other Bases are Belong to Us (https://www.mersenneforum.org/showthread.php?t=16061)

 Stargate38 2011-09-17 23:23

Numbers in Other Bases are Belong to Us

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):
[URL]http://home.ccil.org/~remlaps/DispConWeb/index.html[/URL]

Here's an example:

12345ABCabc`~^&*[SUB]94[/SUB]=403838401633020116561588357495[SUB]10[/SUB]

I used the GIMPS Homepage and found it to be composite (divisible by 2)

www.mersenne.org[SUB]94[/SUB]=23173464146192129623556375950544[SUB]10[/SUB]

Examples of Base-36 primes:

HEART
101 (Largest known Generalized fermat prime base-36)
=1297[SUB]10[/SUB]

I found some rules for the last "digit" (base-94):
[LIST][*]A prime number cannot end in any of the following:[/LIST]0 2 4 6 8 A C E G I K M O Q S U W Y a c e g i k l m o q s u w y ` ! # % & ( - = [ { \ ; , < / '

Divisibility rules are also different:[LIST][*]47: If the last digit is 0 or l, then 47 divides the original number:[/LIST]456r5l is divisible by 47 because it ends in l.[LIST][*]3 or 31: add the digits. If divisible by 3 and/or 31, so is the original number.[*]78914411: add the digits in groups of 5. If the result is divisible by 78914411, so is the original number:[/LIST]V?~17OPfwq3an[sub]94[/sub] is divisible by 78914411 because V?~ + 17OPf + wq3an is divisible by 78914411[LIST][*]5: add every other digit and subtract the rest. If the result is 0 or a multiple of 5, so is the original. For example:[/LIST]1tRaaRt1[sub]94[/sub] is divisible by 5 since 1+R+a+t-t-a-R-1=0

Divisibility Rules for base-19:[LIST][*]911: add digits in groups of 5. If the result is divisible by 911, so is the original number. Also works for 151[*]3: same rule as base-10 (Add digits)[*]Because 19 is odd, primes can end with any nonzero digit, even 2. For example:[/LIST]14[sub]19[/sub]=23[sub]10[/sub]

 Christenson 2011-09-18 02:08

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.

 retina 2011-09-18 02:38

[QUOTE=Christenson;271928]Theorem: A number ending in zero in any base representation is composite.[/QUOTE]Ternary: 10 is not composite. Also 10[sub]5[/sub] and 10[sub]7[/sub] and more generally 10[sub]p[/sub]

 CRGreathouse 2011-09-18 04:24

[QUOTE=Christenson;271928]Theorem: A number ending in zero in any base representation is composite.[/QUOTE]

In any even base greater than 2, that is.

 LaurV 2011-09-18 09:21

Now that is not so clear, you have a number that you can reprezent as ending in 0, no matter what base you choose? :P (this was a joke!)

Or you have a number that ends in 0 when you represent it in "some" base b? In this case, the number is composite if the base is composite, in any case.... not only even bases. Also, if the base is prime, 10 in base b is always prime. Any other number ending with 0 in base b can't be prime (as is divisible by the base), except in case is 10 in base b, and that b is prime itself.

by the way, how we can characterize even numbers in a odd base? some fast divisibility criteria? for example in any even base, they end with even "digits" in that base (Mr Silverman will be on my head now for illegal use of the word "digits", sorry! my English is far away to be such good). For example, say, in base 5, any number is even if and only if the sum of the last two digits is even, and I believe that is the fastest way (hope is also true, it just popped up into my head now, did not check it). Any general rule? This just informal, or more than like a curiosity....

 Christenson 2011-09-18 13:31

LaurV:
Stand Mr Silverman on *his* head...in that conversation, as in this one, it is very clear from context that we mean a base-something digit here when we say digit. :smile: We're in the lounge, so we can prattle about trivialities, as Mr Silverman might say, as long as we admit that this is (extremely) elementary stuff.

Now,to solve your question about evenness, just write your number in arbitrary base out, and keep track of what happens modulo 2.

 science_man_88 2011-09-18 15:32

[QUOTE=Christenson;271953]LaurV:
Stand Mr Silverman on *his* head...in that conversation, as in this one, it is very clear from context that we mean a base-something digit here when we say digit. :smile: We're in the lounge, so we can prattle about trivialities, as Mr Silverman might say, as long as we admit that this is (extremely) elementary stuff.

Now,to solve your question about evenness, just write your number in arbitrary base out, and keep track of what happens modulo 2.[/QUOTE]

turned base 10 numbers into base 3 and was going to make a hypothesis about it but then realized it didn't hold.

even even
even odd
odd even
odd even
even odd
even even
odd odd

seems to be the pattern for most of the last digits in base 3 but I see it can depend on if the rest of it is an even number. I just looked at base five but I'm not mass generalizing.

 Stargate38 2011-09-18 17:59

Any prime of the form 6x-1 is even in base-3. "GOD" is part of a prime triplet in base-36 (21611, [B][COLOR=deepskyblue]21613[/COLOR][/B], 21617 in base-10). Here's a list of base-94 primes with their base-10 values shown (Case Sensitive):
[LIST][*]Hello_Dad0:[sub]94[/sub]=938855066341243463873[*]Velociraptor0@[sub]94[/sub]=1406109285239757038128316053[*]Velociraptor0"[sub]94[/sub]=1406109285239757038128316081[*]9@[sub]94[/sub]=911[*]Af=n[sub]94[/sub]=8675309[*]abcdef7[sub]94[sub]=25109856202429[/LIST]Note that I chose 2 popular primes (911 and 8675309). 911 divides p+1 for the second number in the list.
Here are some large base-36 primes I found:

[CODE]
VELOCIRAPTOR00000000000000000000000000000000000000000000000000000000000000000000000000001
MYTHBUSTERSDDDDDDDDDDDDDDDDDDDDDDDDD
MYTHBUSTERSDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDB
[/CODE]

Note that they all end in a repeating string (0000...000 or DDD...DDDDD) with a 1, D, or B on the end. They all have the form (k*36[sup]n[/sup]-a)/35 No base-36 prime ends with 2, 3, 4, 6, 8, 9, A, C, E, F, G, I, K, L, M, O, Q, R, S, U, W, X, or Y, because it would be divisible by 2 and/or 3.

 LaurV 2011-09-18 19:16

[QUOTE=Stargate38;271971]Any prime of the form 6x-1 is even in base-3.[/QUOTE]

This can't be, or I was not clear about "even" numbers. The fact a number finishes in 0,2,4 etc means nothing. These are just symbols. An even number is a number that can evenly be split in 2 parts. For example, 8 is an even number. If I have a heap of 8 coins, I can split thm in two heaps of 4 coins.

In any even base, take 10 for example, is easy to characterize any even number, saying that "a number is even if and only if the last digit of the number is 0, 2, 4, etc", i.e. last digit is an even number. I am used to compute in even bases only, like the most of us (binary, octal, decimal, hex) and I can do fast computing in my mind, even with numbers with many digits (being programmer for more then two decades) in these bases. But it never occur to me till now the fact that if the base is not even, then some even numbers won't end... right. This first pops to my head when I read the original post of this thread.

So I asked myself, more like a curiosity, how we would see in a blink of an eye that a number is even, if we would use by default an odd base. We are used to count in base 10, because most probably we have 10 fingers, and that was how our ancestors were counting. How about if they were using the moon phases instead, counting for example in base 7, or so? How we would see in a blink of an eye that a number is even? Same story as we can see (in base 10) that a number is divisible by 3, adding the digits till we got one-digit sum and check if it is 3, 6, or 9 (divisible by 3).

I solve this since the last post, but it wasn't as I assumed (about the last two digits). It is a bit more complicated. All of them have to be summed :D

Now saying that "in base 3 any prime of the form.... is even" sounds wrong to my ears. The only even prime is 2, and this does not depends of the base you write the numbers in. All other primes are odd. You can't split them in two equal heaps. They wont be primes in this case.

Number "12" in base 3 is not even. It is odd. But "112" is even (=14 decimal). And so is "221201011" (= 56226 decimal, even).

 Christenson 2011-09-18 19:19

Umm, beyond 16, I can't keep track of which letter corresponds to what value...guess I didn't learn my multiplication tables well enough in school!

I prefer something like: GOD(94) = (16)(24)(13), or spell it all the way out: (16)*94^2 + 24*(94) + 13. Much clearer that way...

 Christenson 2011-09-18 19:31

Laurv, let's correct a bit of english here....I think it will help the confusion.

[QUOTE=LaurV;271979]This can't be, or I was not clear about "even" numbers. The fact a number finishes in 0,2,4 etc means nothing. These are just symbols. An even number is a number that can evenly [ be split in 2 parts]--> [divided by 2, that is x is even if x divided by 2 leaves no remainder]. For example, 8 is an even number. If I have a heap of 8 coins, I can split[divide] them [evenly] in two heaps of 4 coins.

In any even base, take 10 for example, is easy to characterize any even number, saying that "a number is even if and only if the last digit of the number is 0, 2, 4, etc", i.e. last digit is an even number. I am used to compute in even bases only, like the most of us (binary, octal, decimal, hex) and I can do fast computing in my mind, even with numbers with many digits (being programmer for more then two decades) in these bases. But it never occur to me till now the fact that if the base is not even, then some even numbers won't end... right. This first pops to my head when I read the original post of this thread.

So I asked myself, more like a curiosity, how we would see in a blink of an eye that a number is even, if we would use by default an odd base. We are used to count in base 10, because most probably we have 10 fingers, and that was how our ancestors were counting. How about if they were using the moon phases instead, counting for example in base 7, or so? How we would see in a blink of an eye that a number is even? Same story as we can see (in base 10) that a number is divisible by 3, adding the digits till we got one-digit sum and check if it is 3, 6, or 9 (divisible by 3).

I solve this since the last post, and the answer is indeed as I assumed, if the sum of the last two digits is even, then the number would be even. It is easy to prove. End of story.

Now saying that "in base 3 any prime of the form.... is even[-->ends in an even digit]" sounds wrong to my ears. The only even prime is 2, and this does not depends of the base you write the numbers in. All other primes are odd. You can't split them in two equal heaps. They wont be primes in this case.

Number "12" in base 3 is not even. It is odd.[/QUOTE]

Laurv, your universe isn't quite big enough. "If the sum of the last two digits is even, then the number would be even"...really? let's work in base 5, note that 11 (=1*5+1 = (6)) is even, but 111 (=1*125+1*5+1 = 131) is odd. But the sum of these last two digits is *exactly* the same.

As for the statement "in base 3, any prime of the form 6x-1 ends in an even digit", I have a small prize for the first NUMBER (prime or not) of the form 6x-1 when represented in base 3 that ends in a digit other than 2, assuming the usual conventions of representation for integers.

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