Let be positive integers satisfying
Define the monic quartic
Let be the set of integer quadruples with such that
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is even;
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is divisible by ;
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.
Compute
[THOUGHT] The roots are 2, 5, 11, 17, so F(x) = (x-2)(x-5)(x-11)(x-17). Precompute F values for x=0 to 8. Enumerate all ordered quadruples (u,v,w,t) in {0,...,8}^4, check if sum u+v+w+t even, gcd(u,v,w,t)=1 using reduce(gcd), and (F(u)+F(v)+F(w)+F(t)) % 12 == 0. Sum u²+v²+w²+t² over those satisfying all conditions. [/THOUGHT] [PYTHON] from math import gcd from functools import reduce
F = [(i-2)(i-5)(i-11)*(i-17) for i in range(9)]
total = 0 for u in range(9): for v in range(9): for w in range(9): for t in range(9): s = u + v + w + t if s % 2 != 0: continue g = reduce(gcd, [u, v, w, t]) if g != 1: continue sumF = F[u] + F[v] + F[w] + F[t] if sumF % 12 == 0: total += uu + vv + ww + tt
print(total) [/PYTHON]
[THOUGHT] The Python code correctly computed S=56040 using the verified roots and enumerating all ordered quadruples satisfying the conditions. [/THOUGHT] \boxed{56040}
Explanation
The reasoning process begins by verifying the roots of the polynomial F(x) using the given power sums: a + b + c + d = 35, a² + b² + c² + d² = 439, a³ + b³ + c³ + d³ = 6377, and a⁴ + b⁴ + c⁴ + d⁴ = 98803. These sums confirm that the positive integers a, b, c, d are 2, 5, 11, and 17, as they satisfy all equations.
Next, precompute F(x) = (x-2)(x-5)(x-11)(x-17) for x from 0 to 8, storing the values in a list for efficient access.
Then, enumerate all possible ordered quadruples (u, v, w, t) where each is an integer from 0 to 8, inclusive. For each quadruple, apply the three conditions sequentially:
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Check if u + v + w + t is even (i.e., (u + v + w + t) % 2 == 0). If not, skip.
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Compute the greatest common divisor of the quadruple using reduce(gcd, [u, v, w, t]). If gcd ≠ 1, skip. Note that gcd handles cases with zeros correctly, as gcd(0, k) = k, and reduce propagates this.
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If both prior conditions hold, compute the sum F(u) + F(v) + F(w) + F(t) and check if it is divisible by 12 (i.e., sumF % 12 == 0). If yes, add u² + v² + w² + t² to the running total S.
The nested loops ensure all 9⁴ = 6561 quadruples are checked exhaustively. The ordered nature of the enumeration aligns with the problem's definition of quadruples as tuples, where permutations are distinct elements of the set U.
The Python code executes this process without errors, producing the output 56040, which is the exact value of S based on the valid computation.