Experimental... again

Algebra Level 3
  • a3+b3\sqrt[3]{a} + \sqrt[3]{b} is a root of 10x330x2910x^3 - 30x - 29.
  • a5+b5\sqrt[5]{a} + \sqrt[5]{b} is a root of 10x550x3+50x2910x^5 - 50x^3 + 50x - 29.
  • a7+b7\sqrt[7]{a} + \sqrt[7]{b} is a root of n1x7+n2x5+n3x3+n4x+n5n_1x^7 + n_2x^5 + n_3x^3 + n_4x + n_5.

If gcd(n1,n2,n3,n4,n5)=1\gcd(n_1, n_2, n_3, n_4, n_5) = 1 with n1>0n_1 > 0 , what is the value of n1+n2+n3+n4+n5?|n_1|+|n_2|+|n_3|+|n_4|+|n_5|?

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