MHB Find the sum of all positive integers a

[anemone]

Find the sum of all positive integers $a$ such that $\sqrt{\sqrt{(a+500)^2-250000}-a}$ is an integer.

 

[Albert1]

Find the sum of all positive integers $a$ such that $\sqrt{\sqrt{(a+500)^2-250000}-a}$ is an integer.

Spoiler

let m=$\sqrt{\sqrt{(a+500)^2-250000}-a}----(*)$

n=$\sqrt{(a+500)^2-250000}< a+500$

$m<\sqrt{500}\,\, or\,\, m\leq 22$---(1)

$n=\sqrt {a(a+1000)}=\sqrt{a(a+2^35^3)}$

min(a)=125=$5^3$

and $\sqrt {250}<m \,\, or \, 16\leq m$----(2)

from (1) and (2) put m=16,17,18,19,20,21,22 to (*)

we get m=20 where $a=800=2^55^2$ is the only solution

 

[anemone]

Spoiler

let m=$\sqrt{\sqrt{(a+500)^2-250000}-a}----(*)$

n=$\sqrt{(a+500)^2-250000}< a+500$

$m<\sqrt{500}\,\, or\,\, m\leq 22$---(1)

$n=\sqrt {a(a+1000)}=\sqrt{a(a+2^35^3)}$

min(a)=125=$5^3$

and $\sqrt {250}<m \,\, or \, 16\leq m$----(2)

from (1) and (2) put m=16,17,18,19,20,21,22 to (*)

we get m=20 where $a=800=2^55^2$ is the only solution

Well done Albert! Thanks for participating.:)