Two very interesting sums

Note these two sums:

882 + 332 = 8833
122 + 332 = 1233

Square numbers can be defined as all numbers that result from multiplying an integer by itself once.

They have caught the attention of mathematicians for several centuries, generating numerous problems that are difficult to solve. In fact, it can be shown that these are the only four four-digit integers that enjoy this incredible property, making them true celebrities among the rest.

Moreover, its discovery requires the use of advanced tools of the Elementary Number Theory, generally associated with the search for decomposition into prime factors such as the conditions under which it is guaranteed (theorem) that a positive integer can be written by a sum. of two squares.

In this sense, such numbers are not only a simple sum of two distinct squares, but also coincide with the juxtaposition (concatenation) of their square roots that have the same order of magnitude. It is noteworthy that the reader will encounter huge operational difficulties if he decides to look for other numbers with this feature, randomly, or adopting the trial and error strategy, due to the large number of sums to consider in the context of the problem.

Submitted by Prof. Odair José de Freitas