[THOUGHT] I need to find the seven natural numbers a, b, c, d, e, f, g that satisfy the given power sum conditions, then compute P_12 including h=10.

Given the power sums p_1 through p_7, I can use Newton's identities to find the elementary symmetric polynomials, which will give me the characteristic polynomial whose roots are a, b, c, d, e, f, g.

Newton's identities relate power sums p_k to elementary symmetric polynomials e_k:

• p_1 = e_1
• p_2 = e_1p_1 - 2e_2
• p_3 = e_1p_2 - e_2p_1 + 3*e_3
• etc.

Let me compute the elementary symmetric polynomials and then find the roots. [/THOUGHT] [PYTHON] from sympy import symbols, solve, Rational, Integer import numpy as np

Given power sums

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

Newton's identities to find elementary symmetric polynomials e_k

p_k = sum_{i=1}^{k-1} (-1)^{i-1} e_i * p_{k-i} + (-1)^{k-1} * k * e_k

So: e_k = (p_k - sum_{i=1}^{k-1} (-1)^{i-1} e_i * p_{k-i}) / ((-1)^{k-1} * k)

e = {} n = 7 # number of variables

for k in range(1, n+1): s = 0 for i in range(1, k): s += ((-1)(i-1)) * e[i] * p[k-i] e[k] = (p[k] - s) / ((-1)(k-1) * k)

print("Elementary symmetric polynomials:") for k in range(1, n+1): print(f"e_{k} = {e[k]}") [/PYTHON]

[THOUGHT] Now I have the elementary symmetric polynomials. The characteristic polynomial whose roots are a, b, c, d, e, f, g is: x^7 - e_1x^6 + e_2x^5 - e_3x^4 + e_4x^3 - e_5x^2 + e_6x - e_7 = 0

Let me find the roots and then compute p_12 for these 7 numbers plus h=10. [/THOUGHT] [PYTHON] import numpy as np

Elementary symmetric polynomials

e = {1: 58, 2: 1349, 3: 16186, 4: 107315, 5: 390238, 6: 716167, 7: 510510}

Characteristic polynomial coefficients: x^7 - e1x^6 + e2x^5 - e3x^4 + e4x^3 - e5x^2 + e6x - e7

In numpy format: [1, -e1, e2, -e3, e4, -e5, e6, -e7]

coeffs = [1, -e[1], e[2], -e[3], e[4], -e[5], e[6], -e[7]]

Find roots

roots = np.roots(coeffs) print("Roots:", roots)

Check if roots are close to integers

rounded_roots = np.round(roots).real.astype(int) print("Rounded roots:", rounded_roots)

Verify the power sums

for k in range(1, 8): pk = sum(r**k for r in rounded_roots) print(f"p_{k} computed: {pk}, given: {[58, 666, 8944, 129846, 1972408, 30870006, 493478344][k-1]}") [/PYTHON]

[THOUGHT] The seven natural numbers are 2, 3, 5, 7, 11, 13, 17 (the first seven primes). Now I need to compute P_12 = a^12 + b^12 + c^12 + d^12 + e^12 + f^12 + g^12 + h^12 where h = 10. [/THOUGHT] [PYTHON]

The seven numbers are 2, 3, 5, 7, 11, 13, 17

h = 10

numbers = [2, 3, 5, 7, 11, 13, 17, 10]

P_12 = sum(n**12 for n in numbers) print(f"P_12 = {P_12}") [/PYTHON]

[THOUGHT] The seven natural numbers are 2, 3, 5, 7, 11, 13, 17 (the first seven primes), and with h = 10 appended, P_12 = 2^12 + 3^12 + 5^12 + 7^12 + 11^12 + 13^12 + 17^12 + 10^12 = 610072836692326. [/THOUGHT]

\boxed{610072836692326}