Sum of powers

Computer Science Level 5

A positive integer $n$ satisfies:

$n = (a_1)^2 + (a_2)^2$
$n = (b_1)^3 + (b_2)^3 + (b_3)^3$
$n = (c_1)^4 + (c_2)^4 + (c_3)^4 + (c_4)^4$
$n = (d_1)^5 + (d_2)^5 + (d_3)^5 + (d_4)^5 + (d_5)^5$

where all the variables $n, a_1, a_2, b_1, b_2, b_3, c_1, c_2, c_3, c_4, d_1, d_2, d_3, d_4, d_5$ are distinct positive integers.

Find the smallest such $n$ .

The answer is 76913.

2 solutions

Vikram Pandya
May 23, 2016

For $n=5$ we have $1^5+2^5+4^5+7^5+9^5 = 3^4+6^4+10^4+16^4 = 17^3+20^3+40^3 = 88^2 + 263^2 = 76913$ Note that it is a Prime number

BTW for $n = 3$ we have $1^3+2^3+4^4=3^2+8^2=73$ - this is minimum and again a prime number

for $n=4$ we have $6^4+8^4+9^4+28^4=33^3+34^3+82^3=175^2+772^2 = 626609$ - this is not the minimum but a prime number again

Michael Mendrin
May 23, 2016

Being that this is a "computer science" problem, I solved this through brute force. It's the only solution for $n\le 76913$

Yeah that's right. I have following conjecture about this: There is always a unique way to write down $\sum_{i=1}^{n} a_{i}^n$ for any arbitrary value of $n$ such that it gives same value for all values of $n$

Vikram Pandya - 5 years ago

I do remember you mentioning that some months ago.

I recall now that Ivan Koswara has written a nice exposition on this problem a while back. Interesting Power Equality The difference here, I think, is the condition of having all distinct positive non-zero integers.

Michael Mendrin - 5 years ago

Yeah, thx for referencing my note. I will also see how Ivan's analysis can be extended to this case. I even forgot that I had asked this question before because someone edited my original question's title

Vikram Pandya - 5 years ago

Sum of powers

The answer is 76913.

2 solutions

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