Given four equations

\(a^2+b^2=d^2\)

\(a^2+c^2=e^2\)

\(b^2+c^2=f^2\)

\(a^2+b^2+c^2=g^2\)

a,b,c,d,e,f and g are positive integers.

And the problem is to find a triplet (hard) or prove that there are no pairs. (harder)

This problem is calles the perfect cuboid as if a,b and c are the sides of a cuboid, d,e and f are the face diagonnals and g is the space diagonnal. And all of these are integers.

So lets try to solve this?

For a start, I have shown that for integers x, y and z,

\(a = 2xz\)

\(b = 2yz\)

\(c = x^2+y^2-z^2\)

\(d = 2z \sqrt{x^2+y^2}\)

\(e = \sqrt {[(y+z)^2+x^2][(y-z)^2+x^2]}\)

\(f = \sqrt {[(x+z)^2+y^2][(x-z)^2+y^2]}\)

\(g = x^2+y^2+z^2\)

This is because for \(a^2+b^2+c^2=g^2\), a, b, c and g must be in the form given above.

Then, \(d = \sqrt{g^2-c^2}, e = \sqrt{g^2-b^2}, f = \sqrt{g^2-a^2}\)

No vote yet

1 vote

×

Problem Loading...

Note Loading...

Set Loading...

Easy Math Editor

`*italics*`

or`_italics_`

italics`**bold**`

or`__bold__`

boldNote: you must add a full line of space before and after lists for them to show up correctlyparagraph 1

paragraph 2

`[example link](https://brilliant.org)`

`> This is a quote`

Remember to wrap math in \( ... \) or \[ ... \] to ensure proper formatting.`2 \times 3`

`2^{34}`

`a_{i-1}`

`\frac{2}{3}`

`\sqrt{2}`

`\sum_{i=1}^3`

`\sin \theta`

`\boxed{123}`

## Comments

Sort by:

TopNewestNo perfect cuboids have been found with side lengths under \(10^{10}\). Not to discourage you, but things like these are unsolved for a reason.

Log in to reply

I know... Thats why im trying some other methods...

Log in to reply

Also, see that d^2+e^2+f^2=2*g^2. So if a,b,c is a perfect cuboid combination, then it is not possible for d,e,f to also form a perfect cuboid combination. Where d,e,f are the sides of a new cuboid and d^2+e^2+f^2=L^2 such that L is the space diagonal.

Log in to reply

I think it is conjecture that four dimensional Euler Bricks have no solution but a three dimensional Euler Brick has solutions smallest one is(44,117,240),where face diagonals are 125,244,267.This problem not fully similar to Euler Brick as Euler Brick's Diophantine equations was was the first the three equations you written not the fourth one a^2+b^2+c^2=g^2

Log in to reply

Sorry, I confused the names between an Euler Brick and a perfect cuboid.

Log in to reply

Yes it is okay,they are almost similar but not fully similar.

Log in to reply