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Symmetrical combination of the sums and products of trigonometric functions giving integers. (English) Zbl 0880.11024

Systematic methods are given for evaluating various trigonometric sums that have integer values. Some striking examples are \[ \sum^{100}_{k=1} \left(2 \cos {k\pi \over 202} \right)^6 =\sum^{100}_{k=1} \left(2\sin {k\pi\over 202} \right)^6=1988, \]
\[ \sum^5_{k= 1} \left(2 \cos {k\pi\over 12} \right)^{10} =\sum^5_{k=1} \left(2 \sin {k\pi \over 12} \right)^{10} = 1000. \]

MSC:

11B99 Sequences and sets
33B10 Exponential and trigonometric functions
11L03 Trigonometric and exponential sums (general theory)
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