A question on Smarandache function

[Ghost Post]

It has been conjectured that the equation S(n)=S(n+1) has no solution, where S(n) is the Smarandache function, i.e. the smallest integer such that S(n)! is divisible by n.
No one has solved this conjecture so far...

[Mercuria]

the question being...?

[HyToFry]

That is the question, he's asking the Minotaurs to solve it.
I can't, Kevin's gone, looks like it's up to you mith 5.5!

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I took a walk with my pain down memory lane; I still haven't found my way back.

--Oasis lyric... hyMogrifide.

[mithrandir]

wish i could, if it's unsolved that would be pretty impressive of me

let me get the conjecture straight...

S(n) is the lowest number such that S(n)! is divisible by n

and the conjecture is that S(n)=S(n+1) is never true?

code:

---

n   S(n)


1   1


2   2


3   3


4   4


5   5


6   3


7   7


8   4


9   6


10  5

---

etc.? seems like the conjecture would have to be right, but i don't know about proving it... i'll have to look at it.

[HyToFry]

He'll have it done yesterday.

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I took a walk with my pain down memory lane; I still haven't found my way back.

--Oasis lyric... hyMogrifide.

[mithrandir]

you sure this is unsolved? the proof seems fairly basic, but maybe i'm missing something...

for any number n, let the prime factorization of n be represented as:

n=p1^f1*p2^f2*...*pk^fk

taking each prime factor separately, in order for S(n)! to be divisible by px^fx, S(n) must be at least:

px, for fx=1
2px, for 1<fx<4
3px, for 5<fx<8
.
.
.
z*px, for 2^(z-1)+1<fx<2^z

anyway, the point is that for each prime number in the factorization of n, S(n) must be at least mx*px, for some integer mx. for the actual n, S(n) must be the max of all mx*px.

So, S(n) is divisible by px, some prime factor of n.

However, S(n+1) must be divisible by py, some prime factor of n+1. Since n and n1 are coprime, S(n) can not be equal to S(n+1).

[mithrandir]

Ah, the conjecture i found on the website you posted in VSN is different. It talks about Z(n), where Z(n) is the smallest integer such that 1+2+3+...+Z(n) (in a sense, Z(n)! for addition) is divisible by n. That's an entirely different problem.

[HyToFry]

WHOOOSH!

[Lepton]

Mith, you're quite intelligent, eh?

[Luna]

~gazes at mith adoringly~

[mithrandir]

*~blushes~*

[OcularGold]

okay, math genious mith, i got a new one. i asked a mathematician friend of mine, and he really couldnt solve it:

Does every possible finite number combination have to exist somewhere in pi?

so, in other words.. is there some point in point where there are 5 gazillion 3's in a row? or really any other number combination you can think of.

pi certainly has the room for em (since its infinite).. but do they HAVE to exist? prove it or disprove it if you can :-P


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I've seen water. It's water that's all.

[mithrandir]

no, they don't *have* to exist. not all numbers that are infinite have to contain every natural number in them. not even the irrationals. not sure about transcedentals, i dont know enough about them to give an example.

here's an irrational number that doesn't contain every natural number:

.12101112202122100101...

and, of course, here's one that does:

.123456789101112131415...

to prove that pi does (or doesn't) you would have to show that pi is normal (or not). in otherwords, that the digits of pi are statistically random. i think most mathematicians think that pi is normal, but it hasn't be proven yet:
http://www.er.doe.gov/feature_articles_2001/july/Digits_of_Pi/Digits%20of%20Pi.htm

that's all i have to say about that

[CrystyB]

mith, how the hell can you say "x! for addition"??? That's definitely x(x+1)/2, and the proof still makes me blush because its simplicity and my little intuition at that time. I suppose the term "factorial" was invented because there were no terms to describe that, nor any formulae to help you get to it. But for addition?

P.S. I got a bunch of SE's but the post is yet here. You got to solve that problem? If so, Good for you, TYVM!

[This message has been edited by CrystyB (edited 08-20-2001).]

[mithrandir]

hehe, i know what the formula is cb, i was just trying to figure out how he managed to get the two mixed up in the first post.

[OcularGold]

so... an irrational number that isnt normal would have to contain every finite number combination? i still dont see how this makes sense. please explain.

thanks in advance,

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I've seen water. It's water that's all.




[This message has been edited by OcularGold (edited 08-20-2001).]