Square root and positive integer

Number Theory Level 4

$\large k= \sqrt{(n+223)(n+521)}$

Given that $n$ and $k$ are both positive integer, find the value of $n$ .

The answer is 10729.

2 solutions

Chan Lye Lee
Apr 30, 2017

Since $(n+223)(n+521)=k^2$ , after completing the square of the left-hand side, we have $(n+372)^2-22201=k^2$ . Now $(n+372)^2-k^2=(n+372+k)(n+372-k)=22201=149^2$ . As 149 is a prime number, $n$ and $k$ are positive integers, $n+372+k=22201$ and $n+372-k=1$ . Solve the equations, we obtain $n+372=11101$ which means $n=10729$ .

Note: You should mention that 149 is a prime.

Calvin Lin Staff - 4 years, 1 month ago

Added. Thanks for pointing that. This makes the explanation clearer.

Chan Lye Lee - 4 years, 1 month ago

Note: we can easily reject $n+372+k=149$ , since if so, then $n+372-k = 149$ ; but it's impossible, since $(n,k) >0$ . Second, if $n+ 372 + k =149$ , then we can clearly see that $n$ or $k$ is negative. So, the possible equations are $n+372 +k = 22201$ and $n+372 -k =1$ .

Fidel Simanjuntak - 4 years, 1 month ago

Edwin Gray
Aug 25, 2018

Squaring both sides, k^2 = n^2 + 744n + 116183 = n^2 + 744n + 138384 - 22201, or transposing, k^2 + 149^2 = (n + 372)^2. This is a Pythagorean Triplet, so there exists integers a and b such that: (1) n + 372 = a^2 + b^2, (2) 149 = a^2 - b^2, k = 2ab. Since 149 is prime, a - b = 1, a + b = 149. Then a = 75, b = 74. Then n + 372 = 75^2 + 74^2 = 11101, so n = 11101 - 372 = 10729. Ed Gray

Square root and positive integer

The answer is 10729.

2 solutions

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