Question Number 92673 by Ar Brandon last updated on 08/May/20
$$\mathrm{Show}\:\mathrm{that}\:\mathrm{if}\:\mathrm{3}\:\mathrm{prime}\:\mathrm{numbers},\:\mathrm{all}\:\mathrm{greater} \\ $$$$\mathrm{than}\:\mathrm{3},\:\mathrm{form}\:\mathrm{an}\:\mathrm{arithmetic}\:\mathrm{progression}\:\mathrm{then}\:\mathrm{the}\:\mathrm{common} \\ $$$$\mathrm{difference}\:\mathrm{of}\:\mathrm{the}\:\mathrm{progression}\:\mathrm{is}\:\mathrm{divisible}\:\mathrm{by}\:\mathrm{6}. \\ $$
Commented by Rasheed.Sindhi last updated on 09/May/20
$${In}\:{other}\:{words}: \\ $$$${If}\:\:{a}\:\:\&\:\:{b}\:{are}\:{two}\:{primes}\:{such}\:{that} \\ $$$$\mathrm{6}\:\nmid\:\left({b}−{a}\right)\:{then}\:\mathrm{2}{b}−{a}\:{is}\:{composite}. \\ $$
Commented by Ar Brandon last updated on 09/May/20
Interesting Sir. I'll like to get more details please.
Answered by Rasheed.Sindhi last updated on 10/May/20
$${Let}\:{a},{b}\in\mathbb{P}\:{are}\:{first}\:{two}\:{terms} \\ $$$$\therefore\:{b}−{a}\:{is}\:{common}\:{difference} \\ $$$$\therefore\:{The}\:{Third}\:{is}\:{b}+\left({b}−{a}\right)=\mathrm{2}{b}−{a} \\ $$$${According}\:{to}\:{the}\:{given}\:{if}\:\left(\mathrm{2}{b}−{a}\right)\in\mathbb{P} \\ $$$${with}\:{a},{b}\in\mathbb{P},\:{then}\:\mathrm{6}\mid\left({b}−{a}\right). \\ $$$${In}\:{other}\:{words}\:{if}\:\mathrm{6}\:\nmid\:\left({b}−{a}\right)\:{then} \\ $$$$\mathrm{2}{b}−{a}\:{is}\:{not}\:{prime}\:{i}-{e}\:{composite}. \\ $$$${Proof}: \\ $$$$\:^{\bullet} {All}\:{integers}\:\left({w}.\:{r}.\:{t}\:{division}\:{by}\:\mathrm{6}\right) \\ $$$$\:{can}\:{be}\:{categarized}\:{into}\:{six}\:{types}: \\ $$$$\:\mathrm{6}{k},\mathrm{6}{k}+\mathrm{1},\mathrm{6}{k}+\mathrm{2},...,\mathrm{6}{k}+\mathrm{5}. \\ $$$$\:\:{Among}\:{these}\:\mathrm{6}{k},\mathrm{6}{k}+\mathrm{2},\mathrm{6}{k}+\mathrm{3}\:\& \\ $$$$\:\:\mathrm{6}{k}+\mathrm{4}\:{are}\:{obviously}\:{composite}. \\ $$$$\:\:{This}\:{means}\:{primes}\:{are}\:{only}\:{of} \\ $$$$\:{two}\:{types}:\:\mathrm{6}{k}+\mathrm{1}\:\&\:\mathrm{6}{k}+\mathrm{5} \\ $$$$\:{Possibility}\:{for}\:{prime}\:{pair}\:\left({a},{b}\right) \\ $$$$\:\:{are}\:\left(\mathrm{6}{k}+\mathrm{1},\mathrm{6}{k}+\mathrm{1}\right)\:,\:\left(\mathrm{6}{k}+\mathrm{1},\mathrm{6}{k}+\mathrm{5}\right) \\ $$$$\:\:\left(\mathrm{6}{k}+\mathrm{5},\mathrm{6}{k}+\mathrm{1}\right)\:\&\:\left(\mathrm{6}{k}+\mathrm{5},\mathrm{6}{k}+\mathrm{5}\right) \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:{and} \\ $$$$\:\:{b}−{a}\:{will}\:\:{respectively}\:{befall}\:{in} \\ $$$$\:{types}:\mathrm{6}{k},\mathrm{6}{k}+\mathrm{2},\mathrm{6}{k}+\mathrm{4}\:\&\mathrm{6}{k}. \\ $$$$\:{That}\:{means}\:{the}\:{pairs}\:{for}\:{which}\: \\ $$$$\:\:{a},{b}\in\mathbb{P}\:{and}\:\mathrm{6}\:\nmid\:\left({b}−{a}\right)\: \\ $$$$\:\:\left(\mathrm{6}{k}+\mathrm{1},\mathrm{6}{k}+\mathrm{5}\right)\:\&\:\left(\mathrm{6}{k}+\mathrm{5},\mathrm{6}{k}+\mathrm{1}\right) \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:{AND} \\ $$$$\:\:\:{we}'{ve}\:{to}\:{prove}\:{for}\:{these}\:{pairs} \\ $$$$\:\:\:{that}\:\mathrm{2}{b}−{a}\left({the}\:{third}\:{term}\right)\:{is}\: \\ $$$$\:\:{not}\:{prime}\:{i}-{e}\:{composite}.\:{see} \\ $$$$\:\:\:{the}\:{table} \\ $$$$\:\begin{vmatrix}{{a}}&{{b}}&{{b}−{a}}&{\mathrm{2}{b}−{a}}\\{\mathrm{6}{k}+\mathrm{1}}&{\mathrm{6}{k}+\mathrm{1}}&{\underset{\mathrm{6}\mid\left({b}−{a}\right)} {\mathrm{6}{k}}}&{\mathrm{6}{k}+\mathrm{1}}\\{\mathrm{6}{k}+\mathrm{1}}&{\mathrm{6}{k}+\mathrm{5}}&{\underset{\mathrm{6}\:\nmid\:\left({b}−{a}\right)} {\mathrm{6}{k}+\mathrm{4}}}&{\underset{{NonPrime}} {\mathrm{6}{k}+\mathrm{3}^{\blacksquare} }}\\{\mathrm{6}{k}+\mathrm{5}}&{\mathrm{6}{k}+\mathrm{1}}&{\underset{\mathrm{6}\:\nmid\:\left({b}−{a}\right)} {\mathrm{6}{k}+\mathrm{2}}}&{\underset{{NonPrime}} {\mathrm{6}{k}+\mathrm{3}^{\blacksquare} }}\\{\mathrm{6}{k}+\mathrm{5}}&{\mathrm{6}{k}+\mathrm{5}}&{\underset{\mathrm{6}\mid\left({b}−{a}\right)} {\mathrm{6}{k}}}&{\mathrm{6}{k}+\mathrm{5}}\end{vmatrix} \\ $$$$\:^{\blacksquare} \:\mathrm{6}{k}+\mathrm{3}\:{is}\:{nonprime}\:{for}\:{k}>\mathrm{0} \\ $$$$\:\:\:\:\:\:{that}\:{is}\:{for}\:{a},{b}>\mathrm{3} \\ $$$${Red}\:{rows}\:{represent}\:{that}\:{if}\:{b}−{a} \\ $$$$\left({common}\:{difference}\right)\:{can}'{t} \\ $$$${be}\:{divided}\:{by}\:\mathrm{6},\:{the}\:{third}\:{term} \\ $$$${can}'{t}\:{be}\:{prime}. \\ $$$${That}\:{is}\:{if}\:{three}\:{prime}\:{numbers}\left(>\mathrm{3}\right) \\ $$$${are}\:{in}\:{AP}\:,{then}\:\mathrm{6}\:{must}\:{divide}\:{common} \\ $$$${difference}. \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:{Q}.\mathcal{E}.{D} \\ $$
Commented by Ar Brandon last updated on 10/May/20
Thank you ��