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Question-193886

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Question Number 193886 by Rupesh123 last updated on 22/Jun/23
Answered by MM42 last updated on 22/Jun/23
a=521(521^n −521^(n−1) +1)=521m “521” is prime number.therefore “m” must be a multiple “521”. which is valid only for “n=1”
$${a}=\mathrm{521}\left(\mathrm{521}^{{n}} −\mathrm{521}^{{n}−\mathrm{1}} +\mathrm{1}\right)=\mathrm{521}{m} \\ $$$$“\mathrm{521}''\:{is}\:{prime}\:{number}.{therefore}\:“{m}'' \\ $$$$\:{must}\:{be}\:{a}\:{multiple}\:“\mathrm{521}''. \\ $$$${which}\:{is}\:{valid}\:{only}\:{for}\:\:“{n}=\mathrm{1}'' \\ $$
Commented by Rupesh123 last updated on 22/Jun/23
Perfect ��
Answered by Subhi last updated on 22/Jun/23
521((521−1)521^(n−1) +1) 521(521^n −521^(n−1) +1) to be a perfect square 521^n −521^(n−1) +1 = 521^(2m+1) , where m≥0 n≥1 or 521^n −521^(n−1) +1 = 521^(2k−1) ,where k≤0 n≥1 521^(2m+1) −521^n +521^(n−1) =1 521(521^(2m) −521^(n−1) +521^(n−2) )=1 521^(2m) −521^(n−1) +521^(n−2) = (1/(521))=(521)^(−1) ∴ n−2≤0 ⇛ n≤2 n−1≤0 ⇛n≤1 m≤0 ∴ m = 0 , n = 1 521^(2k−1) +521^(n−1) −521^n =1 521(521^(2k−2) +521^(n−2) −521^(n−1) )=1 521^(2k−2) +521^(n−2) −521^(n−1) = (1/(521)) n−1≤0 ⇛ n≤1 ⇛ n=1 ⇛ k = 1
$$\mathrm{521}\left(\left(\mathrm{521}−\mathrm{1}\right)\mathrm{521}^{{n}−\mathrm{1}} +\mathrm{1}\right) \\ $$$$\mathrm{521}\left(\mathrm{521}^{{n}} −\mathrm{521}^{{n}−\mathrm{1}} +\mathrm{1}\right) \\ $$$${to}\:{be}\:{a}\:{perfect}\:{square} \\ $$$$\mathrm{521}^{{n}} −\mathrm{521}^{{n}−\mathrm{1}} +\mathrm{1}\:=\:\mathrm{521}^{\mathrm{2}{m}+\mathrm{1}} \:,\:{where}\:{m}\geqslant\mathrm{0} \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:{n}\geqslant\mathrm{1} \\ $$$${or}\:\mathrm{521}^{{n}} −\mathrm{521}^{{n}−\mathrm{1}} +\mathrm{1}\:=\:\mathrm{521}^{\mathrm{2}{k}−\mathrm{1}} \:,{where}\:{k}\leqslant\mathrm{0} \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:{n}\geqslant\mathrm{1} \\ $$$$\mathrm{521}^{\mathrm{2}{m}+\mathrm{1}} −\mathrm{521}^{{n}} +\mathrm{521}^{{n}−\mathrm{1}} =\mathrm{1} \\ $$$$\mathrm{521}\left(\mathrm{521}^{\mathrm{2}{m}} −\mathrm{521}^{{n}−\mathrm{1}} +\mathrm{521}^{{n}−\mathrm{2}} \right)=\mathrm{1} \\ $$$$\mathrm{521}^{\mathrm{2}{m}} −\mathrm{521}^{{n}−\mathrm{1}} +\mathrm{521}^{{n}−\mathrm{2}} \:=\:\frac{\mathrm{1}}{\mathrm{521}}=\left(\mathrm{521}\right)^{−\mathrm{1}} \\ $$$$\therefore\:{n}−\mathrm{2}\leqslant\mathrm{0}\:\:\Rrightarrow\:{n}\leqslant\mathrm{2}\:\: \\ $$$${n}−\mathrm{1}\leqslant\mathrm{0}\:\:\:\:\:\:\Rrightarrow{n}\leqslant\mathrm{1} \\ $$$${m}\leqslant\mathrm{0} \\ $$$$\therefore\:{m}\:=\:\mathrm{0}\:,\:{n}\:=\:\mathrm{1} \\ $$$$\mathrm{521}^{\mathrm{2}{k}−\mathrm{1}} +\mathrm{521}^{{n}−\mathrm{1}} −\mathrm{521}^{{n}} =\mathrm{1} \\ $$$$\mathrm{521}\left(\mathrm{521}^{\mathrm{2}{k}−\mathrm{2}} +\mathrm{521}^{{n}−\mathrm{2}} −\mathrm{521}^{{n}−\mathrm{1}} \right)=\mathrm{1} \\ $$$$\mathrm{521}^{\mathrm{2}{k}−\mathrm{2}} +\mathrm{521}^{{n}−\mathrm{2}} −\mathrm{521}^{{n}−\mathrm{1}} \:=\:\frac{\mathrm{1}}{\mathrm{521}} \\ $$$${n}−\mathrm{1}\leqslant\mathrm{0}\:\:\Rrightarrow\:{n}\leqslant\mathrm{1}\:\Rrightarrow\:{n}=\mathrm{1}\:\Rrightarrow\:{k}\:=\:\mathrm{1} \\ $$
Commented by Rupesh123 last updated on 23/Jun/23
Perfect ��

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