Diophantine rectangles

 For any n, if n squares (tiles) are given, we can lay out a rectangle in them. If n is a prime number, then the only rectangle that can be unloaded is only formally rectangle, in fact, a row.

Meanwhile, it is true that for any of n, this quantity of squares can always be unloaded in a row. Therefore, and because of the usual aesthetic requirements, we could try to lay out a rectangle that most closely resembles a square. It is clear that a square can be laid out if and only if n is itself a square number.

If n is neither a prime nor a square, we can lay out as many different rectangles as the number of proper divisers of n divided by 2 (this number does not include the case of squares placed in a row). It is easy to decide which rectangle is considered to be closest to the square.

A special case if n is a so-called semiprime (i.e., the product of two prime numbers). Then we can unload a single “prime rectangle” from this.

An interesting question is when we can unload another prime rectangle by adding 2 squares to the original quantity.

Several similar, very interesting questions can be raised on this topic. For example, if the height of the desired rectangle is specified in advance, which is the smallest width?

For such questions, a solution formula cannot usually be given, but the results can be well calculated in function of n, in the head or with a program.

We will provide several examples of these soon. But we are also happy to share the results with others.

 

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