Siam's inequality

Since we have to find out the square roots of $x$ and $(x+34)$ , basically we want to find two consecutive numbers whose squares don't differ by less than $34$ . Let $y$ and $y+1$ be the consecutive numbers, so the difference of their squares is

$(y+1)^2 - y^2 = 2y+1$

Now, we want this equal to $34$ , which yields $y = 16.5$ . Now let $x = y^2$ . But $y^2$ is not an integer. The closest number to $16.5$ whose square is an integer is $17$ . So $y = 17$ and hence $x = y^2 = 289$ .

I solved this problem using perfect squares . Let $p(n)=n^2$ be the $n$ 'th perfect square. If $p(n) \leq x < p(n+1)$ that means that $\lfloor\sqrt{x}\rfloor=n$ . Since $\lfloor\sqrt{x}\rfloor$ has to be equal to $\lfloor\sqrt{x+34}\rfloor$ , we must find two perfect primes, that lie at least $34$ apart (if they don't, $\lfloor\sqrt{x}\rfloor$ will be floored towards $n$ , and $\lfloor\sqrt{x+34}\rfloor$ , will be floored towards $(n+1)$ . The first pair of perfect primes with this property is $p(17)=289$ and $p(18)=324$ , as their difference is $35$ . Checking the equation with $x=289$ confirms, that this is a solution.

x must be a perfect square.

Using mean value theorem, for some u, x < u < x+ 34

[\sqrt { x+ 34 }] -\sqrt{x} = 34/(2 \sqrt{u} ) < 1 when \sqrt{u}> 17

according to the ques , while adding 34 with x , the square root doesnt increase.how is it possible? we know that , the difference between two conjecutive integer is the sum of them. so the least difference should be odd. so , we have to calculate the next odd difference after 34 and that is 35.now how we can get 35? yes , its 18+17.so , root of x+35 should be 18.hence x is 18^2-35 or 289

Now, bases on the assumptions, we can say that, the difference of the 2 perfect squares is less than 34, hence the square root symbol. So, we need to find 2 perfect squares that have a difference of 34 and/or less. The 2 numbers are 256 and 289. Therefore, the answer is 289.

Now, let's do this algebraically. It's basically $y = x^2; (x+1)^2 - x^2 \geq 34$ . Basically, when solved, we get y = 17, so x = 289.

jst see the perfect square after which adding 34 to it stops the result before the next perfect square as it is 289. ... 289+34 is less than 364 that is the next