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Solve a puzzle using FindMinimum?

Posted 6 months ago
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I am trying to minimize y=a+b+c+...+x (all integers) for the function 25^25=a24^25+b23^25+c22^25+...+x1^25

I have tried multiple forms of the FindMinimum() but Mathematica keeps giving me errors I don't understand I use FindMinimum because I also want to know each individual value for a to x where y is minimal.

This is currently my best code:

FindMinimum[{a + b + c + d + e + f + g + h + i + j + k + l + m + n + o + p + q + r + s + t + u + v + w + x,     
  25^25 = a*24^25 - b*23^25 - c*22^25 - d*21^25 - e*20^25 - f*19^25 - 
     g*18^25 - h*17^25 - i*16^25 - j*15^25 - k*14^25 - l*13^25 - 
     m*12^25 - n*11^25 - o*10^25 - p*9^25 - q*8^25 - r*7^25 - s*6^25 -
      t*5^25 - u*4^25 - v*3^25 - w*2^25 - x*1^25 &&     
    a \[Element] Integers && b \[Element] Integers && 
    c \[Element] Integers && d \[Element] Integers && 
    e \[Element] Integers && f \[Element] Integers && 
    g \[Element] Integers && h \[Element] Integers && 
    i \[Element] Integers && j \[Element] Integers && 
    k \[Element] Integers && l \[Element] Integers && 
    m \[Element] Integers && n \[Element] Integers && 
    o \[Element] Integers && p \[Element] Integers && 
    q \[Element] Integers && r \[Element] Integers && 
    s \[Element] Integers && t \[Element] Integers && 
    u \[Element] Integers && v \[Element] Integers && 
    w \[Element] Integers && x \[Element] Integers },     
{a, b, c, d, e, f, g, h, i, j, k, l, m, n, o, p, q, r, s, t, u, v, w, x}]

Did you intend to restrict to nonnegative integers?

If there is no sign restriction, one "small" answer is

{2, -23, 26, -15, -12, -3, -3, -1, 10, 4, 12, -13, 12, -20, 5, 3, -5,  7, -1, -11, 13, 5, -12, 6}
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