When is THIS an integer?

For natural numbers $a,b,c$ when is $\dfrac{abc}{\sqrt{(a^2+b^2+c^2)^2-2(a^4+b^4+c^4)}}$ a natural number?

#NumberTheory

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Math	Appears as
Remember to wrap math in `$` ... `$` or `\[` ... `\]` to ensure proper formatting.
`2 \times 3`	$2 \times 3$
`2^{34}`	$2^{34}$
`a_{i-1}`	$a_{i-1}$
`\frac{2}{3}`	$\frac{2}{3}$
`\sqrt{2}`	$\sqrt{2}$
`\sum_{i=1}^3`	$\sum_{i=1}^3$
`\sin \theta`	$\sin \theta$
`\boxed{123}`	$\boxed{123}$

Comments

Expanding the denominator and comparing it with the expanded form of heron's formula, we can say that if a triangle with side lengths a, b, c exists , then the denominator is equal to 4(Area of the triangle). It can be easily proven that a, b, c satisfy the triangle inequality by a proof by contradiction. If they do not satisfy the inequality, then one of s-a, s-b, s-c is negative while the other expressions are positive which makes the expression under the square root negative which is not possible. Here, s is the semi-perimeter. Hence, proven. Also, the numerator is equal to 4(Area)(Radius of Circumcircle of the triangle) by the formula R = (abc)/(4Δ) where Δ is the Area of the triangle and R is the radius of the circumcircle. Therefore, the given expression reduces to R. Now, we are given that a, b, c and R are natural numbers. A trivial solution would be all Pythagorean triplets where the hypotenuse is even. But these are not the only solutions. For example, a solution is 104, 112, 120. Another way of finding solution is to "compose" two Pythagorean-triplet right triangles having a common length (not the hypotenuse).For example, we can "compose" the 5, 12, 13 and the 9, 12, 15 triangles to get the 13, 14, 15 triangle. Sometimes (including the above case), R will be a rational number and not an integer. In these cases, we can simply take the triangle similar to the obtained triangle by multiplying the sides with the denominator of R. This process should give all possible solutions of the given question as the cosines and sines of all angles must be rational.

A Former Brilliant Member - 1 month, 3 weeks ago

@Inquisitor Math – Do you have the full solution?

A Former Brilliant Member - 1 month, 3 weeks ago

No. I just came up with this problem myself.

I was hoping someone could share a solution :)

Inquisitor Math - 1 month, 3 weeks ago

@Inquisitor Math – I think my comment covers up all possible solutions. Do tell me if you find an example unobtainable by my described process.

A Former Brilliant Member - 1 month, 3 weeks ago

Math	Appears as
Remember to wrap math in `\(` ... `\)` or `\[` ... `\]` to ensure proper formatting.
`2 \times 3`	$2 \times 3$
`2^{34}`	$2^{34}$
`a_{i-1}`	$a_{i-1}$
`\frac{2}{3}`	$\frac{2}{3}$
`\sqrt{2}`	$\sqrt{2}$
`\sum_{i=1}^3`	$\sum_{i=1}^3$
`\sin \theta`	$\sin \theta$
`\boxed{123}`	$\boxed{123}$