Timeline for Are there any Fibonacci numbers that are sandwiched between twin primes?
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| Dec 31, 2022 at 20:23 | history | edited | Ofir Gorodetsky | CC BY-SA 4.0 |
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| Dec 31, 2022 at 20:21 | comment | added | Ofir Gorodetsky | @YCor Cool! Indeed, from your empirical observations and Guy's identities one might guess $F_{4n+2\varepsilon}-1 = F_{2n-1+\varepsilon} L_{2n+1+\varepsilon}$ ($\varepsilon=0,1$) and $F_{4n\pm 1}-1= F_{2n} L_{2n \pm1}$ hold, both of which are easily confirmed (and surely known -- yes, I checked the OEIS page oeis.org/A000071 and they are stated by Peter Bala). We see $F_m\pm 1$ are both composite for $m \ge 7$. (The OEIS page also contains the statement that $2$ and $7$ are the only primes in the sequence.) | |
| Dec 31, 2022 at 19:52 | comment | added | YCor | Empirically (this is certainly known too) $F_{2n}$ divides $F_{4n+2}-1$, $F_{4n+1}-1$, $F_{4n-1}-1$ and $F_{2n+1}$ divides $F_{4n}-1$ for all $n$. For $n\ge 2$, the divisor is $\ge 2$ and smaller than the larger number, and hence in this way we see that $F_m-1$ is also non-prime for all $m\ge 7$. (Thus $F_m-1$ is prime only for $m=4,6$, namely equals $2,7$). | |
| Dec 31, 2022 at 18:48 | vote | accept | Nandakumar R | ||
| S Dec 31, 2022 at 18:35 | history | suggested | mathworker21 | CC BY-SA 4.0 |
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| Dec 31, 2022 at 18:29 | review | Suggested edits | |||
| S Dec 31, 2022 at 18:35 | |||||
| Dec 31, 2022 at 13:05 | history | edited | Ofir Gorodetsky | CC BY-SA 4.0 |
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| Dec 31, 2022 at 12:59 | history | edited | Ofir Gorodetsky | CC BY-SA 4.0 |
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| Dec 31, 2022 at 12:53 | history | answered | Ofir Gorodetsky | CC BY-SA 4.0 |