Dangerous Squares

Number Theory Level 4

Find all positive integers $n$ such that $12n -119$ and $75n -539$ are both perfect squares.

Let $N$ be the sum of all possible values of $n$ . Find $N$

The answer is 32.

1 solution

Soham Karwa
Aug 20, 2014

Let $75n - 539 = l^2$ and $12n - 119 = k^2$ . where $n \in \mathbb{N}$

Multiply $75n - 539 = l^2$ by $4$ to give $300n - 2156 =4l^2$ and $12n - 119 = k^2$ by $25$ to give $300n - 2975 = 25k^2$ .

Subtract the two new expressions to give $4l^2 - 25k^2 = 819$ which can be factorised (using the difference of two squares) to give $(2l - 5k)(2l + 5k) = 819$ .

The prime factorisation of 819 is $3^2 * 7 * 13$ There are five cases to consider. Dealing with the cases (noting that $2l - 5k < 2l +5k$ ) yields that $n$ can only be $20 or 12$

hence $N = 20 + 12 = \boxed{32}$

Same Solution

Kushagra Sahni - 5 years, 11 months ago

How did 20 and 12 come and which are the five cases to be considered

Gagan Jain - 6 years, 9 months ago

The five cases to consider are:

$2l + 5k = 819$ and $2l - 5k = 1$

$2l + 5k = 273$ and $2l - 5k = 3$

$2l + 5k = 117$ and $2l - 5k = 7$

$2l + 5k = 63$ and $2l - 5k = 13$

$2l + 5k = 39$ and $2l - 5k = 21$ .

When we solve these simultaneous equations, and plug our values of $l$ or $k$ into the original equations, only two give integer values for n. ( as required by the question) - the other three are extraneous solutions.

Soham Karwa - 6 years, 9 months ago

This was really a nice question. @Soham Karwa , Can you tell me some resources for learning Diophantine equations?

Jayakumar Krishnan - 6 years, 9 months ago

@Jayakumar Krishnan – Hi Jayakumar!

I would to love to be of service!

When I started to teach myself Number Theory, I too was lost, and didn't really understand how to solve problems.

What really helped me was to dive into a lot of problems, and really understand the solution, even if you get the answer right. That's what helped me!

In terms of resources, I believe Titu Andreescu's book '104 Number Theory Problems' is a great read - it provides a set of in-depth notes to begin with, and then some problems to get your hands dirty!

Also check out AoPs - it's an awesome resource for questions and notes.

I hope that was of help!

Soham Karwa - 6 years, 9 months ago

@Soham Karwa – Thanks! That of course..was..of..great..help...Yes..that is a great book...I've heard of it and kinda used it too :D. Well, anything to with Diophantines more ? (104NT has only 2 pages about DIophantines and that too only linear..) Maybe, the other book which ANdreescu has written about NT (theory, structure, examples)....should do! :D..Should you know something...please add in ! (Personal suggestions!_)

Jayakumar Krishnan - 6 years, 9 months ago

@Jayakumar Krishnan – Hi Jayakumar!

No problem at all - you just reminded me about another of Titu's books (I can't believe I forgot about it, given the nature of the question).

Here it is. - I think there might be a free copy somewhere.

Also, I thought I'd share a few tips about solving diophantine equations:

Always try and look for a factorisation , with variable(s) on one side, and an integer on the right.
If the question involves perfect squares, try and use the 'difference of two squares' identity.
Finally, a trick I learned quite recently, is to try and eliminate a variable (mainly used when the question is a mix between a diophantine and divisibility question), by addition, subtraction or multiplication. (There's a problem by Finn Hulse called I'm back which is quite difficult, but have a go nonetheless!)

Soham Karwa - 6 years, 9 months ago

@Soham Karwa – Thanks a lot again for replying :D.

Jayakumar Krishnan - 6 years, 9 months ago

Thank you for the explanation.

Gagan Jain - 6 years, 9 months ago

Hey! @Soham Karwa Thanks a lot for your suggestion of books.

Anandhu Raj - 6 years, 3 months ago

Dangerous Squares

The answer is 32.

1 solution

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