NT marathon!!!!!!!

For discussing Olympiad Level Number Theory problems
Dustan
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NT marathon!!!!!!!

Unread post by Dustan » Fri Mar 19, 2021 3:07 pm

HI , NT MARATHON is here. Rules are same as the previous marathons. Happy Problem Solving!

$Problem 1 $: Find all positive integer x,y such that
$x^3=2y^2+1$ :D :D

Dustan
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Re: NT marathon!!!!!!!

Unread post by Dustan » Sat Mar 20, 2021 7:43 pm

$x$ is odd.

Now,
$2y^2+1$=$(1+y\sqrt{-2})(1-y\sqrt{-2}$
this two are coprime elements of $\mathbb{Z}[\sqrt{-2}]$
so ,both of them are perfect cube
So, there exist a,b in integer such that
$(a+b\sqrt{-2})=(1+y\sqrt{-2})$
Comparing the coefficients at 1 and $\sqrt{-2}$ yields
$a^3-6ab^2=1$
hence $(a,b)=(1,0)$
$x=1$ and $y=0$ but $y$ isn't a positive integer.

THERE IS NO SOLUTION.

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Mehrab4226
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Re: NT marathon!!!!!!!

Unread post by Mehrab4226 » Sat Mar 20, 2021 8:11 pm

Dustan wrote:
Sat Mar 20, 2021 7:43 pm
$x$ is odd.

Now,
$2y^2+1$=$(1+y\sqrt{-2})(1-y\sqrt{-2}$
this two are coprime elements of $\mathbb{Z}[\sqrt{-2}]$
Could you explain what the notation $\mathbb{Z}[\sqrt{-2}]$ means?
So, there exist a,b in integer such that
$(a+b\sqrt{-2})=(1+y\sqrt{-2})$
And how did you get that?
The Mathematician does not study math because it is useful; he studies it because he delights in it, and he delights in it because it is beautiful.
-Henri Poincaré

Asif Hossain
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Re: NT marathon!!!!!!!

Unread post by Asif Hossain » Sat Mar 20, 2021 8:55 pm

Mehrab4226 wrote:
Sat Mar 20, 2021 8:11 pm
Dustan wrote:
Sat Mar 20, 2021 7:43 pm
$x$ is odd.

Now,
$2y^2+1$=$(1+y\sqrt{-2})(1-y\sqrt{-2}$
this two are coprime elements of $\mathbb{Z}[\sqrt{-2}]$
Could you explain what the notation $\mathbb{Z}[\sqrt{-2}]$ means?
So, there exist a,b in integer such that
$(a+b\sqrt{-2})=(1+y\sqrt{-2})$
And how did you get that?
Maybe he is referring
about gaussian integers.
Hmm..Hammer...Treat everything as nail

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Anindya Biswas
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Problem 2

Unread post by Anindya Biswas » Sat Mar 20, 2021 9:41 pm

Let $\tau:\mathbb{N}\to\mathbb{N}$ such that $\tau(n)$ is the number of positive divisors of $n$. Prove that $\tau(n)<2\sqrt{n}$.
"If people do not believe that mathematics is simple, it is only because they do not realize how complicated life is."
John von Neumann

Asif Hossain
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Re: NT marathon!!!!!!!

Unread post by Asif Hossain » Sat Mar 20, 2021 10:35 pm

This can maybe a solution of prob 1(I don't know really NT noob)(Kindly correct me if i am wrong)
Taking $(mod 4)$ implies $x$ is odd and $y$ is even.Let $x=2k_1 +1$ and $y=2k_2$
Plugging it into the original equation
$\Rightarrow 8k_{1}^{3}+12k_{1}^{2}+6k_1=8k_{2}^{2}$
Now taking mod 4 both side implies $6k_1 \equiv 0 \Rightarrow 2k_1 \equiv 0 (mod 4)$ which implies $k_1$ is even since all positive integer
Then also taking modulo 8 implies $k_1(12k_1+6) \equiv 0 (mod 8)$ one case is $12k_1+6 \equiv 0 (mod 8)$ which implies $k_1$ is odd so contradiction
the other case is $k_1 \equiv 0 (mod 8)$ or $(mod4)$ WLOG that is not possible since iterating the process avoiding odd value of $k_1$ would then mean $k_1$ is divisible by any power of $2$ so then it would have no solution.$\square$
Last edited by Asif Hossain on Sun Mar 21, 2021 10:20 am, edited 3 times in total.
Hmm..Hammer...Treat everything as nail

Dustan
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Re: NT marathon!!!!!!!

Unread post by Dustan » Sat Mar 20, 2021 11:04 pm

One of my friend gave me cool solution 😀.

Here is the soln
$2y^2+1=8a^3+12a^2+6a+1$
Or,$Y^2=a(4a^2+6a+3)$
$(a, 4a^2+6a+3)=1\ or 3$
Case1: , gcd is 1
Both of them are perfect square.
But $(2a+1)^2<4a^2+6a+3<(2a+2)^2$
Contradiction.
Case 2: gcd is 3
$a=3b$
$3b(36a^2+18a+3)=9b(12b^2+6b+1)$
As,$(b, 12b^2+6b+1)=1$
Both of them are square.
From the discriminate, we can say $12b^2+6b+1$ is not a square.
Hence we are done
Last edited by Dustan on Sat Mar 20, 2021 11:57 pm, edited 1 time in total.

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Anindya Biswas
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Re: NT marathon!!!!!!!

Unread post by Anindya Biswas » Sat Mar 20, 2021 11:49 pm

Asif Hossain wrote:
Sat Mar 20, 2021 10:35 pm
Then also taking modulo 8 implies $k_1(12k_1+6) \equiv 0 \Rightarrow 12k_1+6 \equiv 0 (mod 8)$
I don't think $k_1$ can be cancelled like that. Cause in order to cancel it, you must need $\text{gcd}(k_1,8)=1$ which is not the case since you proved $k_1$ is even.
"If people do not believe that mathematics is simple, it is only because they do not realize how complicated life is."
John von Neumann

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Anindya Biswas
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Re: NT marathon!!!!!!!

Unread post by Anindya Biswas » Sun Mar 21, 2021 12:05 am

Dustan wrote:
Sat Mar 20, 2021 11:04 pm
From the discriminate, we can say $12b^2+6b+1$ is not a square.
I think this sentence needs some clearance... How can we say that $12b^2+6b+1$ is never going to be a perfect square?
"If people do not believe that mathematics is simple, it is only because they do not realize how complicated life is."
John von Neumann

Asif Hossain
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Re: NT marathon!!!!!!!

Unread post by Asif Hossain » Sun Mar 21, 2021 7:54 am

Anindya Biswas wrote:
Sat Mar 20, 2021 11:49 pm
Asif Hossain wrote:
Sat Mar 20, 2021 10:35 pm
Then also taking modulo 8 implies $k_1(12k_1+6) \equiv 0 \Rightarrow 12k_1+6 \equiv 0 (mod 8)$
I don't think $k_1$ can be cancelled like that. Cause in order to cancel it, you must need $\text{gcd}(k_1,8)=1$ which is not the case since you proved $k_1$ is even.
Sorry it need some clarification :oops: now
I also mentioned the case $k \equiv 0 (mod 8)$ or $(mod4)$ afterwards
Hmm..Hammer...Treat everything as nail

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