Prove that following equation has no non-zero real solution. $$ \sum_{ 1 \leq n \leq 120,\, 2|n \;\textrm{or}\; 3|n } x^n = 0$$
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2026-04-06 01:20:44.1775438444
Prove that following polynomial has no non-zero real solution.
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It is easy to see that we can rewrite the equation in the following forms $$ 0 = \sum_{ 1 \leq n \leq 120,\, 6|n } x^n + \sum_{ 1 \leq n \leq 120,\, 6|n-2 } x^n +\sum_{ 1 \leq n \leq 120,\, 6|n-3 } x^n +\sum_{ 1 \leq n \leq 120,\, 6|n-4 } x^n , $$ and $$ 0 = x^2 ( 1 + x + x^2 + x^4 ) (1+\sum_{ 1 \leq n < 120,\, 6|n } x^n) . $$ Now, note that $$ 1 + \sum_{ 1 \leq n < 120,\, 6|n } x^n > 0$$ and (since $1 + x + x^2>0$) $$ 1 + x + x^2 + x^4 > 0 $$ So $$ ( 1 + x + x^2 + x^4 ) \sum_{ 1 \leq n < 120,\, 6|n } x^n > 0 . $$