Perfect numbers

Number Theory Level 3

Given that $m^3 - m^2 + n^3 - n^2$ is a perfect number.

$m$ and $n$ are distinct positive integers. What is the minimum positive value of $m + n?$

The answer is 12.

1 solution

Nazmus Sakib
Nov 17, 2017

$\large4^3-4^{2}+8^3-8^{2}=496(\text{perfect number})$

so, $m=4$ and $n=8$

so,the value of $m+n$ is $12.$

How do you know that there isn't a pair $(m,n)$ whose sum is smaller than 12 and also satisfies this condition?

Pi Han Goh - 3 years, 6 months ago

I was doing this repeatedly by m=1,2,3,4,5,6,7,8,9,10 and n=10,9,4,8,7,6,5,4.. then I suddenly found 496 by m=8 and n=4..then I put it on the submit box.And I found it correct.

Nazmus sakib - 3 years, 6 months ago

If we write the perfect number's series, we get, $6,24,496,8128.........$ . IF I take $m^3 - m^2 + n^3 - n^2=6$ or $m^3 - m^2 + n^3 - n^2=24$ then we can't get any value of $m,n$ if I take $m^3 - m^2 + n^3 - n^2=496$ then we only get $(m,n)=(4,8),(8,4)$ . For other value it does't satisfy the equation. So I think there are no any value less than $12$

Md Mehedi Hasan - 3 years, 6 months ago

Perfect numbers

The answer is 12.

1 solution

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