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Unsolved Great Common Divisor and Divisibility Problem

I'm having a hard time with this problem. Need help.

Problem 1

If \(N\) and \(M\) are natural number such that \[\frac{N}{M} = 1 - \frac{1}{2} + \frac{1}{3} - \frac{1}{4}+ \cdots - \frac{1}{2016} + \frac{1}{2017}\] Prove that \(759\) divides \(N\).

I don't even have an idea where to begin. I'm stuck at the problem.

Problem 2

If \(a\) and \(b\) are natural number such that \(a+b\sqrt{2} = (1+\sqrt{2})^{2017}\). Prove that \(\gcd(a,b) = 1\).

I get \(a = \binom{2017}{0} + 2 \binom{2017}{2} + 4\binom{2017}{4} + 6\binom{2017}{6} + \cdots + 2014 \binom{2017}{2014}+ 2016\binom{2017}{2016}\) and \(b=\binom{2017}{1} + 2 \binom{2017}{3} + 4\binom{2017}{5} + 6\binom{2017}{7} + \cdots + 2014 \binom{2017}{2015}+ 2016\binom{2017}{2017}\). Then i don't know what to do.

Note by Jason Chrysoprase
2 months ago

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There must be some mistake in problem 1. I'm pretty sure that the numerators of the alternating harmonic numbers are never divisible by 3, let alone 759. For this particular one, the argument is: after taking out the two terms with denominators divisible by the maximal possible power of \(3\) (in this case, \(3^6 = 729\)), write \(\dfrac{N}{M}\) as \[ \frac1{729} - \frac1{1458} + \frac{A}{243B} \] where \(A\) and \(B\) are integers and \(3 \nmid B.\) After getting a common denominator, this becomes \[ \frac{B+6A}{1458B} \] so the numerator is not divisible by 3.

This pretty clearly generalizes to any other alternating harmonic number. Patrick Corn · 1 month, 4 weeks ago

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Problem 2: Potential method Let \(\left(1+\sqrt{2}\right)^n=A_n+B_n\sqrt{2}\).

Then

\[A_n = \frac{1}{2}\left(\left(1-\sqrt{2}\right)^n+\left(1+\sqrt{2}\right)^n\right) \\ B_n = \frac{1}{2\sqrt{2}}\left(\left(1+\sqrt{2}\right)^n-\left(1-\sqrt{2}\right)^n\right)\]

It follows from this that

\[A_{n+1}=2A_n+A_{n-1}, \quad A_0=A_1=1 \\ B_{n+1}=2B_n+B_{n-1}, \quad B_0=0,\ B_1=1\]

Then what is left is to proof that \(\gcd \left(A_n,B_n\right)=1\), which appears to be the case but I haven't found one yet.

EDIT: I've read the rest of the comments. This solution is kinda dumb Julian Poon · 1 month, 3 weeks ago

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@Julian Poon A comment regarding your work:

The actual recurrence relation is that \(A_{n+1}=A_n+2B_n\) and \(B_{n+1}=A_n+B_n\) from which it follows that \(A_{n+1}=1\times B_{n+1}+B_n\). Then notice that we have,

\[A_{n+1}=1\times B_{n+1}+B_n\\ B_{n+1}=1\times B_n+A_n\]

From the above, we note that since \(\gcd(a,b)=\gcd(b,a-b)\), we have,

\[\gcd(A_{n+1},B_{n+1})=\gcd(B_{n+1},B_n)=\gcd(A_n+B_n,B_n)=\gcd(B_n,A_n)=\gcd(A_n,B_n)~\forall~n\geq 1\]

Hence, we conclude that \(\gcd(A_n,B_n)=\gcd(A_{n-1},B_{n-1})=\cdots =\gcd(A_1,B_1)=1\). Prasun Biswas · 1 month, 3 weeks ago

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@Prasun Biswas Very nicely done!

That's one reason why it's hard to definitely conclude "this particular approach will yield no result". I agree that it looks slightly unapproachable at first, and am pleasantly amazed by what @Prasun Biswas has done :) Calvin Lin Staff · 1 month, 3 weeks ago

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A minor comment:

It's actually \(a=\sum\limits_{k=0}^{1008}\dbinom {2017}{2k}2^k\), not \(a=\dbinom{2017}0+\sum\limits_{k=1}^{1008}\dbinom{2017}{2k}2k\)

Similar comment for \(b\). Prasun Biswas · 2 months ago

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For problem 3 Try to substitute \(n\) with \(n=2k\) or \(n=2k+1\). And then, find out when k is congruent 0 mod 10, congruent 1 mod 10, ... Rizky Souleekeen · 2 months ago

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What have you tried? Calvin Lin Staff · 2 months ago

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@Calvin Lin I have told everything i tried. I have no idea about problem 1. I have solved for \(a\) and \(b\) for problem 2, but i don't know how to find their gcd. I know that \(\frac{n(n+1)}{2} = 1+2+3+\cdots+n\), what do i do next ? Jason Chrysoprase · 2 months ago

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@Jason Chrysoprase

  1. Work on understanding \(M\) in many different ways. For example, can \( 759 \mid M \)? Why, or why not?

  2. Work on undersatnding \(a, b \) in many different ways. The current way that you have doesn't allow for easy manipulation. Is there another relationship that exists between \(a\) and \(b\) directly? E.g. if we can show that \( 2017 a + 7102 b = 1 \), then it follows that \( \gcd(a,b) = 1 \).

Calvin Lin Staff · 2 months ago

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@Calvin Lin

  1. It's obvious that \(759 \mid M\). Let \(a = \mathrm{lcm}(1,2,3,...,2017)\). Since \(1 - \frac{1}{2} + \frac{1}{3} - \frac{1}{4}+ \cdots - \frac{1}{2016} + \frac{1}{2017}\) can be expressed as \(\large \frac{\sum_{n=1}^{2017} (-1)^{n+1} \frac{a}{n}}{a}\) with positive integer numerator and denominator. How does that will help solving for \(N\). Wait, isn't \(\large \frac{\sum_{n=1}^{2017} (-1)^{n+1} \frac{a}{n}}{a}\) gives us the most simplified expression ? \(a\) it self is in form of \(M\) which is divisible by \(759\), and the problem state that \(N\) is divisible by \(759\). But then we can cancel out \(759\) and hence \(\large \frac{\sum_{n=1}^{2017} (-1)^{n+1} \frac{a}{n}}{a}\) is not the most simplified fraction in form of \(\frac{N}{M}\), which is a contradiction. Also i do check for the exact value of \(M\) and \(N\) for the most simplified \(\frac{N}{M}\) on wolfram alpha and \(759\) did not divides \(N\). But i still wonder, since \(N\) is divisible by some prime number greater than \(2017\), can we find the prime number ?

  2. I'm not quite sure for this moment. Any more hints ?

Jason Chrysoprase · 2 months ago

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@Jason Chrysoprase

  1. Great, so you realize that in the naive / unsimplified numerator, we must have \( 759^2 \mid N' \). However, \( N' \) still isn't a very nice number to manipulate. What can we do with it?

  2. Hint: \( x^2 - y^2 = (x-y)(x+y) \).

Calvin Lin Staff · 2 months ago

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@Calvin Lin

  1. I don't understand your statement. Since for the most simplified \(\frac{N}{M}\), \(N\) is not divisible by \(759\), it should be divisible by some prime number greater than \(2017\). Then for non simplified \(\frac{N}{M}\), we can just multiply \(759\) on numerator and denominator. Isn't it that easy ? I'm super confused. Shouldn't the problem itself was wrong ?

  2. So since \((1+\sqrt{2})^{2017} = a+b \sqrt{2}\) and \((1-\sqrt{2})^{2017} = a-b\sqrt{2}\), multiply both equation gives us \(\left((1+\sqrt{2})(1-\sqrt{2})\right)^{2017}=(a+b \sqrt{2})(a-b \sqrt{2})\). Hence \((-1)^{2017}=-1=a^2 - 2b^2\). So \(2b^2 = a^2(1) + 1\). By Euclidean Algorithm, \(a^2 = 1(a^2) + 0\). Hence their great common divisor is equal \(1\)

Jason Chrysoprase · 2 months ago

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@Jason Chrysoprase

  1. When I said "simplified", I did not mean "in reduced form". Currently, the values \( \frac{ N} { M } \) are not easily understood. In particular, you want to claim that \(M = lcm(\ldots ) \), but if the problem statement is true, then that can't be since \( 759 \mid M \). So, let's choose an equivalent form \( \frac{N}{M} = \frac{ A}{B} \) that is not completely reduced, where \(A\) and \(B\) are easy for us to calculate and understand. The choice of \( B = \lcm ( \ldots ) \) doesn't work, so what potential (naive) forms can we try? The goal here is to show that (say) \( 759 ^3 \not \mid B \) and \( 759 ^3 \mid A \), from which we can conclude that \( 759 \mid N \).

  2. Great.

Calvin Lin Staff · 2 months ago

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