If $N=6$ then we have a nice identity: $$(x_1+x_2)(x_1+x_3)(x_2+x_3)=(x_4+x_5)(x_4+x_6)(x_5+x_6).$$ (Loo-Keng Hua used such identities in his "Additive Theory of Prime Numbers".) In particular if $x_5=x_6=0$ then LHS vanishes. From this observation easeky follows that $N>4$.
Alexey Ustinov
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