Problem Analysis #3

[THOUGHT] Let $p_k = a^k+b^k+c^k$. We are given $p_1=151$, $p_2=10939$, and $p_3=957871$. Let $e_1 = a+b+c$, $e_2 = ab+bc+ca$, and $e_3 = abc$. We have the following Newton's sums: $p_1 = e_1$ $p_2 = e_1 p_1 - 2e_2$ $p_3 = e_1 p_2 - e_2 p_1 + 3e_3$ From the given equations: $e_1 = p_1 = 151$ $p_2 = e_1 p_1 - 2e_2 \Rightarrow 10939 = 151^2 - 2e_2 \Rightarrow 10939 = 22801 - 2e_2 \Rightarrow 2e_2 = 22801-10939=11862 \Rightarrow e_2 = 5931$ $p_3 = e_1 p_2 - e_2 p_1 + 3e_3 \Rightarrow 957871 = 151(10939) - 5931(151) + 3e_3 \Rightarrow 957871 = 1651789 - 895581 + 3e_3 \Rightarrow 957871 = 756208 + 3e_3 \Rightarrow 3e_3 = 957871 - 756208 = 201663 \Rightarrow e_3 = 67221$ Now we want to find $a^4+b^4+c^4 = p_4$. We have the recurrence relation: $p_4 = e_1 p_3 - e_2 p_2 + e_3 p_1$ $p_4 = 151(957871) - 5931(10939) + 67221(151)$ $p_4 = 144638521 - 64897000 + 10150371$ $p_4 = 89866521 + 10150371$ $p_4 = 100016892$

Now we verify the solution. We have $a+b+c=151$, $ab+bc+ca = 5931$ and $abc=67221$. Then $a,b,c$ are the roots of the cubic equation: $x^3 - 151 x^2 + 5931 x - 67221 = 0$ We can try integer roots. Since $abc=67221 = 3 \cdot 22407 = 3^2 \cdot 7469$, we can try $x=3$. $27 - 151(9) + 5931(3) - 67221 = 27-1359+17793-67221 = 17820 - 68580 \neq 0$ Try $x=9$. $729 - 151(81) + 5931(9) - 67221 = 729-12231+53379-67221 = 54108 - 79452 \neq 0$ Let's try to find the roots numerically. The roots are approximately $13, 47, 91$. $13+47+91 = 151$, $13(47)+47(91)+91(13) = 611+4277+1183=6071 \neq 5931$ Something is wrong.

Let's recalculate $e_2$. $p_2 = e_1 p_1 - 2e_2 \Rightarrow 10939 = (151)(151) - 2e_2 \Rightarrow 10939 = 22801 - 2e_2 \Rightarrow 2e_2 = 22801-10939=11862 \Rightarrow e_2 = 5931$ Let's recalculate $e_3$. $p_3 = e_1 p_2 - e_2 p_1 + 3e_3 \Rightarrow 957871 = (151)(10939) - (5931)(151) + 3e_3 \Rightarrow 957871 = 1651789 - 895581 + 3e_3 \Rightarrow 957871 = 756208 + 3e_3 \Rightarrow 3e_3 = 957871 - 756208 = 201663 \Rightarrow e_3 = 67221$ Let's recalculate $p_4$. $p_4 = e_1 p_3 - e_2 p_2 + e_3 p_1 \Rightarrow p_4 = (151)(957871) - (5931)(10939) + (67221)(151)$ $p_4 = 144638521 - 64897000 + 10150371 = 89866521+10150371=100016892$

[PYTHON]

def solve():
    p1 = 151
    p2 = 10939
    p3 = 957871

    e1 = p1
    e2 = (e1*e1 - p2) / 2
    e3 = (p3 - e1*p2 + e2*p1) / 3

    p4 = e1 * p3 - e2 * p2 + e3 * p1
    print(int(p4))

solve()

[THOUGHT] The Python code calculates $p_4 = 89909683$.

\boxed{89909683}