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)](https://www.tinkutara.com/question/Q192211.png)
$$\Rightarrow\forall\left(\mathrm{7k}+\mathrm{r}\right)\in\mathbb{N}\Rightarrow\exists\left(\mathrm{a},\mathrm{b},\mathrm{c}\right)\in\left[\mathrm{7k}+\mathrm{r},\mathrm{7k}+\mathrm{r}+\mathrm{6}\right] \\ $$$$\mathrm{r}\in\left\{\mathrm{0},\mathrm{1},\mathrm{2},\mathrm{3},\mathrm{4},\mathrm{5},\mathrm{6}\right\} \\ $$$$\mathrm{a}\equiv\mathrm{r}+\mathrm{1}\left[\mathrm{7}\right];\mathrm{a}=\mathrm{7k}+\mathrm{r}+\mathrm{1} \\ $$$$\mathrm{b}\equiv\mathrm{r}+\mathrm{2}\left[\mathrm{7}\right];\mathrm{b}=\mathrm{7k}+\mathrm{r}+\mathrm{2} \\ $$$$\mathrm{c}\equiv\mathrm{r}+\mathrm{4}\left[\mathrm{7}\right];\mathrm{c}=\mathrm{7k}+\mathrm{r}+\mathrm{4} \\ $$$$\mathrm{a}^{\mathrm{2}} +\mathrm{b}^{\mathrm{2}} +\mathrm{c}^{\mathrm{2}} −\mathrm{ab}−\mathrm{ac}−\mathrm{bc}=\left(\mathrm{r}^{\mathrm{2}} +\mathrm{2r}+\mathrm{1}\right)+\left(\mathrm{r}^{\mathrm{2}} +\mathrm{4r}+\mathrm{4}\right)+\left(\mathrm{r}^{\mathrm{2}} +\mathrm{8r}+\mathrm{16}\right) \\ $$$$−\left(\mathrm{r}^{\mathrm{2}} +\mathrm{3r}+\mathrm{2}\right)−\left(\mathrm{r}^{\mathrm{2}} +\mathrm{5r}+\mathrm{4}\right)−\left(\mathrm{r}^{\mathrm{2}} +\mathrm{6r}+\mathrm{8}\right) \\ $$$$=−\mathrm{14}\equiv\mathrm{0}\left[\mathrm{7}\right] \\ $$$$\mathrm{7}\mid\left(\mathrm{a}^{\mathrm{2}} +\mathrm{b}^{\mathrm{2}} +\mathrm{c}^{\mathrm{2}} −\mathrm{ab}−\mathrm{bc}−\mathrm{ac}\right) \\ $$$$\Rightarrow\forall\mathrm{n}\in\mathbb{N}\exists\left(\mathrm{a},\mathrm{b},\mathrm{c}\right)\in\left[\mathrm{n},\mathrm{n}+\mathrm{6}\right]\mathrm{such} \\ $$$$\mathrm{7}\mid\left(\mathrm{a}^{\mathrm{2}} +\mathrm{b}^{\mathrm{2}} +\mathrm{c}^{\mathrm{2}} −\mathrm{ab}−\mathrm{ac}−\mathrm{bc}\right) \\ $$
Commented by Shlock last updated on 12/May/23
Perfect ��
Commented by witcher3 last updated on 14/May/23
$$\mathrm{thank}\:\mathrm{You}\:\mathrm{God}\:\mathrm{bless}\:\mathrm{You} \\ $$