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Abstract If Perfect cuboid exists, the squares of 3 its face diagonals should construct a Heronian triangle. Proof: photo_2021-02-12_10-19-32.jpg Note: Heronian triangle is a triangle that has side lengths and area that are all integers. At the moment it is known about only two Heronian triangles w...

Zak,

Zak,

Zak,

x3mEn I know about you had showed. But I told about other equation whith new integer variable x, which included sqr(16a^2b^2c^2g^2). The equation will be TRUE only in case when d,e,f will correspond perfect brick's face diagonals. By means of transformations and finding the roots of the equation, w...

Decision of equation in integers will be equal to full task, because if we will have d,e,f , then we can find all the rest. We can find g and then we can find a,b,c. You are not right. When you have a system of Diophantine X equations with Y variables, theoretically you can simplify it to an equati...

the fourh exponent is not too big? It's time to talk about how Math works. :) With the help of strict math logic of equivalent expressions you transfered from the given system of 4 Diophantine equations to consequent (and, btw, not necessary equivalent to input system) expression, which says: "If t...

Zak, you are muddling up 'necessary' and 'enough'. If undersquare expression is 2f^2e^2, yes, it's enough to aquire z as a full square of the sum or the difference of f^2 and e^2. But, is it necessary? No. There are a lot of other expressions binding f and e which are a full square too. The world is...

Some ideas regarding the last equation. As far as One edge, two face diagonals and the body diagonal must be odd, one edge and the remaining face diagonal must be divisible by 4, and the remaining edge must be divisible by 16. Two edges must have length divisible by 3 and at least one of those edges...

Zak, no problem, buу Wolfram|Alpha Pro to get the proof.

Zak, before publish anything, please-please-please verify all your ideas on some trivial samples known as Edge and Face cuboids.

Perfect Cuboid 3rd (loooooong) Batch has finished If a perfect cuboid exists, its body diagonal exceeds 2^53. During this batch we were investigating the range of body diagonal from 2^51 (2'251'799'813'685'248) to 2^53 (9'007'199'254'740'992). 17'408 face cuboids, 34'816 imaginary cuboids, 156'672 t...