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Discussion on Exam 1

Posted: Tue Aug 25, 2015 11:07 pm
by Masum
Ok, the first exam is over. You can discuss on the problems now. Solutions have been posted here. Until then, post your solutions and opinions on the problems or other solutions. And there will be no exam tomorrow. But there will one the next day after that. So be prepared for that.

Re: Discussion on Exam 1

Posted: Tue Aug 25, 2015 11:16 pm
by Masum
Let me give you a hint on how to solve problem 1.2. Though I forgot to mention, but it should be kind of obvious that $a_0>1$.
Hint: If $s(a)$ is the smallest prime divisor of $a$, then $a+s(a)$ is always even.

Re: Discussion on Exam 1

Posted: Wed Aug 26, 2015 12:48 am
by rubabredwan
..........

Re: Discussion on Exam 1

Posted: Wed Aug 26, 2015 12:52 am
by badass0
Can anybody give me hints on how to solve problem 1.4?

Re: Discussion on Exam 1

Posted: Wed Aug 26, 2015 1:10 am
by Masum
rubabredwan wrote:Problem 1.2
$a_{0}$ can't be equal to 1.
let,
$a_{0}=2$ then,
$a_{1} = 4$
$a_{2} = 6/8$
$a_{3} = 8/9/10/12$
$a_{n} = ......$

if $a_{0}=3$ then,
$a_{1} = 6$
$a_{2} = 8/9$
$a_{3} = 10/12$
$a_{n} = ......$

we can see that integers 5, 7, 11 etc. aren't coming in this recurrence. because $a_{n}$ is a multiple of $d(a_{n-1})$ which states that no prime other than $a_{0}$ will come.this recurrence follows a pattern like seive of erastothenes(not fully). if $a_{0}$ is not a prime, then every $a_{n}$ will be composite.

the number of prime numbers are infinite. So there is no such value of $k$.
$a_0$ is a fixed positive integer which is given and greater than $1$. Try it again. And think about the hint too.

Re: Discussion on Exam 1

Posted: Wed Aug 26, 2015 1:11 am
by Masum
badass0 wrote:Can anybody give me hints on how to solve problem 1.4?
$5$ divides $6^n-1$.

Re: Discussion on Exam 1

Posted: Wed Aug 26, 2015 5:45 pm
by shanto00
in problem 1.3 we just have to show that they are a primitive Pythagorean triple .

Re: Discussion on Exam 1

Posted: Wed Aug 26, 2015 7:33 pm
by Masum
No, it is given that they are a Pythagorean Triple. You have to prove $2(b+p)$ is a perfect square.

Re: Discussion on Exam 1

Posted: Thu Aug 27, 2015 9:21 pm
by shanto00
Masum wrote:No, it is given that they are a Pythagorean Triple. You have to prove $2(b+p)$ is a perfect square.
i meant to say that if we can prove that they are primitive , then the work is almost done.