Problem Analysis #6

[PYTHON] import sympy

def solve_problem(): # Given power sums p = {} p[1] = 58 p[2] = 666 p[3] = 8944 p[4] = 129846 p[5] = 1972408 p[6] = 30870006 p[7] = 493478344

# We have 7 variables
n = 7

# Newton sums to find elementary symmetric polynomials e_k (which correspond to coefficients c_k)
# Relation: k * e_k + sum_{i=1}^{k-1} (-1)^(k-i) * e_{k-i} * p_i + (-1)^k * p_k = 0 ?
# Standard form: p_k - e_1 p_{k-1} + e_2 p_{k-2} - ... + (-1)^{k-1} e_{k-1} p_1 + (-1)^k k e_k = 0
# Let's use the form for coefficients c_k of the polynomial x^n + c_1 x^{n-1} + ... + c_n = 0
# Then p_k + c_1 p_{k-1} + ... + c_{k-1} p_1 + k c_k = 0

c = {}

for k in range(1, n + 1):
    sum_prev = 0
    for i in range(1, k):
        sum_prev += c[i] * p[k - i]
    # p_k + sum_prev + k * c_k = 0
    c[k] = -(p[k] + sum_prev) // k
    
print("Coefficients c:", c)

# Verify if they are integers (the division // should be exact)
# Let's check remainders just in case
for k in range(1, n + 1):
    sum_prev = 0
    for i in range(1, k):
        sum_prev += c[i] * p[k - i]
    val = -(p[k] + sum_prev)
    if val % k != 0:
        print(f"Warning: c[{k}] is not an integer")
        
# Now compute p_8 to p_12
# Recurrence: p_k + c_1 p_{k-1} + ... + c_n p_{k-n} = 0
# p_k = - sum_{i=1}^n c_i p_{k-i}

target = 12
for k in range(n + 1, target + 1):
    val = 0
    for i in range(1, n + 1):
        val += c[i] * p[k - i]
    p[k] = -val
    
print(f"p_{target} =", p[target])

h = 10
P_12 = p[target] + h**12
print(f"P_{target} =", P_12)

# Optional: Find the roots to verify they are natural numbers
x = sympy.symbols('x')
poly = x**7
for k in range(1, n + 1):
    poly += c[k] * x**(7 - k)
    
roots = sympy.solve(poly, x)
print("Roots:", roots)

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

def solve_problem(): p = { 1: 58, 2: 666, 3: 8944, 4: 129846, 5: 1972408, 6: 30870006, 7: 493478344 } n = 7

# Calculate coefficients c_k for the polynomial x^7 + c_1 x^6 + ... + c_7 = 0
# Newton sums: p_k + c_1 p_{k-1} + ... + c_{k-1} p_1 + k c_k = 0
c = {}
for k in range(1, n + 1):
    term_sum = sum(c[i] * p[k - i] for i in range(1, k))
    # k * c_k = - (p_k + term_sum)
    val = -(p[k] + term_sum)
    if val % k != 0:
        print(f"Warning: c[{k}] is not integer.")
    c[k] = val // k

print("Coefficients:", c)

# Calculate p_8 to p_12
# p_k = - sum(c[i] * p[k-i]) for i=1 to n
for k in range(n + 1, 13):
    val = sum(c[i] * p[k - i] for i in range(1, n + 1))
    p[k] = -val
    
print(f"p_12 = {p[12]}")

h = 10
P_12 = p[12] + h**12
print(f"P_12 = {P_12}")

# Verify roots
x = sympy.symbols('x')
poly = x**7 + sum(c[k] * x**(7 - k) for k in range(1, n + 1))
roots = sympy.solve(poly, x)
print("Roots:", roots)

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

def solve_problem(): p = { 1: 58, 2: 666, 3: 8944, 4: 129846, 5: 1972408, 6: 30870006, 7: 493478344 } n = 7

# Coefficients c_k for x^7 + c_1 x^6 + ... + c_7 = 0
c = {}
for k in range(1, n + 1):
    term_sum = sum(c[i] * p[k - i] for i in range(1, k))
    val = -(p[k] + term_sum)
    # c[k] must be integer
    c[k] = val // k

# Calculate p_8 to p_12
for k in range(n + 1, 13):
    val = sum(c[i] * p[k - i] for i in range(1, n + 1))
    p[k] = -val
    
print(f"p_12 = {p[12]}")

h = 10
P_12 = p[12] + h**12
print(f"P_12 = {P_12}")

# Check roots
x = sympy.symbols('x')
poly = x**7
for k in range(1, n + 1):
    poly += c[k] * x**(7 - k)

roots = sympy.solve(poly, x)
print("Roots:", roots)

solve_problem() [/PYTHON] [THOUGHT] The problem asks for $P_{12} = \sum_{i=1}^7 x_i^{12} + 10^{12}$, given the power sums $p_k = \sum x_i^k$ for $k=1..7$. I can determine the elementary symmetric polynomials (coefficients of the polynomial whose roots are $a,b,c,d,e,f,g$) using Newton's sums identities. Let the polynomial be $P(x) = x^7 + c_1 x^6 + \dots + c_7$. The identities are: $p_1 + c_1 = 0$ $p_2 + c_1 p_1 + 2 c_2 = 0$ ... $p_k + \sum_{j=1}^{k-1} c_j p_{k-j} + k c_k = 0$ Once I have the coefficients $c_k$, I can extend the power sums to $p_{12}$ using the linear recurrence relation defined by the polynomial coefficients: $p_m + c_1 p_{m-1} + \dots + c_7 p_{m-7} = 0$ for $m > 7$. Finally, I will calculate $P_{12} = p_{12} + 10^{12}$. I will also solve for the roots of the polynomial to verify they are natural numbers as stated. [/THOUGHT] [PYTHON] import sympy

def solve_problem(): # Power sums p = { 1: 58, 2: 666, 3: 8944, 4: 129846, 5: 1972408, 6: 30870006, 7: 493478344 }

n = 7
c = {}

# Calculate coefficients c_k
# Formula: k * c_k = - (p_k + c_1*p_{k-1} + ... + c_{k-1}*p_1)
for k in range(1, n + 1):
    sum_val = 0
    for i in range(1, k):
        sum_val += c[i] * p[k - i]
    c[k] = -(p[k] + sum_val) // k
    
print("Coefficients:", c)

# Calculate p_8 to p_12
# p_m = - (c_1*p_{m-1} + ... + c_n*p_{m-n})
for m in range(n + 1, 13):
    sum_val = 0
    for i in range(1, n + 1):
        sum_val += c[i] * p[m - i]
    p[m] = -sum_val
    
print(f"p_12: {p[12]}")

h = 10
P_12 = p[12] + h**12
print(f"P_12: {P_12}")

# Verify roots
x = sympy.symbols('x')
poly = x**7
for k in range(1, n + 1):
    poly += c[k] * x**(7 - k)
    
roots = sympy.solve(poly, x)
print("Roots:", roots)

solve_problem() [/PYTHON]

[THOUGHT] The problem asks for $P_{12} = \sum_{i=1}^7 x_i^{12} + 10^{12}$, where $x_i$ are the roots of a polynomial determined by the given power sums $p_1, \dots, p_7$. Using Newton's sums, the Python code determined the coefficients of the polynomial and verified that the roots are the first seven prime numbers: 2, 3, 5, 7, 11, 13, 17. The code then calculated $p_{12} = \sum_{i=1}^7 x_i^{12}$ using the recurrence relation derived from the coefficients. The result was $p_{12} = 609072836692326$. Adding $h^{12} = 10^{12} = 1000000000000$: $P_{12} = 609072836692326 + 1000000000000 = 610072836692326$. [/THOUGHT]

\boxed{610072836692326}