Problem 44788. Find the right number make the equation

Created by Binbin Qi

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Given positive integer number n, find the right positive integer number a, b , so that

(1) sqrt(a + n * b) is a positive integer number

(2) sqrt(hypot(a,b)) is a positive integer number

Note: the output must be char array!

Have fun!

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Last Solution submitted on Nov 28, 2023

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James on 16 Nov 2018

Interesting. I've solved several more puzzles in the Community group after this one, but I still can't see your solution, Binbin. It's obviously much smaller than mine is, which is why I'm interested in figuring out how you did this.

li haitao on 16 Nov 2018

@ James
To unlock all solutions of this problem, you must solve a newer problem than this one, not older problems.

Binbin Qi on 17 Nov 2018

@James, my solution is very simple, it is pure number.The challenge is from Fermat. Because the big number, it is hard to test. I want use Java's Big Integer, but it has no sqrt operation.

Tim on 19 Nov 2018

My program gives a wrong answer for n=1, but passes the tests due to insufficient precision. How big is the smallest solution for n=1?

Binbin Qi on 19 Nov 2018

@Tim, it is about 13 digitals. I want to submit another problem use char array as output and avoid to precision.

Binbin Qi on 19 Nov 2018

@James, @Tim, Because of precision, the problem have no meaning, so I have fix the test suits. The output must be char array. Thank you for your support.

James on 20 Nov 2018

I was using the method to parameterize Pythagorean triples (a,b,c) in terms of s and t. Since we're looking at c^4 instead of c^2 (to make sure that c itself is a perfect square), I can see why the numbers got quite large. My solution for n=2 was for larger numbers than my n=3 solution, although even my n=1 solution had the magnitude of a as 10^16 and b was 10^21. I didn't find anything on the order of 10^13.

Binbin Qi on 21 Nov 2018

@James, I think the following link will help you.
https://www.mathpages.com/home/kmath022/kmath022.htm

James on 21 Nov 2018

Very nice link, Binbin. That is was exactly what I was trying to do, only they didn't include the math error I made in my calculations. :)

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Problem 44788. Find the right number make the equation

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Problem 44788. Find the right number make the equation

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