[THOUGHT] We need to find a^4 + b^4 + c^4 given:

1. a + b + c = 151
2. a^2 + b^2 + c^2 = 10939
3. a^3 + b^3 + c^3 = 957871

We can use symmetric polynomial identities: Let p1 = a + b + c = 151 p2 = a^2 + b^2 + c^2 = 10939 p3 = a^3 + b^3 + c^3 = 957871

We can compute elementary symmetric sums: Let s1 = a + b + c = p1 = 151 Let s2 = ab + bc + ca Let s3 = abc

We know: p2 = a^2 + b^2 + c^2 = (a+b+c)^2 - 2(ab+bc+ca) = s1^2 - 2 s2 So s2 = (s1^2 - p2)/2

Similarly: p3 = a^3 + b^3 + c^3 = (a+b+c)^3 - 3(a+b+c)(ab+bc+ca) + 3abc = s1^3 - 3 s1 s2 + 3 s3 So s3 = (p3 - s1^3 + 3 s1 s2)/3

Then we can compute p4 = a^4 + b^4 + c^4 using recurrence or Newton's identities: Newton's identities: p1 = s1 p2 = s1 p1 - 2 s2 p3 = s1 p2 - s2 p1 + 3 s3 p4 = s1 p3 - s2 p2 + s3 p1

But careful: Newton's identities for power sums p_k = sum a_i^k and elementary symmetric sums e_k: p1 = e1 p2 = e1 p1 - 2 e2 p3 = e1 p2 - e2 p1 + 3 e3 p4 = e1 p3 - e2 p2 + e3 p1

Yes, that's correct. So we can compute s2, s3, then p4.

Let's compute with Python to verify.

We'll compute step by step.

[PYTHON] import sympy as sp

Given values

p1 = 151 p2 = 10939 p3 = 957871

Compute s2 = ab + bc + ca

s2 = (p1**2 - p2) / 2

Compute s3 = abc

s3 = (p3 - p1**3 + 3 * p1 * s2) / 3

Now compute p4 = a^4 + b^4 + c^4 using Newton's identities:

p4 = s1 * p3 - s2 * p2 + s3 * s1

p4 = p1 * p3 - s2 * p2 + s3 * p1

print("s2 =", s2) print("s3 =", s3) print("p4 =", p4) [/PYTHON]

[THOUGHT] The Python output shows p4 = 89909683.0, which is an integer. Therefore, a^4 + b^4 + c^4 = 89909683.

\boxed{89909683}