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Question Number 212242 by York12 last updated on 08/Oct/24

$$\mathrm{Help} \\ $$

Commented by York12 last updated on 08/Oct/24

Commented by Frix last updated on 08/Oct/24

Fibonacci Sequences: a_n =(1/(2(√2)))((3+2(√2))^n −(3−2(√2))^n ) a_1 =0; a_2 =2; a_n =−a_(n−2) +6a_(n−1) b_n =(1/2)((3+2(√2))^n +(3−2(√2))^n ) b_1 =1; b_2 =3; b_n =−b_(n−2) +6b_(n−1) ⇒ gcd=1

$$\mathrm{Fibonacci}\:\mathrm{Sequences}: \\ $$$${a}_{{n}} =\frac{\mathrm{1}}{\mathrm{2}\sqrt{\mathrm{2}}}\left(\left(\mathrm{3}+\mathrm{2}\sqrt{\mathrm{2}}\right)^{{n}} −\left(\mathrm{3}−\mathrm{2}\sqrt{\mathrm{2}}\right)^{{n}} \right) \\ $$$${a}_{\mathrm{1}} =\mathrm{0};\:{a}_{\mathrm{2}} =\mathrm{2};\:{a}_{{n}} =−{a}_{{n}−\mathrm{2}} +\mathrm{6}{a}_{{n}−\mathrm{1}} \\ $$$$ \\ $$$${b}_{{n}} =\frac{\mathrm{1}}{\mathrm{2}}\left(\left(\mathrm{3}+\mathrm{2}\sqrt{\mathrm{2}}\right)^{{n}} +\left(\mathrm{3}−\mathrm{2}\sqrt{\mathrm{2}}\right)^{{n}} \right) \\ $$$${b}_{\mathrm{1}} =\mathrm{1};\:{b}_{\mathrm{2}} =\mathrm{3};\:{b}_{{n}} =−{b}_{{n}−\mathrm{2}} +\mathrm{6}{b}_{{n}−\mathrm{1}} \\ $$$$ \\ $$$$\Rightarrow\:\mathrm{gcd}=\mathrm{1} \\ $$

Commented by York12 last updated on 08/Oct/24

$$\mathrm{Thank}\:\mathrm{you}\: \\ $$

Commented by York12 last updated on 08/Oct/24

TBH sir I do not understand why it′s neccesarliy means that the gcd =1 if possible can you elaborate please.

$$\mathrm{TBH}\:\mathrm{sir}\:\mathrm{I}\:\mathrm{do}\:\mathrm{not}\:\mathrm{understand}\:\mathrm{why}\:\mathrm{it}'\mathrm{s} \\ $$$$\mathrm{neccesarliy}\:\mathrm{means}\:\mathrm{that}\:\mathrm{the}\:{gcd}\:=\mathrm{1} \\ $$$$\mathrm{if}\:\mathrm{possible}\:\mathrm{can}\:\mathrm{you}\:\mathrm{elaborate}\:\mathrm{please}. \\ $$

Commented by Frix last updated on 08/Oct/24

I tried the following: a_1 =0; a_2 =2 b_1 =1; b_2 =1+2 a_3 =12; b_3 =5+12 a_4 =70; b_4 =29+70 a_5 =408; b_5 =169+408 ... a_k ; b_k =a_k +n gcd (a_k ; a_k +n) =q ⇒ q∣a_k ∧q∣n n is odd ⇒ q is odd I found no q≠1 up to k=100

$$\mathrm{I}\:\mathrm{tried}\:\mathrm{the}\:\mathrm{following}: \\ $$$${a}_{\mathrm{1}} =\mathrm{0};\:{a}_{\mathrm{2}} =\mathrm{2} \\ $$$${b}_{\mathrm{1}} =\mathrm{1};\:{b}_{\mathrm{2}} =\mathrm{1}+\mathrm{2} \\ $$$${a}_{\mathrm{3}} =\mathrm{12};\:{b}_{\mathrm{3}} =\mathrm{5}+\mathrm{12} \\ $$$${a}_{\mathrm{4}} =\mathrm{70};\:{b}_{\mathrm{4}} =\mathrm{29}+\mathrm{70} \\ $$$${a}_{\mathrm{5}} =\mathrm{408};\:{b}_{\mathrm{5}} =\mathrm{169}+\mathrm{408} \\ $$$$… \\ $$$${a}_{{k}} ;\:{b}_{{k}} ={a}_{{k}} +{n} \\ $$$$\mathrm{gcd}\:\left({a}_{{k}} ;\:{a}_{{k}} +{n}\right)\:={q}\:\Rightarrow\:{q}\mid{a}_{{k}} \wedge{q}\mid{n} \\ $$$${n}\:\mathrm{is}\:\mathrm{odd}\:\Rightarrow\:{q}\:\mathrm{is}\:\mathrm{odd} \\ $$$$\mathrm{I}\:\mathrm{found}\:\mathrm{no}\:{q}\neq\mathrm{1}\:\mathrm{up}\:\mathrm{to}\:{k}=\mathrm{100} \\ $$

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