Testwiki:Reference desk/Archives/Mathematics/2012 September 5

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September 5

Project Euler question

There is a Project Euler problem (whose number I am not quoting so the answer is not given away directly here), where one needs integer solutions to 2b^2 - 2b - n^2 + n = 0. b=3 and n=4 is one solution to this equation. The equations b' = 3b + 2n - 2 and n' = 4b +3n - 3 generate b' and n' which are also solutions to the equation. Two questions please: (1) How do I prove that the b' and n' equations are valid? (2) How does one do Diophantine magic and generate those equations in the first place? -- SGBailey (talk) 08:46, 5 September 2012 (UTC)

The equation is equivalent to Template:Nowrap. Now it becomes obvious Template:Nowrap is another solution... --CiaPan (talk) 09:22, 5 September 2012 (UTC)

I know the equations work, I've got a dozen solutions and could have a dozen more if I wanted them (they get quite big...). But I'm interested in the proof and creation of the "next set of solutions" equations. Yes 1,1 is a solution. So is 0,1. But so is 3,4. -- SGBailey (talk) 09:36, 5 September 2012 (UTC)

I don't know where they come from, but it's simple enough to show that they work: just plug your formulas for b' and n' into the original equation and simplify. You'll get 2b^2 - 2b - n^2 + n = 0. Under the assumption that you started with a solution, this shows that b' and n' are a solution.--121.73.35.181 (talk) 10:25, 5 September 2012 (UTC)

(1) solved. You are right thanks - when I tried that previously, I went wrong; but it worked ok second time. So (2) where did the generating equations come from??? -- SGBailey (talk) 11:26, 5 September 2012 (UTC)

Convergents (p,q) to the continued fraction expansion of sqrt(2) are (1,1), (3,2), (7,5), (17,12), (41,29) etc (see Pell number). Alternate convergents (1,1), (7,5), (41,29) etc. satisfy

$p^2=2q^2-1$

Successive convergents in this sequence are related by the recurrence relations

$p^{\prime }=4q+3p{q}^{\prime }=3q+2p$

Use the transform

$n=\frac{p+1}{2}b=\frac{q+1}{2}$

and you get the (n,b) sequence (1,1), (4,3), (21, 15) etc. which satisfies

$(2n-1{)}^2=2(2b-1{)}^2-1\Rightarrow 4n^2-4n+1=8b^2-8b+1\Rightarrow n^2-n=2b^2-2b$

with the recurrence relations

$n^{\prime }=4b+3n-3b^{\prime }=3b+2n-2$

Gandalf61 (talk) 12:08, 5 September 2012 (UTC)

Thank you (I think). I didn't understand most of that, but I'll study it for a few days and see if it makes any sense then. -- SGBailey (talk) 13:11, 5 September 2012 (UTC)

Complex variables inequality

Hi. I'm working in a book, and looking at a claim that goes as follows: For $|e^z-1|\le \frac{1}{2}$, we have $\frac{1}{2}|z|\le |e^z-1|\le 2|z|$. I've checked this by doing a change of variables $\zeta =e^z-1$, parameterizing the circle $|\zeta |=\frac{1}{2}$, and checking that $\frac{1}{2}|\log (\zeta +1)|\le |\zeta |=\frac{1}{2}\le 2|\log (\zeta +1)|$ all around the circle (for the principal branch). I checked just by graphing the left and right as functions of a real variable, and sure enough, they stayed away from $\frac{1}{2}$. This is terribly awkward, though. Does anyone know an easier way to see this? Thanks in advance. -GTBacchus^(talk) 19:54, 5 September 2012 (UTC)

Unless I’m missing something, the first part of the inequality is false.

I will type * for times, and ^ for ”to the power of”.

Because, suppose that z = 2pi*i. Then e^z = e^(2pi*i) = cos(2pi) + i*sin(2pi) = 1 + 0 = 1. So then |e^z - 1| = 0, which is less than 1/2. So z = 2pi*i satisfies the assumption.

But then the first part of the inequality, (1/2)|z| <= |e^z -1|, becomes (1/2)(2pi) <= 0, or pi <= 0, which is just not true. Cardamon (talk) 07:00, 8 September 2012 (UTC)

Yeah, I noticed that too. The inequality applies for z close to 0, not close to other multiples of 2pi*i. It can be fixed by adding the condition that |z|<1 or something.

I seem to have it figured out. The trick is to do the substitution $\zeta =e^z-1$, and assume that $|\zeta |\le \frac{1}{2}$; this obviates the problem you mention. Then you really want to show that $\frac{1}{2}|\zeta |\le |\log (\zeta +1)|\le 2|\zeta |$. This can be accomplished by playing with the series expansion of $\log (\zeta +1)$ and the triangle inequality. -GTBacchus^(talk) 01:59, 11 September 2012 (UTC)