Question Number 70525 by Rasheed.Sindhi last updated on 05/Oct/19
$${If}\:{x},{y},{z}\:{is}\:{a}\:{primitive}\:{Pythagorean} \\ $$$${triple},{prove}\:{that}\:{x}+{y}\:{and}\:{x}−{y}\:{are} \\ $$$${congruent}\:{modulo}\:\mathrm{8}\:{to}\:{either}\:\mathrm{1}\:{or}\:\mathrm{7}. \\ $$
Answered by mind is power last updated on 07/Oct/19
$${x}=\mathrm{2}{ab} \\ $$$${y}={a}^{\mathrm{2}} −{b}^{\mathrm{2}} \\ $$$${z}={a}^{\mathrm{2}} +{b}^{\mathrm{2}} \\ $$$${x},{y},{z}\:{primitive}\:\Rightarrow{gcd}\left({a},{b}\right)=\mathrm{1}\:{and}\:\:{either}\:{a}=\:\mathrm{2}{k}\:\&{b}=\mathrm{2}{w}+\mathrm{1}\: \\ $$$${because}\:{if}\:{gcd}\left({a},{b}\right)={a}\geqslant\mathrm{2} \\ $$$$\Rightarrow{a}\mid{x},{y},{z}\:{abvious} \\ $$$${and}\:{if}\:{a}={b}\left[\mathrm{2}\right]\Rightarrow\mathrm{2}\mid{x},{y},{z}\:{abvious} \\ $$$${x}=\mathrm{2}\left(\mathrm{2}{k}\right)\left(\mathrm{2}{w}+\mathrm{1}\right)=\mathrm{8}{kw}+\mathrm{4}{k}=\mathrm{4}{k}\left(\mathrm{8}\right) \\ $$$${y}=\left(\mathrm{4}{k}^{\mathrm{2}} \right)−\mathrm{4}{w}^{\mathrm{2}} −\mathrm{4}{w}−\mathrm{1} \\ $$$${y}=\mathrm{4}{k}^{\mathrm{2}} −\mathrm{4}{w}\left({w}+\mathrm{1}\right)−\mathrm{1}=\mathrm{4}{k}^{\mathrm{2}} −\mathrm{1}\left(\mathrm{8}\right) \\ $$$${cause}\:\mathrm{2}\mid{w}\left({w}+\mathrm{1}\right),\forall{w}\in{IN}\:{just}\:{use}\:{either}\:{w}=\mathrm{2}{s}\:{or}\:{w}=\mathrm{2}{s}+\mathrm{1}\:{to}\:{see}\:{that} \\ $$$$\boldsymbol{{if}}\:\boldsymbol{{ko}}{r}\:{k}=\mathrm{0}\left(\mathrm{2}\right)\Rightarrow{x}=\mathrm{0}\left(\mathrm{8}\right),{y}=−\mathrm{1}\left(\mathrm{8}\right)\Rightarrow\begin{cases}{{x}−{y}=\mathrm{1}\left(\mathrm{8}\right)}\\{{x}+{y}=−\mathrm{1}\left(\mathrm{8}\right)=\mathrm{7}\left(\mathrm{8}\right)}\end{cases} \\ $$$$ \\ $$$${k}=\mathrm{1}\left(\mathrm{2}\right)\Rightarrow{x}=\mathrm{4}\left(\mathrm{8}\right),{y}=\mathrm{3}\left(\mathrm{8}\right)\Rightarrow\begin{cases}{{x}−{y}=\mathrm{1}\left(\mathrm{8}\right)}\\{{x}+{y}=\mathrm{7}\left(\mathrm{8}\right)}\end{cases} \\ $$$$\Rightarrow{x}+{y}\:,{x}−{y}\:{congurent}\:{modulo}\:\mathrm{8}\:{to}\:{either}\:\mathrm{1}\:{or}\:\mathrm{7} \\ $$$$ \\ $$$$ \\ $$$$ \\ $$$$ \\ $$$$ \\ $$$$ \\ $$$$ \\ $$$$ \\ $$
Commented by Rasheed.Sindhi last updated on 08/Oct/19
$$\mathcal{N}{o}\:{doubt},\:{mind}\:{is}\:{power}! \\ $$