An easy problem for... Leonhard EULER // Ein einfaches Problem für... Leonhard Euler

 I've messed up my blogs (petrell-man and/or mathematyka2), and none of them know how to type in LaTeX. And then I made a third one matemattica0, where a shy LaTeX form goes. 

              EULER is famous for elucidating Fermat's number  $F_5=2^{2^5}+1=641 \times 6700417$. He is the one who saw that  $641=2^7\times 5+1=2^4+5^4$.

That's why I thought that only he could help me with the problem:

              "1. (page 51)  We will call a positive integer  $n\;cool$  if it is a perfect square and there exists positive integers  $a,\;b,\;c,\;d$  different from each other, with  $a>b\;and\;c>d$ ,  such that  $n=a^3-b^3+c^3-d^3$.  For instance,  225  is cool because  $225=15^2\;\;and\;\;225=7^3-5^3+2^3-1^3$.  Show that :

a)  2025 is cool ;

b)  there are infinitely many cool numbers.

(Relu CIUPEA)" 

ANSWER CiP

a) $2025=45^2=11^3-3^3+9^3-2^3=16^3-15^3+11^3-3^3$

b)  $225\cdot k^6=(15k^3)^2=(7k^2)^3-(5k^2)^3+(2k^2)^3-(k^2)^3,\;k\in\mathbb{N}^*$ 

Solution CiP

                   a) was solved by ChatGPT. I chose the simplest of several equalities.

                    b) We multiply the example provided in the statement by  $k^6$.

$\blacksquare$

                   An infinite number of remarks can be made, but I still haven't managed to find a simple way, accessible to a junior, to achieve the result.

            At first I thought it was easy, starting from

$2025=9\cdot 225=(2^3+1^3)(7^3-5^3+2^3-1^3)=14^3-10^3+7^3-5^3+4^3-1^3.$

But there is no further path to the result in sight.

             Let's write 225 differently then. We know  $225=1^3+2^3+3^3+4^3+5^3+6^3$, and $3^3+4^3+5^3=6^3$, so  $225=1^3+2^3+6^3$  but even that doesn't get us where we want to go.