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Question Number 192208 by Shlock last updated on 11/May/23

	Answered by witcher3 last updated on 11/May/23
	$⇒∀(7k+r)∈N⇒∃(a,b,c)∈[7k+r,7k+r+6] r∈{0,1,2,3,4,5,6} a≡r+1[7];a=7k+r+1 b≡r+2[7];b=7k+r+2 c≡r+4[7];c=7k+r+4 a^2 +b^2 +c^2 −ab−ac−bc=(r^2 +2r+1)+(r^2 +4r+4)+(r^2 +8r+16) −(r^2 +3r+2)−(r^2 +5r+4)−(r^2 +6r+8) =−14≡0[7] 7∣(a^2 +b^2 +c^2 −ab−bc−ac) ⇒∀n∈N∃(a,b,c)∈[n,n+6]such 7∣(a^2 +b^2 +c^2 −ab−ac−bc)$
	$\Rightarrow \forall (7 k + r) \in N \Rightarrow \exists (a, b, c) \in [7 k + r, 7 k + r + 6]$ $r \in {0, 1, 2, 3, 4, 5, 6}$ $a \equiv r + 1 [7]; a = 7 k + r + 1$ $b \equiv r + 2 [7]; b = 7 k + r + 2$ $c \equiv r + 4 [7]; c = 7 k + r + 4$ $a^{2} + b^{2} + c^{2} - ab - ac - bc = (r^{2} + 2 r + 1) + (r^{2} + 4 r + 4) + (r^{2} + 8 r + 16)$ $- (r^{2} + 3 r + 2) - (r^{2} + 5 r + 4) - (r^{2} + 6 r + 8)$ $= - 14 \equiv 0 [7]$ $7 ∣ (a^{2} + b^{2} + c^{2} - ab - bc - ac)$ $\Rightarrow \forall n \in N \exists (a, b, c) \in [n, n + 6] such$ $7 ∣ (a^{2} + b^{2} + c^{2} - ab - ac - bc)$
	Commented by Shlock last updated on 12/May/23
	Perfect ��
	Commented by witcher3 last updated on 14/May/23

	$thank You God bless You$
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