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Alexey Ustinov
Alexey Ustinov
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If $N=6$ then we have a nice identity: $$(x_1+x_2)(x_1+x_3)(x_2+x_3)=(x_4+x_5)(x_4+x_6)(x_5+x_6).$$$$(x_1+x_2)(x_1+x_3)(x_2+x_3)=-(x_4+x_5)(x_4+x_6)(x_5+x_6).$$ (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 easekyeasily follows that $N>4$.

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 $N>4$.

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 easily follows that $N>4$.

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Alexey Ustinov
Alexey Ustinov
  • 12.3k
  • 7
  • 87
  • 119

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 $N>4$.