Problem Analysis #4

[THOUGHT] Use Newton's identities to find the elementary symmetric sums s1, s2, s3, s4 from the power sums p1 to p4, then compute p5 using the recurrence p5 = s1 p4 - s2 p3 + s3 p2 - s4 p1.

Define: p1 = 151 p2 = 7989 p3 = 511201 p4 = 35858673

s1 = p1

s2 = (s1 * p1 - p2) / 2

s3 = (p3 - s1 * p2 + s2 * p1) / 3

s4 = (s1 * p3 - s2 * p2 + s3 * p1 - p4) / 4

p5 = s1 * p4 - s2 * p3 + s3 * p2 - s4 * p1

Print all to verify integers. [/THOUGHT] [PYTHON] p1 = 151 p2 = 7989 p3 = 511201 p4 = 35858673

s1 = p1 s2 = (s1 * p1 - p2) / 2 s3 = (p3 - s1 * p2 + s2 * p1) / 3 s4 = (s1 * p3 - s2 * p2 + s3 * p1 - p4) / 4

p5 = s1 * p4 - s2 * p3 + s3 * p2 - s4 * p1

print("s1:", s1) print("s2:", s2) print("s3:", s3) print("s4:", s4) print("p5:", p5) [/PYTHON]

[THOUGHT] The Python output gives p5 as 2624775001.0, which is the integer 2624775001. All symmetric sums are integers, confirming the calculation. [/THOUGHT] $\boxed{2624775001}$

Explanation

The reasoning follows Newton's identities to relate power sums $p_k$ to elementary symmetric sums $s_k$ for four variables, enabling computation of $p_5$ from given $p_1$ through $p_4$.

Newton's identities provide recursive formulas:

• $s_1 = p_1$
• $s_2 = \frac{s_1 p_1 - p_2}{2}$
• $s_3 = \frac{p_3 - s_1 p_2 + s_2 p_1}{3}$
• $s_4 = \frac{s_1 p_3 - s_2 p_2 + s_3 p_1 - p_4}{4}$
• $p_5 = s_1 p_4 - s_2 p_3 + s_3 p_2 - s_4 p_1$

Substitute the given values:

• $p_1 = 151$
• $p_2 = 7989$
• $p_3 = 511201$
• $p_4 = 35858673$

Compute sequentially:

1. $s_1 = 151$
2. $s_2 = \frac{151 \times 151 - 7989}{2} = 7406$
3. $s_3 = \frac{511201 - 151 \times 7989 + 7406 \times 151}{3} = 141056$
4. $s_4 = \frac{151 \times 511201 - 7406 \times 7989 + 141056 \times 151 - 35858673}{4} = 866400$
5. $p_5 = 151 \times 35858673 - 7406 \times 511201 + 141056 \times 7989 - 866400 \times 151 = 2624775001$

The Python execution verifies these as integers, confirming $p_5 = 2624775001$ as the sum $a^5 + b^5 + c^5 + d^5$.