Problem Analysis #3

[THOUGHT]
We are given the symmetric sums of three natural numbers $a, b, c$:

$$\begin{aligned} S_1 &= a + b + c = 151 \\ S_2 &= a^2 + b^2 + c^2 = 10939 \\ S_3 &= a^3 + b^3 + c^3 = 957871 \end{aligned}$$

We are to find $a^4 + b^4 + c^4$.

We can use Newton's identities to relate power sums to elementary symmetric sums.

Let:

• $e_1 = a + b + c = S_1 = 151$
• $e_2 = ab + bc + ca$
• $e_3 = abc$

We know:

$$a^2 + b^2 + c^2 = (a + b + c)^2 - 2(ab + bc + ca) \Rightarrow S_2 = e_1^2 - 2e_2$$

So:

$$10939 = 151^2 - 2e_2 \Rightarrow 10939 = 22801 - 2e_2 \Rightarrow 2e_2 = 22801 - 10939 = 11862 \Rightarrow e_2 = 5931$$

Now use Newton's identity for $S_3$:

$$S_3 = a^3 + b^3 + c^3 = e_1^3 - 3e_1e_2 + 3e_3$$

Plug in known values:

$$957871 = 151^3 - 3 \cdot 151 \cdot 5931 + 3e_3$$

Compute $151^3$:

$$151^2 = 22801 \\ 151^3 = 151 \cdot 22801 = 3,442,951$$

Now:

$$957871 = 3,442,951 - 3 \cdot 151 \cdot 5931 + 3e_3$$

Compute $3 \cdot 151 \cdot 5931$: First, $151 \cdot 5931$:

$$151 \cdot 5931 = 151 \cdot (6000 - 69) = 151 \cdot 6000 - 151 \cdot 69 = 906000 - 10419 = 895,581$$

Then $3 \cdot 895,581 = 2,686,743$

So:

$$957871 = 3,442,951 - 2,686,743 + 3e_3 \\ \Rightarrow 957871 = 756,208 + 3e_3 \\ \Rightarrow 3e_3 = 957871 - 756208 = 201,663 \\ \Rightarrow e_3 = 67,221$$

Now we have:

• $e_1 = 151$
• $e_2 = 5931$
• $e_3 = 67221$

We now use Newton’s identity for $S_4 = a^4 + b^4 + c^4$:

$$S_4 = e_1 S_3 - e_2 S_2 + e_3 S_1$$

Wait — actually, Newton's sums are:

For $n = 4$:

$$S_4 = e_1 S_3 - e_2 S_2 + e_3 S_1$$

Wait — correction: the correct Newton's identity for $S_4$ is:

$$S_4 = e_1 S_3 - e_2 S_2 + e_3 S_1$$

Wait — actually, the general form is:

$$S_4 = e_1 S_3 - e_2 S_2 + e_3 S_1 - 4e_4$$

But since we have only 3 variables, $e_4 = 0$, so:

$$S_4 = e_1 S_3 - e_2 S_2 + e_3 S_1$$

Yes.

So:

$$S_4 = 151 \cdot 957871 - 5931 \cdot 10939 + 67221 \cdot 151$$

We compute each term.

First: $151 \cdot 957871$

Break it down:

$$150 \cdot 957871 = 143,680,650 \\ 1 \cdot 957871 = 957,871 \\ \Rightarrow 151 \cdot 957871 = 143,680,650 + 957,871 = 144,638,521$$

Second: $5931 \cdot 10939$

Compute: Let’s compute $5931 \cdot 10939$

We can write:

$$5931 \cdot 10939 = 5931 \cdot (11000 - 61) = 5931 \cdot 11000 - 5931 \cdot 61$$

First: $5931 \cdot 11000 = 5931 \cdot 11 \cdot 1000 = (65241) \cdot 1000 = 65,241,000$

Second: $5931 \cdot 61$

$5931 \cdot 60 = 355,860$, $5931 \cdot 1 = 5931$, total = $355,860 + 5,931 = 361,791$

So:

$$5931 \cdot 10939 = 65,241,000 - 361,791 = 64,879,209$$

Third: $67221 \cdot 151$

$67221 \cdot 150 = 67221 \cdot 100 + 67221 \cdot 50 = 6,722,100 + 3,361,050 = 10,083,150$

$67221 \cdot 1 = 67,221$

Total: $10,083,150 + 67,221 = 10,150,371$

Now plug into:

$$S_4 = 144,638,521 - 64,879,209 + 10,150,371$$

First: $144,638,521 - 64,879,209 = 79,759,312$

Then: $79,759,312 + 10,150,371 = 89,909,683$

So:

$$a^4 + b^4 + c^4 = 89,909,683$$

We can verify that $a, b, c$ are natural numbers by solving the cubic:

$$x^3 - e_1 x^2 + e_2 x - e_3 = 0 \\ x^3 - 151x^2 + 5931x - 67221 = 0$$

Try rational roots: factors of 67221 over 1. Try small values.

But since the problem says "natural numbers" and the symmetric sums are integers, and we followed Newton’s identities correctly, the result should be valid.

Thus, the answer is $\boxed{89909683}$

[PYTHON]

# Given values
S1 = 151
S2 = 10939
S3 = 957871

# Compute e2 = ab + bc + ca
e2 = (S1**2 - S2) // 2

# Compute e3 = abc
# Using S3 = e1^3 - 3*e1*e2 + 3*e3
e3 = (S3 - S1**3 + 3*S1*e2) // 3

# Compute S4 = e1*S3 - e2*S2 + e3*S1
S4 = S1 * S3 - e2 * S2 + e3 * S1

print(S4)

[THOUGHT]
The Python code executed successfully and returned the value 89909683. This is the computed value of $a^4 + b^4 + c^4$ based on the given symmetric sums and Newton's identities. The result is consistent with the algebraic derivation and confirmed by direct computation.

Thus, the final answer is:

\boxed{89909683}