Problem Analysis #3

[PYTHON] import sympy

def solve_problem(): S1 = 151 S2 = 10939 S3 = 957871

# Calculate elementary symmetric polynomials e1, e2, e3
# S1 = e1
e1 = S1

# S2 - e1*S1 + 2*e2 = 0 => 2*e2 = e1*S1 - S2
e2 = (e1 * S1 - S2) // 2

# S3 - e1*S2 + e2*S1 - 3*e3 = 0 => 3*e3 = S3 - e1*S2 + e2*S1
e3 = (S3 - e1 * S2 + e2 * S1) // 3

# Calculate S4
# S4 - e1*S3 + e2*S2 - e3*S1 = 0 => S4 = e1*S3 - e2*S2 + e3*S1
S4 = e1 * S3 - e2 * S2 + e3 * S1

print(f"{e1=}")
print(f"{e2=}")
print(f"{e3=}")
print(f"{S4=}")

# Verify if roots are natural numbers
x = sympy.symbols('x')
cubic_eq = x**3 - e1*x**2 + e2*x - e3
roots = sympy.solve(cubic_eq, x)
print(f"Roots: {roots}")

# Double check S4 calculation using roots
check_S4 = sum([r**4 for r in roots])
print(f"Check S4: {check_S4}")

solve_problem() [/PYTHON][PYTHON] import sympy

def solve_problem(): S1 = 151 S2 = 10939 S3 = 957871

# e1 = S1
e1 = S1

# 2*e2 = e1*S1 - S2
e2 = (e1 * S1 - S2) // 2

# 3*e3 = S3 - e1*S2 + e2*S1
# Note: S3 - e1*S2 + e2*S1 - 3*e3 = 0
# So 3*e3 = S3 - e1*S2 + e2*S1
# But wait, usually the identity is P_k - e_1 P_{k-1} + ... + (-1)^{k-1} k e_k = 0?
# Let's check Newton sums carefully.
# P1 - e1 = 0
# P2 - e1 P1 + 2e2 = 0
# P3 - e1 P2 + e2 P1 - 3e3 = 0
# P4 - e1 P3 + e2 P2 - e3 P1 + 4e4 = 0
# Yes, signs alternate.

val_for_3e3 = S3 - e1 * S2 + e2 * S1
e3 = val_for_3e3 // 3

print(f"e1: {e1}")
print(f"e2: {e2}")
print(f"e3: {e3}")

# Check if e3 is integer
if val_for_3e3 % 3 != 0:
    print("Warning: e3 is not an integer")
    
# Solve for roots
x = sympy.symbols('x')
poly = x**3 - e1*x**2 + e2*x - e3
roots = sympy.solve(poly, x)
print(f"Roots: {roots}")

# Calculate S4
# S4 - e1 S3 + e2 S2 - e3 S1 = 0
# S4 = e1 S3 - e2 S2 + e3 S1
S4 = e1 * S3 - e2 * S2 + e3 * S1
print(f"S4: {S4}")

solve_problem() [/PYTHON][PYTHON] import sympy

def analyze_roots(): S1 = 151 S2 = 10939 S3 = 957871

e1 = S1
e2 = (e1**2 - S2) // 2
e3 = (S3 - e1*S2 + e2*S1) // 3

print(f"Coefficients: e1={e1}, e2={e2}, e3={e3}")

x = sympy.symbols('x')
poly = x**3 - e1*x**2 + e2*x - e3
roots = sympy.solve(poly, x)
print(f"Symbolic Roots: {roots}")

# Numerical roots
n_roots = sympy.nroots(poly)
print(f"Numerical Roots: {n_roots}")

# Check if roots are integers
is_integer = True
for r in n_roots:
    if abs(r - round(r)) > 1e-9:
        is_integer = False

if is_integer:
    int_roots = [int(round(r)) for r in n_roots]
    print(f"Integer Roots: {int_roots}")
    # Verify sums
    s1_calc = sum(int_roots)
    s2_calc = sum(r**2 for r in int_roots)
    s3_calc = sum(r**3 for r in int_roots)
    print(f"Calculated S1: {s1_calc} (Expected {S1})")
    print(f"Calculated S2: {s2_calc} (Expected {S2})")
    print(f"Calculated S3: {s3_calc} (Expected {S3})")
    
    s4 = sum(r**4 for r in int_roots)
    print(f"Calculated S4 from roots: {s4}")
    
# Calculate S4 using Newton sums
S4_newton = e1*S3 - e2*S2 + e3*S1
print(f"S4 from Newton Sums: {S4_newton}")

analyze_roots() [/PYTHON][PYTHON] def solve(): S1 = 151 S2 = 10939 S3 = 957871

e1 = S1
e2 = (e1**2 - S2) // 2
e3 = (S3 - e1*S2 + e2*S1) // 3

print(f"e1: {e1}")
print(f"e2: {e2}")
print(f"e3: {e3}")

S4 = e1*S3 - e2*S2 + e3*S1
print(f"S4 (Newton): {S4}")

# Roots check
roots = [21, 33, 97]
print(f"Roots: {roots}")
print(f"Sum: {sum(roots)}")
print(f"Sum sq: {sum(x**2 for x in roots)}")
print(f"Sum cu: {sum(x**3 for x in roots)}")
print(f"Sum 4th: {sum(x**4 for x in roots)}")

solve() [/PYTHON]The given system of equations is: [ \begin{cases} a+b+c = 151 \ a^{2}+b^{2}+c^{2} = 10939 \ a^{3}+b^{3}+c^{3} = 957871 \end{cases} ] Let $e_1, e_2, e_3$ be the elementary symmetric polynomials in $a, b, c$. Using Newton's sums, we have:

1. $e_1 = a+b+c = 151$.
2. $S_2 - e_1 S_1 + 2e_2 = 0 \implies 2e_2 = e_1^2 - S_2 = 151^2 - 10939 = 22801 - 10939 = 11862$. Thus, $e_2 = 5931$.
3. $S_3 - e_1 S_2 + e_2 S_1 - 3e_3 = 0 \implies 3e_3 = S_3 - e_1 S_2 + e_2 S_1$. Substituting the values: $3e_3 = 957871 - 151(10939) + 5931(151)$ $3e_3 = 957871 - 1651789 + 895581$ $3e_3 = 201663$ $e_3 = 67221$.

The numbers $a, b, c$ are the roots of the cubic equation $x^3 - e_1 x^2 + e_2 x - e_3 = 0$: [ x^3 - 151x^2 + 5931x - 67221 = 0 ] Solving this cubic equation yields the roots $\{21, 33, 97\}$. These are all natural numbers.

We want to find $S_4 = a^4 + b^4 + c^4$. Using Newton's sums again: $S_4 - e_1 S_3 + e_2 S_2 - e_3 S_1 = 0 \implies S_4 = e_1 S_3 - e_2 S_2 + e_3 S_1$. Substituting the values: $S_4 = 151(957871) - 5931(10939) + 67221(151)$ $S_4 = 144638521 - 64879209 + 10150371$ $S_4 = 89909683$.

Alternatively, calculating directly from the roots: $21^4 + 33^4 + 97^4 = 194481 + 1185921 + 88529281 = 89909683$.

\boxed{89909683}

The system of equations is given by: [ \begin{cases} 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{cases} ] We want to find $S_4 = a^4 + b^4 + c^4$.

Using Newton's sums relations for elementary symmetric polynomials $e_1, e_2, e_3$:

1. $e_1 = S_1 = 151$.
2. $S_2 - e_1 S_1 + 2e_2 = 0 \implies 2e_2 = e_1^2 - S_2 = 151^2 - 10939 = 22801 - 10939 = 11862 \implies e_2 = 5931$.
3. $S_3 - e_1 S_2 + e_2 S_1 - 3e_3 = 0 \implies 3e_3 = S_3 - e_1 S_2 + e_2 S_1$. Substituting the values: $3e_3 = 957871 - 151(10939) + 5931(151) = 957871 - 1651789 + 895581 = 201663$. $e_3 = 67221$.

The values $a, b, c$ are roots of $x^3 - 151x^2 + 5931x - 67221 = 0$, which are $\{21, 33, 97\}$.

To find $S_4$, we use the next Newton sum relation: $S_4 - e_1 S_3 + e_2 S_2 - e_3 S_1 = 0 \implies S_4 = e_1 S_3 - e_2 S_2 + e_3 S_1$. $S_4 = 151(957871) - 5931(10939) + 67221(151)$ $S_4 = 144638521 - 64879209 + 10150371$ $S_4 = 89909683$.

Alternatively, calculating directly: $21^4 + 33^4 + 97^4 = 194481 + 1185921 + 88529281 = 89909683$.

\boxed{89909683}