fun math problem

[lio45]

Ignoring the division by n for now, we can see that the two last terms of the formula are always canceling the first and last items of the expansion: - 2ⁿxⁿ /n and - (-1)ⁿyⁿ.

So, each of the two "side ones" in the Pascal triangle disappears in the end. (The right one would have been 1/n, so always a fraction, capping n at 1 if it stayed there.)

So, the triangle:

First line (2)1 (-1)1

Second line 2²1 2¹(-1)¹2 (-1)²1

Third line 2³1 2²(-1)¹3 2¹(-1)²3 (-1)²1

We can see we can easily start ignoring the minus ones regardless of the power as they have no bearing on the fractionality of the coefficient.

Fourth line 2⁴1 2³4 2²6 2¹4 2⁰1

(Bringing back the division by n that we left off the screen for clarity, we can easily verify at this stage that the terms that matter on the fourth line are 32/4 (integer), 24/4 (integer), 8/4 (integer). So, all integers.)


On the nth line of the triangle, the rth coefficient is 2^n-r times the rth number of the nth line of the Pascal triangle which is n!/(r!(n-r)!). Divided by n, of course.

i.e.

2^n-r(n-1)!
----------
r!(n-r)!


So, the problem becomes, this, above, has to be an integer for all values of r from 1 to n-1 (we can ignore the first and last terms as they're cancelled in the formula).

2^n-r is always an integer so it doesn't bring anything more to the table, so we can ignore it.

I am pretty sure there's no upper limit to n satisfying the condition of this quotient being an integer:
(n-1)!
-------
r!(n-r)!

So, at first sight, no upper limit to n.

Take everything I've written with a huge grain of salt, it's been well over a decade since I last did such things and found out I was extremely rusty.

Edit: It's obvious that Jay R knows what he's talking about!! You're right, let's keep (n)!/(r!(n-r)!) and the division by n separate.

Problem reduces itself to, for the n you pick, for all possible values of r from 1 to n-1, (n)!/(r!(n-r)!) has to be divisible by n. Correct?

[Asta Kask]

QED.

QCD.

Ten characters minimum.

[lio45]

If you are correct then I am wrong (looks like it might work but thanks to before stated deadlines don't have time to check).

Be honest, are you making us do (or at least double-check) your school work?

[Asta Kask]

Be honest, are you making us do (or at least double-check) your school work?

Is that even allowed here?

[Lightlawbliss]

Be honest, are you making us do (or at least double-check) your school work?

no. I enjoyed the problem when an old friend gave it to me and felt others would enjoy it too.
(not to mention i'm in dif eq)

[Marcivo]

All right. This whole thread is bugging me.

I hold two BSci degrees in Maths and Physics. I'm the math nerd among all of my friends who are math nerds. I've learned to trust my gut when it comes to math and math-people, and my gut's telling me, right now, that this is either somebody's homework or a random solved problem pulled off the internet.

But if it's not, and this is the part I keep coming back to, then why post it here? If anything, we have a forum specifically devoted to science and technology, a forum where the problem might be better presented as an object for serious mutual discourse and not a cryptic, unsolicited challenge to weed out those who "get it" and those who don't.

[Lightlawbliss]

All right. This whole thread is bugging me.

I hold two BSci degrees in Maths and Physics. I'm the math nerd among all of my friends who are math nerds. I've learned to trust my gut when it comes to math and math-people, and my gut's telling me, right now, that this is either somebody's homework or a random solved problem pulled off the internet.

But if it's not, and this is the part I keep coming back to, then why post it here? If anything, we have a forum specifically devoted to science and technology, a forum where the problem might be better presented as an object for serious mutual discourse and not a cryptic, unsolicited challenge to weed out those who "get it" and those who don't.

...There is a science and technology section?... *looks a forum main page* That's what I get for stopping at the word mad.

It may have been a random problem from the internet before I got it, but if so then my google fu can't find it.

[Jay R]

no. I enjoyed the problem when an old friend gave it to me and felt others would enjoy it too.
(not to mention i'm in dif eq)

Then where did the deadline you mentioned come from?

[Lightlawbliss]

Then where did the deadline you mentioned come from?

three papers due today and a test monday.