The minimal non-zero absolute value of sums of n fifth roots of unity is computed exactly; it is monotone non-increasing within each residue class modulo 5 and jumps only at n equal to 5F_m, L_m or 2L_m for Fibonacci and Lucas numbers.
Small sums of five roots of unity
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The Minimal Absolute Value of Sums of Fifth Roots of Unity
The minimal non-zero absolute value of sums of n fifth roots of unity is computed exactly; it is monotone non-increasing within each residue class modulo 5 and jumps only at n equal to 5F_m, L_m or 2L_m for Fibonacci and Lucas numbers.