Euler brick: box with edges a,b,c and integer face diagonals. Smallest: (44,117,240). All three face diagonals d_ab=√(a²+b²), d_ac=√(a²+c²), d_bc=√(b²+c²) must be integers. The perfect Euler brick (space diagonal √(a²+b²+c²) also integer) is an unsolved problem.
How to Use
Enter edges (a,b,c):
- Diagonals: Face and space
- Integer?: Check each diagonal
- Perfect?: All four integer
Perfect Brick
No perfect Euler brick has ever been found! If a,b,c and ALL FOUR diagonals (3 face + 1 space) are integers, it would be perfect. Exhaustive search up to 10^12 found none. Equivalent to finding specific systems of Diophantine equations.
Parametric Families
Saunderson (1740) gave the parametric solution: a=u(4v²−w²), b=v(4u²−w²), c=4uvw where (u,v,w) is a Pythagorean triple. This generates infinitely many Euler bricks but not all.
Step-by-Step Instructions
- 1Enter three edges.
- 2Check face diagonals.
- 3Check space diagonal.
- 4Test perfection.
- 5Try parametric.