Problem E : 17 504

Ever since my girlfriend  ALINA left me, I decided to immerse myself in work... 

( I wasn't even able to finish this post, where I have the complete solution. )

.See the comment on this post...

I can't even solve this 8th grade problem...

"E:17504.  Determine the natural numbers  $a\;\; and \;\;b$ , for which the number

  $A=\sqrt{a^2+b+1}+\sqrt{b^2+a+4}$  is natural.

{Author : } Ştefan GOBEJ, student, Curtea de Argeş "

I have not been able to solve this problem in any reasonable time. Two solutions I have found are :

$a=0,\;b=0\;\;:\;A=\sqrt{1}+\sqrt{4}=3\;\;;$

$a=1,\;b=2\;\;:\;A=\sqrt{1+2+1}+\sqrt{4+1+4}=5$

          I will raise the problem on AOPS, and I will let you know the result.

$\square$

Edit 23.07.2026 17:00      An AOPS user, yayokidoki77_-,  confirmed my answer. I asked him for details.

$\square\;\;\square$