Número de factores primos del producto de los primeros números de Fibonacci

Question

Tratando de encontrar un patrón que sugiera que hay infinitos primos de Fibonacci, considero la secuencia de término general $ a_n=\omega(\prod_{k=1}^{n}F_{k}) $ donde $ F_k $ es el $ k $ -número de Fibonacci y $ \omega(m) $ el número de factores primos del mayor número libre de cuadrados que divide $ m $ . Según los cálculos rápidos realizados a mano, parece que $ a_{n}\sim C.n $ para alguna constante positiva $ C $ posiblemente igual a $ \sqrt{2-\sqrt{2}} $ .

¿Se puede confirmar esto? Si no es así, ¿puede establecerse un límite superior para $ a_{n} $ ¿se puede averiguar?

Answer 1

Observando los factores de los números de Fibonacci (los factores grises son los antiguos, los subrayados son los nuevos):
$F_{1} = 1$ ;
$F_{2} = 1$ ;
$F_{3} = 2: \underline{2}$ ;
$F_{4} = 3: \underline{3}$ ;
$F_{5} = 5: \underline{5}$ ;
$F_{6} = 8: \color{gray}{2}^3\;\;$ ;
$F_{7} = 13: \underline{13}$ ;
$F_{8} = 21: \color{gray}{3}\;\;\underline{7}$ ;
$F_{9} = 34: \color{gray}{2}\;\;\underline{17}$ ;
$F_{10} = 55: \color{gray}{5}\;\;\underline{11}$ ;
$F_{11} = 89: \underline{89}$ ;
$F_{12} = 144: \color{gray}{2}^4\;\;\color{gray}{3}^2\;\;$ ;
$F_{13} = 233: \underline{233}$ ;
$F_{14} = 377: \color{gray}{13}\;\;\underline{29}$ ;
$F_{15} = 610: \color{gray}{2}\;\;\color{gray}{5}\;\;\underline{61}$ ;
$F_{16} = 987: \color{gray}{3}\;\;\color{gray}{7}\;\;\underline{47}$ ;
$F_{17} = 1597: \underline{1597}$ ;
$F_{18} = 2584: \color{gray}{2}^3\;\;\color{gray}{17}\;\;\underline{19}$ ;
$F_{19} = 4181: \underline{37}\;\;\underline{113}$ ;
$F_{20} = 6765: \color{gray}{3}\;\;\color{gray}{5}\;\;\color{gray}{11}\;\;\underline{41}$ ;
$F_{21} = 10946: \color{gray}{2}\;\;\color{gray}{13}\;\;\underline{421}$ ;
$F_{22} = 17711: \color{gray}{89}\;\;\underline{199}$ ;
$F_{23} = 28657: \underline{28657}$ ;
$F_{24} = 46368: \color{gray}{2}^5\;\;\color{gray}{3}^2\;\;\color{gray}{7}\;\;\underline{23}$ ;
$F_{25} = 75025: \color{gray}{5}^2\;\;\underline{3001}$ ;
$F_{26} = 121393: \color{gray}{233}\;\;\underline{521}$ ;
$F_{27} = 196418: \color{gray}{2}\;\;\color{gray}{17}\;\;\underline{53}\;\;\underline{109}$ ;
$F_{28} = 317811: \color{gray}{3}\;\;\color{gray}{13}\;\;\color{gray}{29}\;\;\underline{281}$ ;
$F_{29} = 514229: \underline{514229}$ ;
$F_{30} = 832040: \color{gray}{2}^3\;\;\color{gray}{5}\;\;\color{gray}{11}\;\;\underline{31}\;\;\color{gray}{61}$ ;
$F_{31} = 1346269: \underline{557}\;\;\underline{2417}$ ;
$F_{32} = 2178309: \color{gray}{3}\;\;\color{gray}{7}\;\;\color{gray}{47}\;\;\underline{2207}$ ;
$F_{33} = 3524578: \color{gray}{2}\;\;\color{gray}{89}\;\;\underline{19801}$ ;
$F_{34} = 5702887: \color{gray}{1597}\;\;\underline{3571}$ ;
$F_{35} = 9227465: \color{gray}{5}\;\;\color{gray}{13}\;\;\underline{141961}$ ;
$F_{36} = 14930352: \color{gray}{2}^4\;\;\color{gray}{3}^3\;\;\color{gray}{17}\;\;\color{gray}{19}\;\;\underline{107}$ ;
$F_{37} = 24157817: \underline{73}\;\;\underline{149}\;\;\underline{2221}$ ;
$F_{38} = 39088169: \color{gray}{37}\;\;\color{gray}{113}\;\;\underline{9349}$ ;
$F_{39} = 63245986: \color{gray}{2}\;\;\color{gray}{233}\;\;\underline{135721}$ ;
$F_{40} = 102334155: \color{gray}{3}\;\;\color{gray}{5}\;\;\color{gray}{7}\;\;\color{gray}{11}\;\;\color{gray}{41}\;\;\underline{2161}$ ;
$F_{41} = 165580141: \underline{2789}\;\;\underline{59369}$ ;
$F_{42} = 267914296: \color{gray}{2}^3\;\;\color{gray}{13}\;\;\color{gray}{29}\;\;\underline{211}\;\;\color{gray}{421}$ ;
$F_{43} = 433494437: \underline{433494437}$ ;
$F_{44} = 701408733: \color{gray}{3}\;\;\underline{43}\;\;\color{gray}{89}\;\;\color{gray}{199}\;\;\underline{307}$ ;
$F_{45} = 1134903170: \color{gray}{2}\;\;\color{gray}{5}\;\;\color{gray}{17}\;\;\color{gray}{61}\;\;\underline{109441}$ ;
$F_{46} = 1836311903: \underline{139}\;\;\underline{461}\;\;\color{gray}{28657}$ ;
$F_{47} = 2971215073: \underline{2971215073}$ ;
$F_{48} = 4807526976: \color{gray}{2}^6\;\;\color{gray}{3}^2\;\;\color{gray}{7}\;\;\color{gray}{23}\;\;\color{gray}{47}\;\;\underline{1103}$ ;
$F_{49} = 7778742049: \color{gray}{13}\;\;\underline{97}\;\;\underline{6168709}$ ;
$F_{50} = 12586269025: \color{gray}{5}^2\;\;\color{gray}{11}\;\;\underline{101}\;\;\underline{151}\;\;\color{gray}{3001}$ ;
$F_{51} = 20365011074: \color{gray}{2}\;\;\color{gray}{1597}\;\;\underline{6376021}$ ;
$F_{52} = 32951280099: \color{gray}{3}\;\;\color{gray}{233}\;\;\color{gray}{521}\;\;\underline{90481}$ ;
$F_{53} = 53316291173: \underline{953}\;\;\underline{55945741}$ ;
$F_{54} = 86267571272: \color{gray}{2}^3\;\;\color{gray}{17}\;\;\color{gray}{19}\;\;\color{gray}{53}\;\;\color{gray}{109}\;\;\underline{5779}$ ;
$F_{55} = 139583862445: \color{gray}{5}\;\;\color{gray}{89}\;\;\underline{661}\;\;\underline{474541}$ ;
$F_{56} = 225851433717: \color{gray}{3}\;\;\color{gray}{7}^2\;\;\color{gray}{13}\;\;\color{gray}{29}\;\;\color{gray}{281}\;\;\underline{14503}$ ;
$F_{57} = 365435296162: \color{gray}{2}\;\;\color{gray}{37}\;\;\color{gray}{113}\;\;\underline{797}\;\;\underline{54833}$ ;
$F_{58} = 591286729879: \underline{59}\;\;\underline{19489}\;\;\color{gray}{514229}$ ;
$F_{59} = 956722026041: \underline{353}\;\;\underline{2710260697}$ ;
$F_{60} = 1548008755920: \color{gray}{2}^4\;\;\color{gray}{3}^2\;\;\color{gray}{5}\;\;\color{gray}{11}\;\;\color{gray}{31}\;\;\color{gray}{41}\;\;\color{gray}{61}\;\;\underline{2521}$ ;
$F_{61} = 2504730781961: \underline{4513}\;\;\underline{555003497}$ ;
$F_{62} = 4052739537881: \color{gray}{557}\;\;\color{gray}{2417}\;\;\underline{3010349}$ ;
$F_{63} = 6557470319842: \color{gray}{2}\;\;\color{gray}{13}\;\;\color{gray}{17}\;\;\color{gray}{421}\;\;\underline{35239681}$ ;
$F_{64} = 10610209857723: \color{gray}{3}\;\;\color{gray}{7}\;\;\color{gray}{47}\;\;\underline{1087}\;\;\color{gray}{2207}\;\;\underline{4481}$ ;
$F_{65} = 17167680177565: \color{gray}{5}\;\;\color{gray}{233}\;\;\underline{14736206161}$ ;
$F_{66} = 27777890035288: \color{gray}{2}^3\;\;\color{gray}{89}\;\;\color{gray}{199}\;\;\underline{9901}\;\;\color{gray}{19801}$ ;
$F_{67} = 44945570212853: \underline{269}\;\;\underline{116849}\;\;\underline{1429913}$ ;
$F_{68} = 72723460248141: \color{gray}{3}\;\;\underline{67}\;\;\color{gray}{1597}\;\;\color{gray}{3571}\;\;\underline{63443}$ ;
$F_{69} = 117669030460994: \color{gray}{2}\;\;\underline{137}\;\;\underline{829}\;\;\underline{18077}\;\;\color{gray}{28657}$ ;
$F_{70} = 190392490709135: \color{gray}{5}\;\;\color{gray}{11}\;\;\color{gray}{13}\;\;\color{gray}{29}\;\;\underline{71}\;\;\underline{911}\;\;\color{gray}{141961}$ ;
$F_{71} = 308061521170129: \underline{6673}\;\;\underline{46165371073}$ ;
$F_{72} = 498454011879264: \color{gray}{2}^5\;\;\color{gray}{3}^3\;\;\color{gray}{7}\;\;\color{gray}{17}\;\;\color{gray}{19}\;\;\color{gray}{23}\;\;\color{gray}{107}\;\;\underline{103681}$ ;
$F_{73} = 806515533049393: \underline{9375829}\;\;\underline{86020717}$ ;
$F_{74} = 1304969544928657: \color{gray}{73}\;\;\color{gray}{149}\;\;\color{gray}{2221}\;\;\underline{54018521}$ ;
$F_{75} = 2111485077978050: \color{gray}{2}\;\;\color{gray}{5}^2\;\;\color{gray}{61}\;\;\color{gray}{3001}\;\;\underline{230686501}$ ;
$F_{76} = 3416454622906707: \color{gray}{3}\;\;\color{gray}{37}\;\;\color{gray}{113}\;\;\color{gray}{9349}\;\;\underline{29134601}$ ;
$F_{77} = 5527939700884757: \color{gray}{13}\;\;\color{gray}{89}\;\;\underline{988681}\;\;\underline{4832521}$ ;
$F_{78} = 8944394323791464: \color{gray}{2}^3\;\;\underline{79}\;\;\color{gray}{233}\;\;\color{gray}{521}\;\;\underline{859}\;\;\color{gray}{135721}$ ;
$F_{79} = 14472334024676221: \underline{157}\;\;\underline{92180471494753}$ ;
$F_{80} = 23416728348467685: \color{gray}{3}\;\;\color{gray}{5}\;\;\color{gray}{7}\;\;\color{gray}{11}\;\;\color{gray}{41}\;\;\color{gray}{47}\;\;\underline{1601}\;\;\color{gray}{2161}\;\;\underline{3041}$ ;
$F_{81} = 37889062373143906: \color{gray}{2}\;\;\color{gray}{17}\;\;\color{gray}{53}\;\;\color{gray}{109}\;\;\underline{2269}\;\;\underline{4373}\;\;\underline{19441}$ ;
$F_{82} = 61305790721611591: \color{gray}{2789}\;\;\color{gray}{59369}\;\;\underline{370248451}$ ;
$F_{83} = 99194853094755497: \underline{99194853094755497}$ ;
$F_{84} = 160500643816367088: \color{gray}{2}^4\;\;\color{gray}{3}^2\;\;\color{gray}{13}\;\;\color{gray}{29}\;\;\underline{83}\;\;\color{gray}{211}\;\;\color{gray}{281}\;\;\color{gray}{421}\;\;\underline{1427}$ ;
$F_{85} = 259695496911122585: \color{gray}{5}\;\;\color{gray}{1597}\;\;\underline{9521}\;\;\underline{3415914041}$ ;
$F_{86} = 420196140727489673: \underline{6709}\;\;\underline{144481}\;\;\color{gray}{433494437}$ ;
$F_{87} = 679891637638612258: \color{gray}{2}\;\;\underline{173}\;\;\color{gray}{514229}\;\;\underline{3821263937}$ ;
$F_{88} = 1100087778366101931: \color{gray}{3}\;\;\color{gray}{7}\;\;\color{gray}{43}\;\;\color{gray}{89}\;\;\color{gray}{199}\;\;\underline{263}\;\;\color{gray}{307}\;\;\underline{881}\;\;\underline{967}$ ;
$F_{89} = 1779979416004714189: \underline{1069}\;\;\underline{1665088321800481}$ ;
$F_{90} = 2880067194370816120: \color{gray}{2}^3\;\;\color{gray}{5}\;\;\color{gray}{11}\;\;\color{gray}{17}\;\;\color{gray}{19}\;\;\color{gray}{31}\;\;\color{gray}{61}\;\;\underline{181}\;\;\underline{541}\;\;\color{gray}{109441}$ ;
$F_{91} = 4660046610375530309: \color{gray}{13}^2\;\;\color{gray}{233}\;\;\underline{741469}\;\;\underline{159607993}$ ;
$F_{92} = 7540113804746346429: \color{gray}{3}\;\;\color{gray}{139}\;\;\color{gray}{461}\;\;\underline{4969}\;\;\color{gray}{28657}\;\;\underline{275449}$ ;

podemos observar que casi cada número de Fibonacci "aporta" uno (o incluso más) nuevo(s) factor(es) a la lista principal de factores primos.

Como se puede ver, la proporción $\dfrac{a_n}{n}$ crece lentamente con $n$ y para $n=37$ el producto $\prod_{k=1}^nF_n$ tendrá $38$ factores: $2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 47, 53, 61, 73, 89, 107, 109, 113, 149, 199, 233, 281, 421, 521, 557, 1597, 2207, 2221, 2417, 3001, 3571, 19801, 28657, 141961, 514229;$
Así, para $n=37$ tenemos $\dfrac{a_n}{n}>1$ .

Y de esta lista podemos derivar que para $n=92$ tenemos $\dfrac{a_n}{n}\approx 1.40217$ .