MHB Find all pairs (m,k) of integer solutions

  • Thread starter Thread starter anemone
  • Start date Start date
  • Tags Tags
    Integer
AI Thread Summary
The discussion focuses on finding integer solutions to the equation (m^2+k)(m+k^2)=(m-k)^3. After expanding and simplifying, a quadratic equation in k is derived, leading to the discriminant condition that must be a perfect square. The analysis reveals that m can take specific values, resulting in three solutions: (m,k) = (9,-21), (9,-6), and (-1,-1). Additionally, a missed solution (m,k) = (8,-10) is identified, highlighting the importance of considering all factorization scenarios. The conversation emphasizes the mathematical reasoning behind determining valid integer pairs.
anemone
Gold Member
MHB
POTW Director
Messages
3,851
Reaction score
115
Find all solutions to $(m^2+k)(m+k^2)=(m-k)^3$, where $m$ and $k$ are non-zero integers.
 
Mathematics news on Phys.org
anemone said:
Find all solutions to $(m^2+k)(m+k^2)=(m-k)^3$, where $m$ and $k$ are non-zero integers.
[sp]Multiply out the brackets: $$m^3 + m^2k^2 + mk + k^3 = m^3 -3m^2k + 3mk^2 - k^3$$, which (after dividing by the non-zero $k$) simplifies to $2k^2 + (m^2-3m)k + (m+3m^2) = 0.$ The discriminant of that quadratic in $k$ is $(m^2-3m)^2 - 8(m+3m^2)$, and that has to be a square, say $n^2$. So $$n^2 = (m^2-3m)^2 - 8(m+3m^2) = m^4 - 6m^3 - 15m^2 - 8m = m(m-8)(m+1)^2$$ and therefore $m(m-8) = (m-4)^2 - 16$ must also be a square. The only way that can happen is if $m-4 = \pm5$. Thus $m = 9$ or $-1$. If $m=9$ then the quadratic for $k$ has solutions $k= -21$ and $k=-6$. If $m=-1$ then the only solution for $k$ is $k=-1$.

That gives three possible solutions, $(m,k) = (9,-21),\ (9,-6),\ (-1,-1)$.[/sp]
 
Opalg said:
therefore $m(m-8) = (m-4)^2 - 16$ must also be a square. The only way that can happen is if $m-4 = \pm5$.

Why is that?
 
I like Serena said:
Opalg said:
therefore $m(m-8) = (m-4)^2 - 16$ must also be a square. The only way that can happen is if $m-4 = \pm5$.
Why is that?
If $x^2-16 = y^2$ then $16 = x^2-y^2 = (x+y)(x-y)$. The only factorisation of $16$ giving positive integer values for $x$ and $y$ is $x+y=8$, $x-y=2$, so that $x=5$ and $y=3$.
 
Thank you so much for participating, Opalg! And welcome back to the forum as well!:) But Opalg, I am sorry to tell you that you missed another solution set, which is $(m, k)=(8, -10)$.

My solution:
By expanding and simplifying the given equation, we get

$$m^3+m^2k^2+mk+k^3=m^3-3m^2k+3mk^2-k^3$$

$$2k^3+(m^2-3m)k^2+mk+3m^2k=0$$

Since $k \ne 0$, divide through the equation above by $k$, we have

$$2k^2+(m^2-3m)k+m+3m^2=0$$

and solve for $k$ using the quadratic formula yields

$$k=\frac{-(m^2-3m) \pm \sqrt{(m^2-3m)^2-4(2)(m+3m^2)}}{2(2)}$$

$$\;\;\;=\frac{3m-m^2 \pm \sqrt{(m+1)^2(m(m-8))}}{4}$$

$$\;\;\;=\frac{3m-m^2 \pm (m+1)\sqrt{m(m-8)}}{4}$$

Recall that the expression inside the radical must be greater than or equal to zero, thus, $$m(m-8)\ge 0$$ and this gives

$m\le 0$ and $m \ge 8$.

Also, bear in mind that we're told $m, k$ are both integer values, so, $m(m-8)$ has to be a squre.

But if we apply the AM-GM inequality to the terms $m$ and $m-8$, we find

$$\frac{m+m-8}{2} \ge \sqrt{m(m-8)}$$

$$m-4 \ge \sqrt{m(m-8)}$$

$m\ge 8$ implies $m-4 \ge 4$ and this further implies $\sqrt{m(m-8)} \le 4$ and now, we will solve for $m$ by considering $\sqrt{m(m-8)}=0$ or $1$ or $2$ or $3$or $4$ and we will take whichever solution that gives the integer values of $m$.

$$\sqrt{m(m-8)}= 4$$ gives irrational $m$ values.

$$\sqrt{m(m-8)}= 3$$ gives $m=-1, 9$.

$$\sqrt{m(m-8)}= 2$$ gives irrational $m$ values.

$$\sqrt{m(m-8)}= 1$$ gives irrational $m$ values.

$$\sqrt{m(m-8)}= 0$$ gives irrational $m=0, 8$ values but $m>0$, hence $m=8$.

Thus, by substituting each of the value of $m$ to the equation (*) above to find for its corresponding $k$ value, we end up with the following 4 solution sets, namely

$(m, k)=(-1, -1), (8, -10), (9, -6), (9, -21)$
 
I like Serena said:
Opalg said:
therefore $m(m-8) = (m-4)^2 - 16$ must also be a square. The only way that can happen is if $m-4 = \pm5$.
Why is that?
Oops, I should have taken I like Serena's query more seriously. I was thinking that the only way you could have $x^2-16 = y^2$ was in the situation $5^2 - 4^2 = 3^2$. But in the solution to this problem, the situation $4^2 - 4^2 = 0^2$ is also possible. That leads to the solution $m=8$ and $k=-10$.
 
Suppose ,instead of the usual x,y coordinate system with an I basis vector along the x -axis and a corresponding j basis vector along the y-axis we instead have a different pair of basis vectors ,call them e and f along their respective axes. I have seen that this is an important subject in maths My question is what physical applications does such a model apply to? I am asking here because I have devoted quite a lot of time in the past to understanding convectors and the dual...
Insights auto threads is broken atm, so I'm manually creating these for new Insight articles. In Dirac’s Principles of Quantum Mechanics published in 1930 he introduced a “convenient notation” he referred to as a “delta function” which he treated as a continuum analog to the discrete Kronecker delta. The Kronecker delta is simply the indexed components of the identity operator in matrix algebra Source: https://www.physicsforums.com/insights/what-exactly-is-diracs-delta-function/ by...

Similar threads

Back
Top