1. **State the problem:** Given the system of equations: $$a^7 + b^7 = n$$ $$a^3 + b^3 = p^3$$ $$m^3 + 2p^3 = 3mn$$ $$a + b = m$$ We want to analyze relationships between these variables. 2. **Express powers in terms of $a$ and $b$:** We know: $$a + b = m$$ $$a^3 + b^3 = p^3$$ $$a^7 + b^7 = n$$ Recall sum of cubes identity: $$a^3 + b^3 = (a+b)(a^2 - ab + b^2)$$ So, $$p^3 = m (a^2 - ab + b^2)$$ 3. **Express $a^2 - ab + b^2$ in terms of $m$ and $ab$:** Since, $$a^2 + b^2 = (a+b)^2 - 2ab = m^2 - 2ab$$ Therefore, $$a^2 - ab + b^2 = a^2 + b^2 - ab = m^2 - 3ab$$ Substituting into $p^3$: $$p^3 = m (m^2 - 3ab) = m^3 - 3m ab$$ So, $$p^3 = m^3 - 3m ab$$ 4. **Rewrite the third equation $m^3 + 2p^3 = 3mn$ using $n = a^7 + b^7$: $$m^3 + 2p^3 = 3m n$$ Substitute $p^3$ in: $$m^3 + 2(m^3 - 3m ab) = 3m n$$ Simplify left side: $$m^3 + 2m^3 - 6 m ab = 3m n$$ $$3m^3 - 6 m ab = 3m n$$ Divide both sides by $3m$ (assuming $m eq 0$): $$m^2 - 2 ab = n$$ 5. **Final relation:** Therefore, $$n = m^2 - 2 ab$$ Given $m = a + b$ and $ab$ is symmetric product, this expresses $n$ in terms of $m$ and $ab$. --- **Summary:** - From the given system, we derived: $$p^3 = m^3 - 3 m ab$$ $$n = m^2 - 2 ab$$ These identities relate the variables $a,b,m,n,p$ under the given constraints. **Note:** Without additional information on $ab$, we cannot solve numerically but have established key polynomial relations.