Numbers in Other Bases are Belong to Us - Page 2

LaurV · 2011-09-18, 19:50

You were too fast to reply :D, that was a mispost of mine, which I edited already, immediately (2:47AM here, almost sleeping). I was affraid of your fast reply :D I saw you also edited my post, a thing I don't like. Please don't change the meaning of the things I said.

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

science_man_88 · 2011-09-18, 20:27

Quote:

Originally Posted by Christenson

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.

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.

science_man_88 · 2011-09-18, 23:16

Quote:

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.

sorry not -2 it's a typo

Christenson · 2011-09-18, 23:23

Quote:

Originally Posted by LaurV

You were too fast to reply :D, that was a mispost of mine, which I edited already, immediately (2:47AM here, almost sleeping). I was affraid of your fast reply :D I saw you also edited my post, a thing I don't like. Please don't change the meaning of the things I said.

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

1) I was hoping *not* to change your meaning...only to make it clear, and help you a bit in the future. "Splitting into 2 parts" is not quite the same as DIVIDING evenly into two parts -- since I can split 3 into 2+1, but I can't divide it evenly by two.

2) To keep track of whether a number in an odd base is even or not, you don't have to do as much as you said....to see this, just take my favorite expansion and ask what happens modulo 2. Things will simplify.

3) On prime numbers in base 3 ending in 2: At some time BEFORE midnight, work out what 6x-1 looks like in base 3...you'll get the idea..

SM88: You haven't *quite* got a proof yet, but you are close....try a re-write -- start by stating what you are trying to prove.

science_man_88 · 2011-09-18, 23:46

Quote:

Originally Posted by Christenson

SM88: You haven't *quite* got a proof yet, but you are close....try a re-write -- start by stating what you are trying to prove.

first what numbers are you looking at ? because :

12₃-20₃ = (20-1)-20 = -1 = an answer without a ending 2 as should be all numbers in 6x-1 below it.

LaurV · 2011-09-19, 02:12

Quote:

Originally Posted by Christenson

1) I was hoping *not* to change your meaning...only to make it clear, and help you a bit in the future. "Splitting into 2 parts" is not quite the same as DIVIDING evenly into two parts -- since I can split 3 into 2+1, but I can't divide it evenly by two.

2) To keep track of whether a number in an odd base is even or not, you don't have to do as much as you said....to see this, just take my favorite expansion and ask what happens modulo 2. Things will simplify.

3) On prime numbers in base 3 ending in 2: At some time BEFORE midnight, work out what 6x-1 looks like in base 3...you'll get the idea..

SM88: You haven't *quite* got a proof yet, but you are close....try a re-write -- start by stating what you are trying to prove.

1. Reading all that stuff that I posted, seen through your eyes now, I remember my big boss long ago, entering the purchasing department with an invoice in his hands and some red on his face: "this is bloody expensive, we can not buy it! who is the idiot that signed the purchasing order?", the girl give him the PO. It was signed by him, few weeks before. So I am tempted to say "who is the the crank who wrote that?".

(I am talking about MY post, not your comments that I quoted above).

At a second thought, reading my post again, in spite of the fact that I was writing it a 3 o'clock in the night, i clearly stated "evenly split". I may not know the English grammar quite well, but I have the sufficient vocabulary (maybe more then an average native speaker), as I was (a good) scrabble player in my youths and had to learn words by heart, even I didn't know their meaning (learned "heavy" words from a list, that is, words with high scrabble score).

And NO, you can't "evenly split" 3 into 2+1.

2. You are most probably right at this part, and I am tempted to believe you. I was a crack at this part. Mea culpa. Posting that was silly, and even sillier was me editing it later, without effectively computing it. I am fully aware of the fact that saying this now improves my crank score

Still did not figured it out (just woke up), but I promise you that I will take a paper and a pencil and play with it at my coffee.

3. Here we disagree again. There is NO EVEN PRIME NUMBER except 2. No matter what is the base you write the number in, no matter what is the SYMBOL (2, 7, a, w, $\alpha,\ \phi$ etc) that ends the number. The OP stated that "every prime number of the form (whatever) is even". False. And false it will be.

I never contested the fact that numbers of the form kb-1 end in (b-1 as digit) when written in base b.

Like you said... "I give you a bonus if you find an EVEN prime other then 2, no matter what base you write the number in, odd base, even base, prime base, etc..."

Christenson · 2011-09-19, 02:55

Let me try to collect that bonus: My number is 2^43,112,609-1. It's even..it MUST be even.....

...I swear it!

I knew you knew there was a problem with 6x-1...SM88 is being his incoherent self in trying to prove the property you give..the property you gave is exactly what I was driving at.

And I must have lost an even part of your post somewhere...it's now getting late on me!

LaurV · 2011-09-19, 03:16

Now, a number x (see x as a heap of candies, sticks, coins, not as its numerical representation) can be represented in positional notation in any base b by an expression like:

$x=a_nb^n+a_{n-1}b^{n-1}+...+a_2b^2+a_1b+a_0$

To be an even number, this has to be congruent to 0, modulo 2.

$x=a_nb^n+a_{n-1}b^{n-1}+...+a_2b^2+a_1b+a_0\equiv 0\ (mod 2)$

Call this class "the parity" of x. So, the parity is 0, the number is even. The parity is 1, number is odd.

The "parity" of this "sum" is given by the parity of its terms.

When the base b is even, all the terms are even (as being a product of an even number and something else), except the last one, $a_0$ . So, the "parity" of the number is given by this "last digit". Pardon me for "digit", call it figure (I don't like that, I always associate it in my mind with expressions like "a well known figure in society", or "a figure of speech", "the dove is a figure of peace", "let's figure it out", etc, is more confusing). Call it cipher, as in "any of the (Arabic or else) symbols used to represent numbers in positional notation). In our language we have a special word for it, that translates more like "symbol".

Now, when the base is odd, the "parity" of each term is given in fact by its coefficient. If $a_i$ is odd, the term is odd. If $a_i$ is even, the term is even.

Now, going back to adding digits, as in the "criterion of divisibility by 3 in base 10", where we have to add all digits, here it does not make too much sense to add the digits that are even, does it?

They won't change the parity.

Is this what you wanted to say? If so, then I learning something new today, and all our arguing was not in vain. I am quite happy about it. Thanks. This was a good coffee this morning....

Edit: Big revelation: you don't even need to "add" them. Counting is enough. Whaaaa, what a revelation!

For example in base 3 you only need to count how many of 1 do appear in the number. In base 5 you have to count 1's and 3's altogether...

This coffee was for sure a good one.
Leaving for my daily job...

Stargate38 · 2011-09-20, 21:47

Actually, 2^43112609-1 ends in 1 when expressed in base-3. All Mersenne primes do (except 3, which is 10₃). In base b, where b is prime, primes can end in anything except 0 (with the exception of b). 911 is 2m mod 3ⁿ for n<7, where m is a natural number (even or odd):
911 = 1020202₃
911 = 1222₉
911 mod 27 = K₂₇
911 mod 81 = K₈₁
911 mod 243 = 182
911 mod 729 = 182
911 mod 2187 = 911

Notice the pattern. 911 leaves an even remainder for all powers of 3 less than or equal to 729, but not any higher powers of 3.

Quote:

There is NO EVEN PRIME NUMBER except 2

By even, I didn't mean even as in divisible by 2; I meant that the last figure (digit) was even (2, 4, 6, 8, A, C, ...) in that base. However, this only happens in odd bases (bases 3, 5, 7, 9, etc.). All primes are 1 mod 2 (except 2) and therefore all primes (except 2) are odd. It just depends on what base you learned to use. If human math were to start with base-7 instead of base-10, primes would end in 1, 2, 3, 4, 5, and 6. Only multiples of 7 would end in 0. Look at the powers of 10 in base-7:

13
202
2626
41104
564355
11333311
150666343
2322662122
33531600616
50254411644
10140043655335

Note the pattern in the last figure: odd, even, even, even, odd, odd, odd, even, even, etc. repeating with a period of 6. This is because of 1/7's period (0.1428571428571428571428571...). Any prime base p will show this kind of pattern, equal in length to the period of 1/p. This means that all powers of 10 end in 1 when expressed in base 3 because 10ⁿ-1 is always divisible by 9 (3²). Also note that a million is a palindrome in base-7 (11333311) and that 1/911 has a period of 14 when expressed in base 7:

0.000243066642360002430666423600024306664236...

This means that 7⁷+1 is divisible by 911.
911 is also a Sophie-Germain prime, meaning that 2p+1 is prime, in this case 1823.

911 is a palindrome in base-5 (12121).

Did you know that the difference between a number and it's reverse is always divisible by 9
911-119=792=2*2*2*3*3*11
907-709=198=2*3*3*11
9035768-8675309=360459=9*40051=3²*11²*331

Note that I like 911. It's the first 3 digit prime I learned about (U.S. Emergency Phone Number). Before I learned about primes, I actully tried to factor 8675309 (I was only 4 years old at the time) and found it too difficult. I love Tommy Tutone. Did you know that 911#-9110119 is prime? Here it is in base-36 with letter values:

Code:

244 figures (digits)
lb4svav3wehd5ohb1zmy25k24vu41mp0m2jicghe4zsnvbaegtakwg8qh8jsvn2nwmn582atkowqhwzaeo6dgkv41wekvj5kj3gwfwhcgdvgbf7cjvy9875wxe3oob0qnxb01ne79u6csddds841n8c2na15zc4vubbof52nzux5dn2xyyef82ygn6nljaccj3t3260yokaik2w6u6hkr0lt4zr4f8b5udaa02y9q2epy687egfz
a=10, b=11, c=12, d=13, e=14, f=15, g=16, h=17, i=18, j=19, k=20, l=21, m=22, n=23, o=24, p=25, q=26, r=27, s=28, t=29, u=30, v=31, w=32, x=33, y=34 z=35

And here it is in base-10:

Code:

380 digits
32365034185618020010716991513958659767082943236694935729328906136551863378072625698564549664662002553551382345074856987881103473255815956621387223447804841155658305239172609952384690477550897152816216908432126258545416331558137240581301042339607841611309055744941010472200207903859708352679007908322424571268778054926384062108162417863979176373973320401208882140740392521906172351

When it's finished, I'll upload my table of the first 168 primes in base-117 (with symbol table).

kar_bon · 2011-09-20, 21:53

Quote:

Originally Posted by Stargate38

Did you know that 911#-9110119 is prime? Here it is in base-36 with letter values:

Code:

244 figures (digits)
lb4svav3wehd5ohb1zmy25k24vu41mp0m2jicghe4zsnvbaegtakwg8qh8jsvn2nwmn582atkowqhwzaeo6dgkv41wekvj5kj3gwfwhcgdvgbf7cjvy9875wxe3oob0qnxb01ne79u6csddds841n8c2na15zc4vubbof52nzux5dn2xyyef82ygn6nljaccj3t3260yokaik2w6u6hkr0lt4zr4f8b5udaa02y9q2epy687egfz
a=10, b=11, c=12, d=13, e=14, f=15, g=16, h=17, i=18, j=19, k=20, l=21, m=22, n=23, o=24, p=25, q=26, r=27, s=28, t=29, u=30, v=31, w=32, x=33, y=34 z=35

And here it is in base-10:

Code:

380 digits
32365034185618020010716991513958659767082943236694935729328906136551863378072625698564549664662002553551382345074856987881103473255815956621387223447804841155658305239172609952384690477550897152816216908432126258545416331558137240581301042339607841611309055744941010472200207903859708352679007908322424571268778054926384062108162417863979176373973320401208882140740392521906172351

When it's finished, I'll upload my table of the first 168 primes in base-117 (with symbol table).

See here.

LaurV · 2011-09-21, 03:34

The direction of this topic reminds me old times 20 years ago when it was a fashion (at least in my part of the world) to take people names (influential public persons, movie starts, your colleagues, etc), add the ASCII codes of that names, eventually do some other operation (google for Finagle's constant) and "prove" that the respective man/woman, etc. is the reincarnated devil. I think the joke started with some freaks "proving" that Bill Gates is evil. He used to call himself "Bill Gates the Third" after his father and grandfather being also Bill (no idea if this story is true or fabricated) and if you add his ascii codes, voila! The trick is that you can make (almost) anybody's name add to whatever you like, if you give the right rule. I always liked the number 23, I don't know why, since I was quite young, maybe I liked the sound of it, so you take my (real) name, add 5 (2+3) and I am evil too...

So, take "velociraptor", "mythbusters" or other of the given examples, add whatever Finagle's constant you like at the end of them, and voila! one has his primes. This leads to nothing.

Just my two pence. Better talk about some math, so I (and anyone else reading) could learn something new...

2011-09-18, 19:50	#12
LaurV Romulan Interpreter Jun 2011 Thailand 9153₁₀ Posts	You were too fast to reply :D, that was a mispost of mine, which I edited already, immediately (2:47AM here, almost sleeping). I was affraid of your fast reply :D I saw you also edited my post, a thing I don't like. Please don't change the meaning of the things I said. Still did not get the part about prime numbers in base 3 ending in 2. So what? Are they even? Last fiddled with by LaurV on 2011-09-18 at 19:57

2011-09-19, 02:55	#18
Christenson Dec 2010 Monticello 5·359 Posts	Let me try to collect that bonus: My number is 2^43,112,609-1. It's even..it MUST be even........I swear it! I knew you knew there was a problem with 6x-1...SM88 is being his incoherent self in trying to prove the property you give..the property you gave is exactly what I was driving at. And I must have lost an even part of your post somewhere...it's now getting late on me!

2011-09-19, 03:16	#19
LaurV Romulan Interpreter Jun 2011 Thailand 3⁴×113 Posts	Now, a number x (see x as a heap of candies, sticks, coins, not as its numerical representation) can be represented in positional notation in any base b by an expression like: $x=a_nb^n+a_{n-1}b^{n-1}+...+a_2b^2+a_1b+a_0$ To be an even number, this has to be congruent to 0, modulo 2. $x=a_nb^n+a_{n-1}b^{n-1}+...+a_2b^2+a_1b+a_0\equiv 0\ (mod 2)$ Call this class "the parity" of x. So, the parity is 0, the number is even. The parity is 1, number is odd. The "parity" of this "sum" is given by the parity of its terms. When the base b is even, all the terms are even (as being a product of an even number and something else), except the last one, $a_0$ . So, the "parity" of the number is given by this "last digit". Pardon me for "digit", call it figure (I don't like that, I always associate it in my mind with expressions like "a well known figure in society", or "a figure of speech", "the dove is a figure of peace", "let's figure it out", etc, is more confusing). Call it cipher, as in "any of the (Arabic or else) symbols used to represent numbers in positional notation). In our language we have a special word for it, that translates more like "symbol". Now, when the base is odd, the "parity" of each term is given in fact by its coefficient. If $a_i$ is odd, the term is odd. If $a_i$ is even, the term is even. Now, going back to adding digits, as in the "criterion of divisibility by 3 in base 10", where we have to add all digits, here it does not make too much sense to add the digits that are even, does it? They won't change the parity. Is this what you wanted to say? If so, then I learning something new today, and all our arguing was not in vain. I am quite happy about it. Thanks. This was a good coffee this morning.... Edit: Big revelation: you don't even need to "add" them. Counting is enough. Whaaaa, what a revelation! For example in base 3 you only need to count how many of 1 do appear in the number. In base 5 you have to count 1's and 3's altogether... This coffee was for sure a good one. Leaving for my daily job... Last fiddled with by LaurV on 2011-09-19 at 03:28

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2011-09-21, 03:34	#22
LaurV Romulan Interpreter Jun 2011 Thailand 3⁴×113 Posts	The direction of this topic reminds me old times 20 years ago when it was a fashion (at least in my part of the world) to take people names (influential public persons, movie starts, your colleagues, etc), add the ASCII codes of that names, eventually do some other operation (google for Finagle's constant) and "prove" that the respective man/woman, etc. is the reincarnated devil. I think the joke started with some freaks "proving" that Bill Gates is evil. He used to call himself "Bill Gates the Third" after his father and grandfather being also Bill (no idea if this story is true or fabricated) and if you add his ascii codes, voila! The trick is that you can make (almost) anybody's name add to whatever you like, if you give the right rule. I always liked the number 23, I don't know why, since I was quite young, maybe I liked the sound of it, so you take my (real) name, add 5 (2+3) and I am evil too... So, take "velociraptor", "mythbusters" or other of the given examples, add whatever Finagle's constant you like at the end of them, and voila! one has his primes. This leads to nothing. Just my two pence. Better talk about some math, so I (and anyone else reading) could learn something new...