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May 18

Parity of Numbers Given by Polynomials

The numbers given by f(x) = x are 1,2,3,... and have parity odd, even, odd, etc. The numbers given by f(x) = (1/2)(x*(x+1)) are 1,3,6,10,... (the triangle numbers) and have parity odd, odd, even, even, etc. Via Pascal's Triangle, we can see that there are an infinite number of polynomials where a series of n odd numbers is followed by a series of n even numbers where n is a power of two.

However, I am unable to think of a polynomial such that the numbers generated by it are of the form odd, odd, odd, even, even, even, etc, or in the more general case, where a series of n odd numbers is followed by a series of n even numbers where n is not a power of two.

Can anyone think of such a polynomial? Nkot (talk) 18:14, 18 May 2012 (UTC)

Cool problem, but I'm pretty sure the answer is that there isn't one. I don't have a good clue for a way to pull in a parity argument though...

Lets define the forward difference operator $\mathrm{\Delta }$ for function $f(x)$ as $\mathrm{\Delta }f(x)=f(x+1)-f(x)$. If $f(x)$ is a nonzero polynomial, then $\mathrm{\Delta }f(x)$ is a polynomial of a lower degree, so for any polynomial $f(x)$, there is $k$ big enough such that ${\mathrm{\Delta }}^{k}f(x)=0$ for any x. (${\mathrm{\Delta }}^{k}f$ means applying $\mathrm{\Delta }$ to $f$ k times). Suppose $f(x)$ is a polynomial which its results have parities odd,odd,odd,even,even,even,odd,odd,odd,even,even,even,... and it easily seen that no matter how many we apply $\mathrm{\Delta }$ in it, it will "stuck" at the series even,odd,odd,even,odd,odd,even,odd,odd,... which is not constant, therefore $f$ cannot be polynomial. --77.125.208.4 (talk) 12:56, 20 May 2012 (UTC)

my matrix-fu is weak

I never learned much matrix algebra beyond what it takes to solve simultaneous linear equations. What can I read to improve my skilz? —Tamfang (talk) 21:01, 18 May 2012 (UTC)

Any textbook on linear algebra. Looie496 (talk) 00:16, 19 May 2012 (UTC)

For a course on matrices specifically, I would recommend Gilbert Strang's "Linear algebra and its applications", followed by Horne and Johnson's "Matrix analysis". For a course that emphasizes abstract vector spaces and linear transformations, I would suggest Halmos's "Finite dimensional vector spaces". (A book to follow Halmos is also desirable, but mostly I think I would go with a textbook in abstract algebra, such as Serge Lang's tome at that point.) Sławomir Biały (talk) 20:36, 20 May 2012 (UTC)

I love specific advice. Thanks, I've saved your remarks. —Tamfang (talk) 08:09, 22 May 2012 (UTC)

Smallest prime number

What is the smallest prime number that is not divisible by prime numbers smaller than it? 220.239.37.244 (talk) 23:40, 18 May 2012 (UTC)

"not divisible by numbers smaller than it" is the definition of a prime number; so the answer to your question is the smallest prime number, which is 2. —Tamfang (talk) 00:04, 19 May 2012 (UTC)

The number you need is 2 my friend, for I have, am and always will be drt2012 (talk) 19:50, 22 May 2012 (UTC)