System of Sums Of Powers

Algebra Level 2

Given the following system of equations:

$\begin{aligned} x + y + z &= 1\\ x^{2} + y^{2} + z^{2} &= 2\\ x^{3} + y^{3} + z^{3} &= 3, \end{aligned}$

find the smallest positive integer value of $n ~(> 3)$ such that $x^{n} + y^{n} + z^{n}$ is an integer.

The answer is 5.

2 solutions

Chew-Seong Cheong
Jan 19, 2015

Let $S_1 = x + y +z$ , $\space S_2 = xy+yz+zx$ , $\space S_3 = xyz\space$ and $\space P_n = x^n + y^n + z^n$ .

Then, using Newton's Sums method:

$\begin{cases} P_1 = S_1 = 1 & & \\ P_2 = S_1P_1 - 2S_2 & \Rightarrow 2 = 1(1) - 2S_2 & \Rightarrow S_2 = - \frac {1}{2} \\ P_3 = S_1P_2 - S_2P_1 + 3S_3 & \Rightarrow 3 = 1(2) + \frac {1}{2}(1) +3S_3 & \Rightarrow S_3 = \frac {1}{6} \\ P_4 = S_1P_3 - S_2P_2 + S_3P_1 & = 1(3) + \frac {1}{2}(3) +\frac {1}{6}(1) & = \frac {25}{6} \\ P_5 = S_1P_4 - S_2P_3 + S_3P_2 & = 1(\frac{25}{6}) + \frac {1}{2}(3) +\frac {1}{6}(2) & = \frac {25+9+2}{6} = 6 \end {cases}$

Therefore the smallest $n > 3$ such that $x^n+y^n+z^n$ is an integer is $n = \boxed{5}$

suppose if the answer would have been 10 then one have to calculate till 10. isn't there an easy solution

Dheeraj Agarwal - 6 years, 4 months ago

i also have this doubt

nikhil jaiswal - 6 years, 4 months ago

Not that I know of. If we know $S_1$ , $S_2$ and $S_3$ , we can easily compute $P_{10}$ with an Excel spreadsheet or other programming software.

Chew-Seong Cheong - 4 years, 2 months ago

There are 3 equations . And so the elementary symmetric poly need to be taken of order 3 .that is e3. So as your question the ans must not be 10. As we then we need e of order greater than 3. . Which can to be obtained from a poly of degree 3 . So the the problem will run only to p5. . The higher order can not be obtained

Swastik Sanyal - 1 year, 5 months ago

Nope, $n$ can be any positive integer.

Chew-Seong Cheong - 6 months, 3 weeks ago

There is a small error in your solution. P2 = 2 not 3 (in the P4 line )

Ankith A Das - 6 years, 1 month ago

Can you explain the 4th and 5th line? How can you get $P_4 = S_1P_3 - S2P_2 + S_3P_1$ and $P_5 = S_1P_4 - S_2P_3 + S_3P_2$ ?

Fidel Simanjuntak - 4 years, 3 months ago

It is Newton's Sums method . For 3 variables $x, y, z$ , when $n>3$ , $P_n = S_1P_{n-1}-S_2P_{n-2}+S_3P_{n-3}$

Chew-Seong Cheong - 4 years, 2 months ago

If we have 4 variables, does the equation become $P_n = S_1P_{n-1} - S_2P_{n-2} + S_3P_{n-3} - S_4P_{n-4}$ for $n>4$ ?

Fidel Simanjuntak - 4 years, 2 months ago

@Fidel Simanjuntak – Yes, if $n > 4$ . $P_1$ , $P_2$ , and $P_3$ follows that of 3 variables but $P_4 = S_1P_3-S_2P_2 + S_3P_1 - 4S_4$ .

Chew-Seong Cheong - 4 years, 2 months ago

@Chew-Seong Cheong – Thanks a lot! I understand it now. Thanks, sir.

Fidel Simanjuntak - 4 years, 2 months ago

I think a good way to solve the problem is to find the regular between dofferent answer

Mingrui Li - 6 months, 3 weeks ago

Galav Kapoor
May 9, 2016

Let (LaTeX: $\ a_n =x^n+y^n+z^n a_n=a_{n-1}+\frac{a_{n-2}}{2}+\frac{a_{n-3}}{6}$ . Rest is computational.

System of Sums Of Powers

The answer is 5.

2 solutions

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