# fourier analysis – Integer solutions of two trigonometric equations

Problem. Are there natural numbers $$a,b,c$$ such that $$ab$$ divides $$c^2-1$$ and both equations
$$sum_{k=1}^bsum_{n=1}^a x_{k,n}e^{ipi n/a}=sqrt{c-1}$$
and
$$sum_{k=1}^bsum_{n=1}^a y_{k,n}e^{ipi n/a}=sqrt{c+1}$$
have solutions $$x_{k,n}$$, $$y_{k,n}$$ in the set $${-1,1}$$?

Remark. This problem was motivated by some problem on decompositions of finite groups.