Yosue dijo:
Explicación «Por n puntos, pasan al menos n funciones que los contienen».
1- Decir “Al menos” no está acotando el límite superior, porque n puede ser cualquier valor de R, simplemente quise expresar que “era sencillo” hallar al menos n funciones que pasaban por estos n puntos utilizando el método mostrado, sin utilizar la Matemática Superior, etc. Por supuesto que se pueden encontrar más, y eso no entra en contradicción con la teoría.
Sean x1,x2,…,xn valores reales cualesquiera distintos entre sí. Sean y1,y2,….yn valores reales cualquiera. n pertenece a los Reales.
Sean los puntos (x1,y1); (x2,y2); ….(xn,yn) puntos ordenados donde a cada xi le corresponde un yi de los anteriormente señalados.
Entonces existen al menos n funciones que los contienen a todos. (Como n pertenece al dominio de los números reales, (n puede ser tan grande como el infinito) , (pero además en la solución propuesta solo demostraremos de funciones polinómicas que cumplen los requisitos mostrados, pueden ser otras.)
Demostración.
Primero encontremos una de las funciones (polinómicas) que pasa por todos los n puntos anteriores. (La voy a construir paso a paso para no tener que demostrar.)
Para ello utilizaremos la siguiente propiedad (a*x/a ) es igual a x para todo a<>0 para encontrar los sumandos del polinomio. Además de utilizar la función base siguiente r(x)=(x-a)y1/(b-a) + (x-b)y2/(a-b) … función que tiene la propiedad de pasar por los punto (a,y1) y (b,y2) para cualquier valor de x donde a<>b.
Sean ( x-x2)*(x-x3)* … *(x-xn) el producto de todos los (x-xi) , 1<i<=n, excepto para x1, sean además, ( x1-x2)*(x1-x3)* … *(x1-xn) el producto de todos los (x1-xi), 1<i<=n; entonces se puede formar el polinomio P1(x):
P1(x)=(y1*(( x-x2)*(x-x3)* … *(x-xn))/( ( x1-x2)*(x1-x3)*…*(x1-xn)))
“que se anula para todos los x=xi con 1<i<=n”, y que además para x=x1 nos da y1 porque al evaluar esta expresión del numerador (( x-x2)*(x-x3)* … *(x-xn)) para x1, el resultado es igual denominador (( x1-x2)*(x1-x3)* … *(x1-xn)) y por tanto se simplifica y queda y1. Esto garantiza que al evaluar el polinomio para x1 el resultado es y1, y como consecuencia (x1;y1) pertenecería siempre a esta función .
Igualmente procedemos para todo xi: En el numerador excluimos el factor (x-xi) y en el denominador excluimos el factor (xi-xi): y los factores serían de la forma (xi-xj) con (j<>i, 1<=j<=n)
El segundo polinomio quedaría así:
P2(x)=(y2*(( x-x1)*(x-x3)* … *(x-xn))/( ( x2-x1)*(x2-x3)* … *(x2-xn)))
Esto garantiza que este polinomio para cualquier valor de x<>x2 se anula siempre y para x2 el resultado es y2.
Siguiendo esta idea continuamos construyendo los sumandos de nuestra función para cada xi, que siempre se anulara para todo 1<=xj<=n, j<>i; y para xi al ser evaluada nos da yi.
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El polinomio para xi quedaría así:
Pi(x)=(yi*(( x-x1)*(x-x3)* … *(x-xj)*(x-xk)* … *(x-xn))/( ( xi-x1)*(xi-x3)* …. *(xi-xi-1)*(xi-xi+1)*…*(xi-xn)))
Como se puede apreciar en el numerador falta (x-xi) y en el denominador falta (xi-xi), garantizando igualmente que al evaluar la función para xi nos de yi, y para cualquier valor distinto de xi, el polinomio se anula.
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El polinomio para xn quedaría así:
Pn(x)=(yn*(( x-x1)*(x-x3)* … *(x-xn-1))/( ( xi-x1)*(xi-x3)* …. *(xn-xn-1)))
Como se puede apreciar en el numerador falta (x-xn) y en el denominador falta (xn-xn), garantizando igualmente que al evaluar la función para xn nos de yn, y para cualquier valor distinto de xn, el polinomio se anula.
Entonces mi función que incluye a todos los puntos (x1,y1); (x2,y2); ….(xn,yn) sería:
F(x)=P1(x)+P2(x)+ … + Pn(x),
O sea, para concluir
F(x)= (y1*(( x-x2)*(x-x3)* … *(x-xn))/( ( x1-x2)*(x1-x3)*…*(x1-xn))) +
+ (y2*(( x-x1)*(x-x3)* … *(x-xn))/( ( x2-x1)*(x2-x3)* … *(x2-xn))) +
+ …. +
+ (yi*(( x-x1)*(x-x3)* … *(x-xj)*(x-xk)* … *(x-xn))/( ( xi-x1)*(xi-x3)* …. *(xi-xi-1)*(xi-xi+1)*…*(xi-xn))) +
+ (yn*(( x-x1)*(x-x3)* … *(x-xn-1))/( ( xi-x1)*(xi-x3)* …. *(xn-xn-1)))
Creo que hasta aquí se puede entender que esa función F(x) para un valor cualquiera xj con 1<=j<=n, se anula en todos los polinomios Px<>j, y para Pj nos daría yj .
Por tanto F(xj)= P(xj)=yj
Y por tanto la función pasa por todos los puntos vistos anteriormente. (L.q.q.d).
Como segundo paso demostraremos que “por n puntos pasan al menos n funciones”, (n pertenece a los reales).
Creo que esta es la parte más sencilla de todas. Como dije en la respuesta al ejercicio, basta agregarle a los n puntos n-1 puntos más de uno en uno, y determinar la nueva función para cada uno de ellos.
O sea, a la función F(x) anterior le agregamos los polinomios Pn+1 y obtenemos una nueva función F2(x) que pasará por los n puntos anteriores y por el punto (xn+1 , yn+1);
Repetimos el paso anterior y para cada polinomio Pn+i agregado formamos la función Fn+i+1(x) que pasará por todos los puntos anteriores más el punto (xn+j; yn+j). (1<=j<=n+j)
Hasta llegar al polinomio Pn+n-1 agregado, obteniéndose la función Fn(x) que va a pasar por todos los puntos desde (x1, y1) hasta (xn+1; yn+n+1)
Por tanto existen al menos n funciones que pasan por los n puntos. (Digo al menos porque existen mucho más, pero les demostré que al menos existen n funciones polinómicas que pasan por los n puntos, al seguir agregando punto sigo teniendo funciones pero todas del tipo polinómicas, por eso dije el “Al menos”)
Es verdad, podríamos quitarle el “Al menos” pero entonces tendría que demostrar que no solamente pasan los tipos de funciones descritas anteriormente sino, otras que por lo menos yo, no las conozco.
Nota: Este tipo de funciones polinómicas ” r(x)=(x-a)y1/(b-a) + (x-b)y2/(a-b) … función que tiene la propiedad de pasar por los punto (a,y1) y (b,y2) para cualquier valor de x donde a<>b. ” las empleo con frecuencia cuando estoy programando condicionales anidadas, para eliminarlas.
Me explico: si tengo que hacer la siguiente condicional:
si x=x1 entonces y=y1
si x=x2 entonces y=y2
si x=x3 entonces y=y3
Sustituyo todas esas condiciones por:
Y=((x-x2)*(x-x3))*y1/((x1-x2)*(x1-x3)) + ((x-x1)*(x-x3))*y2/((x2-x1)*(x2-x1)) + ((x-x1)*(x-x2))*y3/((x3-x1)*(x3-x2))
esta fue la base para exponer esta teoría.
Segunda Parte
No cabe la menor duda que el reto con los amigos de Para Pensar me ha llevado a pensar cosas que hace 20 años no veía, y mi Amigo Alvy me puso a pensar con su pregunta ¿Podrá esta función contener a todos los números primos?
Mi respuesta es que contendrá a todos los numeros primos conocidos, cada vez que aparezca uno nuevo, entonces hay que agregar una nueva función con ese punto.
Para lograr eso, basta escribir la función descrita anteriormente para los n puntos primos conocidos como valores de yi.
Ejemplo: Los puntos a tomar tener en cuenta son (0,2); (1,3); (2,5); (3,7); (4,11); (5,13); …. ; (19,67); (20,71); …..; (35,149); (36,151); ……. (n;pn) donde pn sea el mayor n’umero primo conocido. Basta enconces interpolar la función que pase por estos n puntos con el método descrito anteriormente. Y así cade vez que evalúe esta función para un número natural <=n tendría como resultado un número primo.
Posteriormente escribo un ejemplo.
No lo realizo aquí por varias razones
- No conozco todos los números primos y no tengo internet para buscarlos, solamente navegación nacional.
- Los cálculos son extensos y para eso tendría que tener un ordenador potente.
Ejemplos:
Ejemplo1: Función que contiene todos los números primos menores de 100.
Yp100 = ((x-1)(x-2)(x-3)(x-4)(x-5)(x-6)(x-7)(x-8)(x-9)(x-10)(x-11)(x-12)(x-13)(x-14)(x-15)(x-16)(x-17)(x-18)(x-19)(x-20)(x-21)(x-22)(x-23)(x-24))*2/((0-1)(0-2)(0-3)(0-4)(0-5)(0-6)(0-7)(0-8)(0-9)(0-10)(0-11)(0-12)(0-13)(0-14)(0-15)(0-16)(0-17)(0-18)(0-19)(0-20)(0-21)(0-22)(0-23)(0-24)) + ((x-0)(x-2)(x-3)(x-4)(x-5)(x-6)(x-7)(x-8)(x-9)(x-10)(x-11)(x-12)(x-13)(x-14)(x-15)(x-16)(x-17)(x-18)(x-19)(x-20)(x-21)(x-22)(x-23)(x-24))*3/((1-0)(1-2)(1-3)(1-4)(1-5)(1-6)(1-7)(1-8)(1-9)(1-10)(1-11)(1-12)(1-13)(1-14)(1-15)(1-16)(1-17)(1-18)(1-19)(1-20)(1-21)(1-22)(1-23)(1-24)) + ((x-0)(x-1)(x-3)(x-4)(x-5)(x-6)(x-7)(x-8)(x-9)(x-10)(x-11)(x-12)(x-13)(x-14)(x-15)(x-16)(x-17)(x-18)(x-19)(x-20)(x-21)(x-22)(x-23)(x-24))*5/((2-0)(2-1)(2-3)(2-4)(2-5)(2-6)(2-7)(2-8)(2-9)(2-10)(2-11)(2-12)(2-13)(2-14)(2-15)(2-16)(2-17)(2-18)(2-19)(2-20)(2-21)(2-22)(2-23)(2-24)) + ((x-0)(x-1)(x-2)(x-4)(x-5)(x-6)(x-7)(x-8)(x-9)(x-10)(x-11)(x-12)(x-13)(x-14)(x-15)(x-16)(x-17)(x-18)(x-19)(x-20)(x-21)(x-22)(x-23)(x-24))*7/((3-0)(3-1)(3-2)(3-4)(3-5)(3-6)(3-7)(3-8)(3-9)(3-10)(3-11)(3-12)(3-13)(3-14)(3-15)(3-16)(3-17)(3-18)(3-19)(3-20)(3-21)(3-22)(3-23)(3-24)) + ((x-0)(x-1)(x-2)(x-3)(x-5)(x-6)(x-7)(x-8)(x-9)(x-10)(x-11)(x-12)(x-13)(x-14)(x-15)(x-16)(x-17)(x-18)(x-19)(x-20)(x-21)(x-22)(x-23)(x-24))*11/((4-0)(4-1)(4-2)(4-3)(4-5)(4-6)(4-7)(4-8)(4-9)(4-10)(4-11)(4-12)(4-13)(4-14)(4-15)(4-16)(4-17)(4-18)(4-19)(4-20)(4-21)(4-22)(4-23)(4-24)) + ((x-0)(x-1)(x-2)(x-3)(x-4)(x-6)(x-7)(x-8)(x-9)(x-10)(x-11)(x-12)(x-13)(x-14)(x-15)(x-16)(x-17)(x-18)(x-19)(x-20)(x-21)(x-22)(x-23)(x-24))*13/((5-0)(5-1)(5-2)(5-3)(5-4)(5-6)(5-7)(5-8)(5-9)(5-10)(5-11)(5-12)(5-13)(5-14)(5-15)(5-16)(5-17)(5-18)(5-19)(5-20)(5-21)(5-22)(5-23)(5-24)) + ((x-0)(x-1)(x-2)(x-3)(x-4)(x-5)(x-7)(x-8)(x-9)(x-10)(x-11)(x-12)(x-13)(x-14)(x-15)(x-16)(x-17)(x-18)(x-19)(x-20)(x-21)(x-22)(x-23)(x-24))*17/((6-0)(6-1)(6-2)(6-3)(6-4)(6-5)(6-7)(6-8)(6-9)(6-10)(6-11)(6-12)(6-13)(6-14)(6-15)(6-16)(6-17)(6-18)(6-19)(6-20)(6-21)(6-22)(6-23)(6-24)) + ((x-0)(x-1)(x-2)(x-3)(x-4)(x-5)(x-6)(x-8)(x-9)(x-10)(x-11)(x-12)(x-13)(x-14)(x-15)(x-16)(x-17)(x-18)(x-19)(x-20)(x-21)(x-22)(x-23)(x-24))*19/((7-0)(7-1)(7-2)(7-3)(7-4)(7-5)(7-6)(7-8)(7-9)(7-10)(7-11)(7-12)(7-13)(7-14)(7-15)(7-16)(7-17)(7-18)(7-19)(7-20)(7-21)(7-22)(7-23)(7-24)) + ((x-0)(x-1)(x-2)(x-3)(x-4)(x-5)(x-6)(x-7)(x-9)(x-10)(x-11)(x-12)(x-13)(x-14)(x-15)(x-16)(x-17)(x-18)(x-19)(x-20)(x-21)(x-22)(x-23)(x-24))*23/((8-0)(8-1)(8-2)(8-3)(8-4)(8-5)(8-6)(8-7)(8-9)(8-10)(8-11)(8-12)(8-13)(8-14)(8-15)(8-16)(8-17)(8-18)(8-19)(8-20)(8-21)(8-22)(8-23)(8-24)) + ((x-0)(x-1)(x-2)(x-3)(x-4)(x-5)(x-6)(x-7)(x-8)(x-10)(x-11)(x-12)(x-13)(x-14)(x-15)(x-16)(x-17)(x-18)(x-19)(x-20)(x-21)(x-22)(x-23)(x-24))*29/((9-0)(9-1)(9-2)(9-3)(9-4)(9-5)(9-6)(9-7)(9-8)(9-10)(9-11)(9-12)(9-13)(9-14)(9-15)(9-16)(9-17)(9-18)(9-19)(9-20)(9-21)(9-22)(9-23)(9-24)) + ((x-0)(x-1)(x-2)(x-3)(x-4)(x-5)(x-6)(x-7)(x-8)(x-9)(x-11)(x-12)(x-13)(x-14)(x-15)(x-16)(x-17)(x-18)(x-19)(x-20)(x-21)(x-22)(x-23)(x-24))*31/((10-0)(10-1)(10-2)(10-3)(10-4)(10-5)(10-6)(10-7)(10-8)(10-9)(10-11)(10-12)(10-13)(10-14)(10-15)(10-16)(10-17)(10-18)(10-19)(10-20)(10-21)(10-22)(10-23)(10-24)) + ((x-0)(x-1)(x-2)(x-3)(x-4)(x-5)(x-6)(x-7)(x-8)(x-9)(x-10)(x-12)(x-13)(x-14)(x-15)(x-16)(x-17)(x-18)(x-19)(x-20)(x-21)(x-22)(x-23)(x-24))*37/((11-0)(11-1)(11-2)(11-3)(11-4)(11-5)(11-6)(11-7)(11-8)(11-9)(11-10)(11-12)(11-13)(11-14)(11-15)(11-16)(11-17)(11-18)(11-19)(11-20)(11-21)(11-22)(11-23)(11-24)) + ((x-0)(x-1)(x-2)(x-3)(x-4)(x-5)(x-6)(x-7)(x-8)(x-9)(x-10)(x-11)(x-13)(x-14)(x-15)(x-16)(x-17)(x-18)(x-19)(x-20)(x-21)(x-22)(x-23)(x-24))*41/((12-0)(12-1)(12-2)(12-3)(12-4)(12-5)(12-6)(12-7)(12-8)(12-9)(12-10)(12-11)(12-13)(12-14)(12-15)(12-16)(12-17)(12-18)(12-19)(12-20)(12-21)(12-22)(12-23)(12-24)) + ((x-0)(x-1)(x-2)(x-3)(x-4)(x-5)(x-6)(x-7)(x-8)(x-9)(x-10)(x-11)(x-12)(x-14)(x-15)(x-16)(x-17)(x-18)(x-19)(x-20)(x-21)(x-22)(x-23)(x-24))*43/((13-0)(13-1)(13-2)(13-3)(13-4)(13-5)(13-6)(13-7)(13-8)(13-9)(13-10)(13-11)(13-12)(13-14)(13-15)(13-16)(13-17)(13-18)(13-19)(13-20)(13-21)(13-22)(13-23)(13-24)) + ((x-0)(x-1)(x-2)(x-3)(x-4)(x-5)(x-6)(x-7)(x-8)(x-9)(x-10)(x-11)(x-12)(x-13)(x-15)(x-16)(x-17)(x-18)(x-19)(x-20)(x-21)(x-22)(x-23)(x-24))*47/((14-0)(14-1)(14-2)(14-3)(14-4)(14-5)(14-6)(14-7)(14-8)(14-9)(14-10)(14-11)(14-12)(14-13)(14-15)(14-16)(14-17)(14-18)(14-19)(14-20)(14-21)(14-22)(14-23)(14-24)) + ((x-0)(x-1)(x-2)(x-3)(x-4)(x-5)(x-6)(x-7)(x-8)(x-9)(x-10)(x-11)(x-12)(x-13)(x-14)(x-16)(x-17)(x-18)(x-19)(x-20)(x-21)(x-22)(x-23)(x-24))*53/((15-0)(15-1)(15-2)(15-3)(15-4)(15-5)(15-6)(15-7)(15-8)(15-9)(15-10)(15-11)(15-12)(15-13)(15-14)(15-16)(15-17)(15-18)(15-19)(15-20)(15-21)(15-22)(15-23)(15-24)) + ((x-0)(x-1)(x-2)(x-3)(x-4)(x-5)(x-6)(x-7)(x-8)(x-9)(x-10)(x-11)(x-12)(x-13)(x-14)(x-15)(x-17)(x-18)(x-19)(x-20)(x-21)(x-22)(x-23)(x-24))*59/((16-0)(16-1)(16-2)(16-3)(16-4)(16-5)(16-6)(16-7)(16-8)(16-9)(16-10)(16-11)(16-12)(16-13)(16-14)(16-15)(16-17)(16-18)(16-19)(16-20)(16-21)(16-22)(16-23)(16-24)) + ((x-0)(x-1)(x-2)(x-3)(x-4)(x-5)(x-6)(x-7)(x-8)(x-9)(x-10)(x-11)(x-12)(x-13)(x-14)(x-15)(x-16)(x-18)(x-19)(x-20)(x-21)(x-22)(x-23)(x-24))*61/((17-0)(17-1)(17-2)(17-3)(17-4)(17-5)(17-6)(17-7)(17-8)(17-9)(17-10)(17-11)(17-12)(17-13)(17-14)(17-15)(17-16)(17-18)(17-19)(17-20)(17-21)(17-22)(17-23)(17-24)) + ((x-0)(x-1)(x-2)(x-3)(x-4)(x-5)(x-6)(x-7)(x-8)(x-9)(x-10)(x-11)(x-12)(x-13)(x-14)(x-15)(x-16)(x-17)(x-19)(x-20)(x-21)(x-22)(x-23)(x-24))*67/((18-0)(18-1)(18-2)(18-3)(18-4)(18-5)(18-6)(18-7)(18-8)(18-9)(18-10)(18-11)(18-12)(18-13)(18-14)(18-15)(18-16)(18-17)(18-19)(18-20)(18-21)(18-22)(18-23)(18-24)) + ((x-0)(x-1)(x-2)(x-3)(x-4)(x-5)(x-6)(x-7)(x-8)(x-9)(x-10)(x-11)(x-12)(x-13)(x-14)(x-15)(x-16)(x-17)(x-18)(x-20)(x-21)(x-22)(x-23)(x-24))*71/((19-0)(19-1)(19-2)(19-3)(19-4)(19-5)(19-6)(19-7)(19-8)(19-9)(19-10)(19-11)(19-12)(19-13)(19-14)(19-15)(19-16)(19-17)(19-18)(19-20)(19-21)(19-22)(19-23)(19-24)) + ((x-0)(x-1)(x-2)(x-3)(x-4)(x-5)(x-6)(x-7)(x-8)(x-9)(x-10)(x-11)(x-12)(x-13)(x-14)(x-15)(x-16)(x-17)(x-18)(x-19)(x-21)(x-22)(x-23)(x-24))*73/((20-0)(20-1)(20-2)(20-3)(20-4)(20-5)(20-6)(20-7)(20-8)(20-9)(20-10)(20-11)(20-12)(20-13)(20-14)(20-15)(20-16)(20-17)(20-18)(20-19)(20-21)(20-22)(20-23)(20-24)) + ((x-0)(x-1)(x-2)(x-3)(x-4)(x-5)(x-6)(x-7)(x-8)(x-9)(x-10)(x-11)(x-12)(x-13)(x-14)(x-15)(x-16)(x-17)(x-18)(x-19)(x-20)(x-22)(x-23)(x-24))*79/((21-0)(21-1)(21-2)(21-3)(21-4)(21-5)(21-6)(21-7)(21-8)(21-9)(21-10)(21-11)(21-12)(21-13)(21-14)(21-15)(21-16)(21-17)(21-18)(21-19)(21-20)(21-22)(21-23)(21-24)) + ((x-0)(x-1)(x-2)(x-3)(x-4)(x-5)(x-6)(x-7)(x-8)(x-9)(x-10)(x-11)(x-12)(x-13)(x-14)(x-15)(x-16)(x-17)(x-18)(x-19)(x-20)(x-21)(x-23)(x-24))*83/((22-0)(22-1)(22-2)(22-3)(22-4)(22-5)(22-6)(22-7)(22-8)(22-9)(22-10)(22-11)(22-12)(22-13)(22-14)(22-15)(22-16)(22-17)(22-18)(22-19)(22-20)(22-21)(22-23)(22-24)) + ((x-0)(x-1)(x-2)(x-3)(x-4)(x-5)(x-6)(x-7)(x-8)(x-9)(x-10)(x-11)(x-12)(x-13)(x-14)(x-15)(x-16)(x-17)(x-18)(x-19)(x-20)(x-21)(x-22)(x-24))*89/((23-0)(23-1)(23-2)(23-3)(23-4)(23-5)(23-6)(23-7)(23-8)(23-9)(23-10)(23-11)(23-12)(23-13)(23-14)(23-15)(23-16)(23-17)(23-18)(23-19)(23-20)(23-21)(23-22)(23-24)) + ((x-0)(x-1)(x-2)(x-3)(x-4)(x-5)(x-6)(x-7)(x-8)(x-9)(x-10)(x-11)(x-12)(x-13)(x-14)(x-15)(x-16)(x-17)(x-18)(x-19)(x-20)(x-21)(x-22)(x-23))*97/((24-0)(24-1)(24-2)(24-3)(24-4)(24-5)(24-6)(24-7)(24-8)(24-9)(24-10)(24-11)(24-12)(24-13)(24-14)(24-15)(24-16)(24-17)(24-18)(24-19)(24-20)(24-21)(24-22)(24-23))
Reduciéndola un poco tendría:
| Y p100 =(((x-1)*(x-2)*(x-3)*(x-4)*(x-5)*(x-6)*(x-7)*(x-8)*(x-9)*(x-10)*(x-11)*(x-12)*(x-13)*(x-14)*(x-15)*(x-16)*(x-17)*(x-18)*(x-19)*(x-20)*(x-21)*(x-22)*(x-23)*(x-24))*2/(-6227020800)) + (((x-0)*(x-2)*(x-3)*(x-4)*(x-5)*(x-6)*(x-7)*(x-8)*(x-9)*(x-10)*(x-11)*(x-12)*(x-13)*(x-14)*(x-15)*(x-16)*(x-17)*(x-18)*(x-19)*(x-20)*(x-21)*(x-22)*(x-23)*(x-24))*3/(479001600)) + (((x-0)*(x-1)*(x-3)*(x-4)*(x-5)*(x-6)*(x-7)*(x-8)*(x-9)*(x-10)*(x-11)*(x-12)*(x-13)*(x-14)*(x-15)*(x-16)*(x-17)*(x-18)*(x-19)*(x-20)*(x-21)*(x-22)*(x-23)*(x-24))*5/(-79833600)) + (((x-0)*(x-1)*(x-2)*(x-4)*(x-5)*(x-6)*(x-7)*(x-8)*(x-9)*(x-10)*(x-11)*(x-12)*(x-13)*(x-14)*(x-15)*(x-16)*(x-17)*(x-18)*(x-19)*(x-20)*(x-21)*(x-22)*(x-23)*(x-24))*7/(21772800)) + (((x-0)*(x-1)*(x-2)*(x-3)*(x-5)*(x-6)*(x-7)*(x-8)*(x-9)*(x-10)*(x-11)*(x-12)*(x-13)*(x-14)*(x-15)*(x-16)*(x-17)*(x-18)*(x-19)*(x-20)*(x-21)*(x-22)*(x-23)*(x-24))*11/(-8709120)) + (((x-0)*(x-1)*(x-2)*(x-3)*(x-4)*(x-6)*(x-7)*(x-8)*(x-9)*(x-10)*(x-11)*(x-12)*(x-13)*(x-14)*(x-15)*(x-16)*(x-17)*(x-18)*(x-19)*(x-20)*(x-21)*(x-22)*(x-23)*(x-24))*13/(4838400)) + (((x-0)*(x-1)*(x-2)*(x-3)*(x-4)*(x-5)*(x-7)*(x-8)*(x-9)*(x-10)*(x-11)*(x-12)*(x-13)*(x-14)*(x-15)*(x-16)*(x-17)*(x-18)*(x-19)*(x-20)*(x-21)*(x-22)*(x-23)*(x-24))*17/(-3628800)) + (((x-0)*(x-1)*(x-2)*(x-3)*(x-4)*(x-5)*(x-6)*(x-8)*(x-9)*(x-10)*(x-11)*(x-12)*(x-13)*(x-14)*(x-15)*(x-16)*(x-17)*(x-18)*(x-19)*(x-20)*(x-21)*(x-22)*(x-23)*(x-24))*19/(3628800)) + (((x-0)*(x-1)*(x-2)*(x-3)*(x-4)*(x-5)*(x-6)*(x-7)*(x-9)*(x-10)*(x-11)*(x-12)*(x-13)*(x-14)*(x-15)*(x-16)*(x-17)*(x-18)*(x-19)*(x-20)*(x-21)*(x-22)*(x-23)*(x-24))*23/(-4838400)) + (((x-0)*(x-1)*(x-2)*(x-3)*(x-4)*(x-5)*(x-6)*(x-7)*(x-8)*(x-10)*(x-11)*(x-12)*(x-13)*(x-14)*(x-15)*(x-16)*(x-17)*(x-18)*(x-19)*(x-20)*(x-21)*(x-22)*(x-23)*(x-24))*29/(8709120)) + (((x-0)*(x-1)*(x-2)*(x-3)*(x-4)*(x-5)*(x-6)*(x-7)*(x-8)*(x-9)*(x-11)*(x-12)*(x-13)*(x-14)*(x-15)*(x-16)*(x-17)*(x-18)*(x-19)*(x-20)*(x-21)*(x-22)*(x-23)*(x-24))*31/(-21772800)) + (((x-0)*(x-1)*(x-2)*(x-3)*(x-4)*(x-5)*(x-6)*(x-7)*(x-8)*(x-9)*(x-10)*(x-12)*(x-13)*(x-14)*(x-15)*(x-16)*(x-17)*(x-18)*(x-19)*(x-20)*(x-21)*(x-22)*(x-23)*(x-24))*37/(79833600)) + (((x-0)*(x-1)*(x-2)*(x-3)*(x-4)*(x-5)*(x-6)*(x-7)*(x-8)*(x-9)*(x-10)*(x-11)*(x-13)*(x-14)*(x-15)*(x-16)*(x-17)*(x-18)*(x-19)*(x-20)*(x-21)*(x-22)*(x-23)*(x-24))*41/(-479001600)) + (((x-0)*(x-1)*(x-2)*(x-3)*(x-4)*(x-5)*(x-6)*(x-7)*(x-8)*(x-9)*(x-10)*(x-11)*(x-12)*(x-14)*(x-15)*(x-16)*(x-17)*(x-18)*(x-19)*(x-20)*(x-21)*(x-22)*(x-23)*(x-24))*43/(6227020800)) + (((x-0)*(x-1)*(x-2)*(x-3)*(x-4)*(x-5)*(x-6)*(x-7)*(x-8)*(x-9)*(x-10)*(x-11)*(x-12)*(x-13)*(x-15)*(x-16)*(x-17)*(x-18)*(x-19)*(x-20)*(x-21)*(x-22)*(x-23)*(x-24))*47/(87178291200)) + (((x-0)*(x-1)*(x-2)*(x-3)*(x-4)*(x-5)*(x-6)*(x-7)*(x-8)*(x-9)*(x-10)*(x-11)*(x-12)*(x-13)*(x-14)*(x-16)*(x-17)*(x-18)*(x-19)*(x-20)*(x-21)*(x-22)*(x-23)*(x-24))*53/(1307674368000)) + (((x-0)*(x-1)*(x-2)*(x-3)*(x-4)*(x-5)*(x-6)*(x-7)*(x-8)*(x-9)*(x-10)*(x-11)*(x-12)*(x-13)*(x-14)*(x-15)*(x-17)*(x-18)*(x-19)*(x-20)*(x-21)*(x-22)*(x-23)*(x-24))*59/(20922789888000)) + (((x-0)*(x-1)*(x-2)*(x-3)*(x-4)*(x-5)*(x-6)*(x-7)*(x-8)*(x-9)*(x-10)*(x-11)*(x-12)*(x-13)*(x-14)*(x-15)*(x-16)*(x-18)*(x-19)*(x-20)*(x-21)*(x-22)*(x-23)*(x-24))*61/(355687428096000)) + (((x-0)*(x-1)*(x-2)*(x-3)*(x-4)*(x-5)*(x-6)*(x-7)*(x-8)*(x-9)*(x-10)*(x-11)*(x-12)*(x-13)*(x-14)*(x-15)*(x-16)*(x-17)*(x-19)*(x-20)*(x-21)*(x-22)*(x-23)*(x-24))*67/(6402373705728000)) + (((x-0)*(x-1)*(x-2)*(x-3)*(x-4)*(x-5)*(x-6)*(x-7)*(x-8)*(x-9)*(x-10)*(x-11)*(x-12)*(x-13)*(x-14)*(x-15)*(x-16)*(x-17)*(x-18)*(x-20)*(x-21)*(x-22)*(x-23)*(x-24))*71/(121645100408832000)) + (((x-0)*(x-1)*(x-2)*(x-3)*(x-4)*(x-5)*(x-6)*(x-7)*(x-8)*(x-9)*(x-10)*(x-11)*(x-12)*(x-13)*(x-14)*(x-15)*(x-16)*(x-17)*(x-18)*(x-19)*(x-21)*(x-22)*(x-23)*(x-24))*73/(2432902008176640000)) + (((x-0)*(x-1)*(x-2)*(x-3)*(x-4)*(x-5)*(x-6)*(x-7)*(x-8)*(x-9)*(x-10)*(x-11)*(x-12)*(x-13)*(x-14)*(x-15)*(x-16)*(x-17)*(x-18)*(x-19)*(x-20)*(x-22)*(x-23)*(x-24))*79/(51090942171709400000)) + (((x-0)*(x-1)*(x-2)*(x-3)*(x-4)*(x-5)*(x-6)*(x-7)*(x-8)*(x-9)*(x-10)*(x-11)*(x-12)*(x-13)*(x-14)*(x-15)*(x-16)*(x-17)*(x-18)*(x-19)*(x-20)*(x-21)*(x-23)*(x-24))*83/(1.12400072777761E+21)) + (((x-0)*(x-1)*(x-2)*(x-3)*(x-4)*(x-5)*(x-6)*(x-7)*(x-8)*(x-9)*(x-10)*(x-11)*(x-12)*(x-13)*(x-14)*(x-15)*(x-16)*(x-17)*(x-18)*(x-19)*(x-20)*(x-21)*(x-22)*(x-24))*89/(2.5852016738885E+22)) + (((x-0)*(x-1)*(x-2)*(x-3)*(x-4)*(x-5)*(x-6)*(x-7)*(x-8)*(x-9)*(x-10)*(x-11)*(x-12)*(x-13)*(x-14)*(x-15)*(x-16)*(x-17)*(x-18)*(x-19)*(x-20)*(x-21)*(x-22)*(x-23))*97/(6.20448401733239E+23)) Esta función pasa por los primeros números primos menores que 100. Easy, verdad. |
Ejemplo 2:
Por los 5 puntos siguientes: (No pongo más porque los cálculos son extensos)
(2,7) ; (3,8) ; (5,9) ; (7,10) ; (11,11)
Pasan al menos las 10 funciones siguientes:
F1(x)= ((x-3)(x-5)(x-7)(x-11))*7/((2-3)(2-5)(2-7)(2-11)) + ((x-2)(x-5)(x-7)(x-11))*8/((3-2)(3-5)(3-7)(3-11)) + ((x-2)(x-3)(x-7)(x-11))*9/((5-2)(5-3)(5-7)(5-11)) + ((x-2)(x-3)(x-5)(x-11))*10/((7-2)(7-3)(7-5)(7-11)) + ((x-2)(x-3)(x-5)(x-7))*11/((11-2)(11-3)(11-5)(11-7))
Reduciendo un poco quedaría:
F1(x)= (((x-3)*(x-5)*(x-7)*(x-11))*7/(135)) + (((x-2)*(x-5)*(x-7)*(x-11))*8/(-64)) + (((x-2)*(x-3)*(x-7)*(x-11))*9/(72)) + (((x-2)*(x-3)*(x-5)*(x-11))*10/(-160)) + (((x-2)*(x-3)*(x-5)*(x-7))*11/(1728))
F2(x)= ((x-3)(x-5)(x-7)(x-11)(x-13))*7/((2-3)(2-5)(2-7)(2-11)(2-13)) + ((x-2)(x-5)(x-7)(x-11)(x-13))*8/((3-2)(3-5)(3-7)(3-11)(3-13)) + ((x-2)(x-3)(x-7)(x-11)(x-13))*9/((5-2)(5-3)(5-7)(5-11)(5-13)) + ((x-2)(x-3)(x-5)(x-11)(x-13))*10/((7-2)(7-3)(7-5)(7-11)(7-13)) + ((x-2)(x-3)(x-5)(x-7)(x-13))*11/((11-2)(11-3)(11-5)(11-7)(11-13)) + ((x-2)(x-3)(x-5)(x-7)(x-11))*12/((13-2)(13-3)(13-5)(13-7)(13-11))
Reduciendo un poco quedaría:
F2(x)= (((x-3)*(x-5)*(x-7)*(x-11)*(x-13))*7/(135)) + (((x-2)*(x-5)*(x-7)*(x-11)*(x-13))*8/(-64)) + (((x-2)*(x-3)*(x-7)*(x-11)*(x-13))*9/(72)) + (((x-2)*(x-3)*(x-5)*(x-11)*(x-13))*10/(-160)) + (((x-2)*(x-3)*(x-5)*(x-7)*(x-13))*11/(1728)) + (((x-2)*(x-3)*(x-5)*(x-7)*(x-11))*12/(10560))
F3(x)= ((x-3)(x-5)(x-7)(x-11)(x-13)(x-17))*7/((2-3)(2-5)(2-7)(2-11)(2-13)(2-17)) + ((x-2)(x-5)(x-7)(x-11)(x-13)(x-17))*8/((3-2)(3-5)(3-7)(3-11)(3-13)(3-17)) + ((x-2)(x-3)(x-7)(x-11)(x-13)(x-17))*9/((5-2)(5-3)(5-7)(5-11)(5-13)(5-17)) + ((x-2)(x-3)(x-5)(x-11)(x-13)(x-17))*10/((7-2)(7-3)(7-5)(7-11)(7-13)(7-17)) + ((x-2)(x-3)(x-5)(x-7)(x-13)(x-17))*11/((11-2)(11-3)(11-5)(11-7)(11-13)(11-17)) + ((x-2)(x-3)(x-5)(x-7)(x-11)(x-17))*12/((13-2)(13-3)(13-5)(13-7)(13-11)(13-17)) + ((x-2)(x-3)(x-5)(x-7)(x-11)(x-13))*13/((17-2)(17-3)(17-5)(17-7)(17-11)(17-13))
Reduciendo un poco quedaría:
F3(x)= (((x-3)*(x-5)*(x-7)*(x-11)*(x-13)*(x-17))*7/(135)) + (((x-2)*(x-5)*(x-7)*(x-11)*(x-13)*(x-17))*8/(-64)) + (((x-2)*(x-3)*(x-7)*(x-11)*(x-13)*(x-17))*9/(72)) + (((x-2)*(x-3)*(x-5)*(x-11)*(x-13)*(x-17))*10/(-160)) + (((x-2)*(x-3)*(x-5)*(x-7)*(x-13)*(x-17))*11/(1728)) + (((x-2)*(x-3)*(x-5)*(x-7)*(x-11)*(x-17))*12/(10560)) + (((x-2)*(x-3)*(x-5)*(x-7)*(x-11)*(x-13))*13/(604800))
F4(x)= ((x-3)(x-5)(x-7)(x-11)(x-13)(x-17)(x-19))*7/((2-3)(2-5)(2-7)(2-11)(2-13)(2-17)(2-19)) + ((x-2)(x-5)(x-7)(x-11)(x-13)(x-17)(x-19))*8/((3-2)(3-5)(3-7)(3-11)(3-13)(3-17)(3-19)) + ((x-2)(x-3)(x-7)(x-11)(x-13)(x-17)(x-19))*9/((5-2)(5-3)(5-7)(5-11)(5-13)(5-17)(5-19)) + ((x-2)(x-3)(x-5)(x-11)(x-13)(x-17)(x-19))*10/((7-2)(7-3)(7-5)(7-11)(7-13)(7-17)(7-19)) + ((x-2)(x-3)(x-5)(x-7)(x-13)(x-17)(x-19))*11/((11-2)(11-3)(11-5)(11-7)(11-13)(11-17)(11-19)) + ((x-2)(x-3)(x-5)(x-7)(x-11)(x-17)(x-19))*12/((13-2)(13-3)(13-5)(13-7)(13-11)(13-17)(13-19)) + ((x-2)(x-3)(x-5)(x-7)(x-11)(x-13)(x-19))*13/((17-2)(17-3)(17-5)(17-7)(17-11)(17-13)(17-19)) + ((x-2)(x-3)(x-5)(x-7)(x-11)(x-13)(x-17))*14/((19-2)(19-3)(19-5)(19-7)(19-11)(19-13)(19-17))
Reduciendo un poco quedaría:
F4(x)= (((x-3)*(x-5)*(x-7)*(x-11)*(x-13)*(x-17)*(x-19))*7/(135)) + (((x-2)*(x-5)*(x-7)*(x-11)*(x-13)*(x-17)*(x-19))*8/(-64)) + (((x-2)*(x-3)*(x-7)*(x-11)*(x-13)*(x-17)*(x-19))*9/(72)) + (((x-2)*(x-3)*(x-5)*(x-11)*(x-13)*(x-17)*(x-19))*10/(-160)) + (((x-2)*(x-3)*(x-5)*(x-7)*(x-13)*(x-17)*(x-19))*11/(1728)) + (((x-2)*(x-3)*(x-5)*(x-7)*(x-11)*(x-17)*(x-19))*12/(10560)) + (((x-2)*(x-3)*(x-5)*(x-7)*(x-11)*(x-13)*(x-19))*13/(604800)) + (((x-2)*(x-3)*(x-5)*(x-7)*(x-11)*(x-13)*(x-17))*14/(4386816))
F5(x)= ((x-3)(x-5)(x-7)(x-11)(x-13)(x-17)(x-19)(x-23))*7/((2-3)(2-5)(2-7)(2-11)(2-13)(2-17)(2-19)(2-23)) + ((x-2)(x-5)(x-7)(x-11)(x-13)(x-17)(x-19)(x-23))*8/((3-2)(3-5)(3-7)(3-11)(3-13)(3-17)(3-19)(3-23)) + ((x-2)(x-3)(x-7)(x-11)(x-13)(x-17)(x-19)(x-23))*9/((5-2)(5-3)(5-7)(5-11)(5-13)(5-17)(5-19)(5-23)) + ((x-2)(x-3)(x-5)(x-11)(x-13)(x-17)(x-19)(x-23))*10/((7-2)(7-3)(7-5)(7-11)(7-13)(7-17)(7-19)(7-23)) + ((x-2)(x-3)(x-5)(x-7)(x-13)(x-17)(x-19)(x-23))*11/((11-2)(11-3)(11-5)(11-7)(11-13)(11-17)(11-19)(11-23)) + ((x-2)(x-3)(x-5)(x-7)(x-11)(x-17)(x-19)(x-23))*12/((13-2)(13-3)(13-5)(13-7)(13-11)(13-17)(13-19)(13-23)) + ((x-2)(x-3)(x-5)(x-7)(x-11)(x-13)(x-19)(x-23))*13/((17-2)(17-3)(17-5)(17-7)(17-11)(17-13)(17-19)(17-23)) + ((x-2)(x-3)(x-5)(x-7)(x-11)(x-13)(x-17)(x-23))*14/((19-2)(19-3)(19-5)(19-7)(19-11)(19-13)(19-17)(19-23)) + ((x-2)(x-3)(x-5)(x-7)(x-11)(x-13)(x-17)(x-19))*15/((23-2)(23-3)(23-5)(23-7)(23-11)(23-13)(23-17)(23-19))
Reduciendo un poco quedaría:
F5(x)= (((x-3)*(x-5)*(x-7)*(x-11)*(x-13)*(x-17)*(x-19)*(x-23))*7/(135)) + (((x-2)*(x-5)*(x-7)*(x-11)*(x-13)*(x-17)*(x-19)*(x-23))*8/(-64)) + (((x-2)*(x-3)*(x-7)*(x-11)*(x-13)*(x-17)*(x-19)*(x-23))*9/(72)) + (((x-2)*(x-3)*(x-5)*(x-11)*(x-13)*(x-17)*(x-19)*(x-23))*10/(-160)) + (((x-2)*(x-3)*(x-5)*(x-7)*(x-13)*(x-17)*(x-19)*(x-23))*11/(1728)) + (((x-2)*(x-3)*(x-5)*(x-7)*(x-11)*(x-17)*(x-19)*(x-23))*12/(10560)) + (((x-2)*(x-3)*(x-5)*(x-7)*(x-11)*(x-13)*(x-19)*(x-23))*13/(604800)) + (((x-2)*(x-3)*(x-5)*(x-7)*(x-11)*(x-13)*(x-17)*(x-23))*14/(4386816)) + (((x-2)*(x-3)*(x-5)*(x-7)*(x-11)*(x-13)*(x-17)*(x-19))*15/(348364800))
F6(x)= ((x-3)(x-5)(x-7)(x-11)(x-13)(x-17)(x-19)(x-23)(x-29))*7/((2-3)(2-5)(2-7)(2-11)(2-13)(2-17)(2-19)(2-23)(2-29)) + ((x-2)(x-5)(x-7)(x-11)(x-13)(x-17)(x-19)(x-23)(x-29))*8/((3-2)(3-5)(3-7)(3-11)(3-13)(3-17)(3-19)(3-23)(3-29)) + ((x-2)(x-3)(x-7)(x-11)(x-13)(x-17)(x-19)(x-23)(x-29))*9/((5-2)(5-3)(5-7)(5-11)(5-13)(5-17)(5-19)(5-23)(5-29)) + ((x-2)(x-3)(x-5)(x-11)(x-13)(x-17)(x-19)(x-23)(x-29))*10/((7-2)(7-3)(7-5)(7-11)(7-13)(7-17)(7-19)(7-23)(7-29)) + ((x-2)(x-3)(x-5)(x-7)(x-13)(x-17)(x-19)(x-23)(x-29))*11/((11-2)(11-3)(11-5)(11-7)(11-13)(11-17)(11-19)(11-23)(11-29)) + ((x-2)(x-3)(x-5)(x-7)(x-11)(x-17)(x-19)(x-23)(x-29))*12/((13-2)(13-3)(13-5)(13-7)(13-11)(13-17)(13-19)(13-23)(13-29)) + ((x-2)(x-3)(x-5)(x-7)(x-11)(x-13)(x-19)(x-23)(x-29))*13/((17-2)(17-3)(17-5)(17-7)(17-11)(17-13)(17-19)(17-23)(17-29)) + ((x-2)(x-3)(x-5)(x-7)(x-11)(x-13)(x-17)(x-23)(x-29))*14/((19-2)(19-3)(19-5)(19-7)(19-11)(19-13)(19-17)(19-23)(19-29)) + ((x-2)(x-3)(x-5)(x-7)(x-11)(x-13)(x-17)(x-19)(x-29))*15/((23-2)(23-3)(23-5)(23-7)(23-11)(23-13)(23-17)(23-19)(23-29)) + ((x-2)(x-3)(x-5)(x-7)(x-11)(x-13)(x-17)(x-19)(x-23))*16/((29-2)(29-3)(29-5)(29-7)(29-11)(29-13)(29-17)(29-19)(29-23))
Reduciendo un poco quedaría:
F6(x)= (((x-3)*(x-5)*(x-7)*(x-11)*(x-13)*(x-17)*(x-19)*(x-23)*(x-29))*7/(135)) + (((x-2)*(x-5)*(x-7)*(x-11)*(x-13)*(x-17)*(x-19)*(x-23)*(x-29))*8/(-64)) + (((x-2)*(x-3)*(x-7)*(x-11)*(x-13)*(x-17)*(x-19)*(x-23)*(x-29))*9/(72)) + (((x-2)*(x-3)*(x-5)*(x-11)*(x-13)*(x-17)*(x-19)*(x-23)*(x-29))*10/(-160)) + (((x-2)*(x-3)*(x-5)*(x-7)*(x-13)*(x-17)*(x-19)*(x-23)*(x-29))*11/(1728)) + (((x-2)*(x-3)*(x-5)*(x-7)*(x-11)*(x-17)*(x-19)*(x-23)*(x-29))*12/(10560)) + (((x-2)*(x-3)*(x-5)*(x-7)*(x-11)*(x-13)*(x-19)*(x-23)*(x-29))*13/(604800)) + (((x-2)*(x-3)*(x-5)*(x-7)*(x-11)*(x-13)*(x-17)*(x-23)*(x-29))*14/(4386816)) + (((x-2)*(x-3)*(x-5)*(x-7)*(x-11)*(x-13)*(x-17)*(x-19)*(x-29))*15/(348364800)) + (((x-2)*(x-3)*(x-5)*(x-7)*(x-11)*(x-13)*(x-17)*(x-19)*(x-23))*16/(76859228160))
F7(x)= ((x-3)(x-5)(x-7)(x-11)(x-13)(x-17)(x-19)(x-23)(x-29)(x-31))*7/((2-3)(2-5)(2-7)(2-11)(2-13)(2-17)(2-19)(2-23)(2-29)(2-31)) + ((x-2)(x-5)(x-7)(x-11)(x-13)(x-17)(x-19)(x-23)(x-29)(x-31))*8/((3-2)(3-5)(3-7)(3-11)(3-13)(3-17)(3-19)(3-23)(3-29)(3-31)) + ((x-2)(x-3)(x-7)(x-11)(x-13)(x-17)(x-19)(x-23)(x-29)(x-31))*9/((5-2)(5-3)(5-7)(5-11)(5-13)(5-17)(5-19)(5-23)(5-29)(5-31)) + ((x-2)(x-3)(x-5)(x-11)(x-13)(x-17)(x-19)(x-23)(x-29)(x-31))*10/((7-2)(7-3)(7-5)(7-11)(7-13)(7-17)(7-19)(7-23)(7-29)(7-31)) + ((x-2)(x-3)(x-5)(x-7)(x-13)(x-17)(x-19)(x-23)(x-29)(x-31))*11/((11-2)(11-3)(11-5)(11-7)(11-13)(11-17)(11-19)(11-23)(11-29)(11-31)) + ((x-2)(x-3)(x-5)(x-7)(x-11)(x-17)(x-19)(x-23)(x-29)(x-31))*12/((13-2)(13-3)(13-5)(13-7)(13-11)(13-17)(13-19)(13-23)(13-29)(13-31)) + ((x-2)(x-3)(x-5)(x-7)(x-11)(x-13)(x-19)(x-23)(x-29)(x-31))*13/((17-2)(17-3)(17-5)(17-7)(17-11)(17-13)(17-19)(17-23)(17-29)(17-31)) + ((x-2)(x-3)(x-5)(x-7)(x-11)(x-13)(x-17)(x-23)(x-29)(x-31))*14/((19-2)(19-3)(19-5)(19-7)(19-11)(19-13)(19-17)(19-23)(19-29)(19-31)) + ((x-2)(x-3)(x-5)(x-7)(x-11)(x-13)(x-17)(x-19)(x-29)(x-31))*15/((23-2)(23-3)(23-5)(23-7)(23-11)(23-13)(23-17)(23-19)(23-29)(23-31)) + ((x-2)(x-3)(x-5)(x-7)(x-11)(x-13)(x-17)(x-19)(x-23)(x-31))*16/((29-2)(29-3)(29-5)(29-7)(29-11)(29-13)(29-17)(29-19)(29-23)(29-31)) + ((x-2)(x-3)(x-5)(x-7)(x-11)(x-13)(x-17)(x-19)(x-23)(x-29))*17/((31-2)(31-3)(31-5)(31-7)(31-11)(31-13)(31-17)(31-19)(31-23)(31-29))
Reduciendo un poco quedaría:
F7(x)= (((x-3)*(x-5)*(x-7)*(x-11)*(x-13)*(x-17)*(x-19)*(x-23)*(x-29)*(x-31))*7/(135)) + (((x-2)*(x-5)*(x-7)*(x-11)*(x-13)*(x-17)*(x-19)*(x-23)*(x-29)*(x-31))*8/(-64)) + (((x-2)*(x-3)*(x-7)*(x-11)*(x-13)*(x-17)*(x-19)*(x-23)*(x-29)*(x-31))*9/(72)) + (((x-2)*(x-3)*(x-5)*(x-11)*(x-13)*(x-17)*(x-19)*(x-23)*(x-29)*(x-31))*10/(-160)) + (((x-2)*(x-3)*(x-5)*(x-7)*(x-13)*(x-17)*(x-19)*(x-23)*(x-29)*(x-31))*11/(1728)) + (((x-2)*(x-3)*(x-5)*(x-7)*(x-11)*(x-17)*(x-19)*(x-23)*(x-29)*(x-31))*12/(10560)) + (((x-2)*(x-3)*(x-5)*(x-7)*(x-11)*(x-13)*(x-19)*(x-23)*(x-29)*(x-31))*13/(604800)) + (((x-2)*(x-3)*(x-5)*(x-7)*(x-11)*(x-13)*(x-17)*(x-23)*(x-29)*(x-31))*14/(4386816)) + (((x-2)*(x-3)*(x-5)*(x-7)*(x-11)*(x-13)*(x-17)*(x-19)*(x-29)*(x-31))*15/(348364800)) + (((x-2)*(x-3)*(x-5)*(x-7)*(x-11)*(x-13)*(x-17)*(x-19)*(x-23)*(x-31))*16/(76859228160)) + (((x-2)*(x-3)*(x-5)*(x-7)*(x-11)*(x-13)*(x-17)*(x-19)*(x-23)*(x-29))*17/(490311843840))
F8(x)= ((x-3)(x-5)(x-7)(x-11)(x-13)(x-17)(x-19)(x-23)(x-29)(x-31)(x-37))*7/((2-3)(2-5)(2-7)(2-11)(2-13)(2-17)(2-19)(2-23)(2-29)(2-31)(2-37)) + ((x-2)(x-5)(x-7)(x-11)(x-13)(x-17)(x-19)(x-23)(x-29)(x-31)(x-37))*8/((3-2)(3-5)(3-7)(3-11)(3-13)(3-17)(3-19)(3-23)(3-29)(3-31)(3-37)) + ((x-2)(x-3)(x-7)(x-11)(x-13)(x-17)(x-19)(x-23)(x-29)(x-31)(x-37))*9/((5-2)(5-3)(5-7)(5-11)(5-13)(5-17)(5-19)(5-23)(5-29)(5-31)(5-37)) + ((x-2)(x-3)(x-5)(x-11)(x-13)(x-17)(x-19)(x-23)(x-29)(x-31)(x-37))*10/((7-2)(7-3)(7-5)(7-11)(7-13)(7-17)(7-19)(7-23)(7-29)(7-31)(7-37)) + ((x-2)(x-3)(x-5)(x-7)(x-13)(x-17)(x-19)(x-23)(x-29)(x-31)(x-37))*11/((11-2)(11-3)(11-5)(11-7)(11-13)(11-17)(11-19)(11-23)(11-29)(11-31)(11-37)) + ((x-2)(x-3)(x-5)(x-7)(x-11)(x-17)(x-19)(x-23)(x-29)(x-31)(x-37))*12/((13-2)(13-3)(13-5)(13-7)(13-11)(13-17)(13-19)(13-23)(13-29)(13-31)(13-37)) + ((x-2)(x-3)(x-5)(x-7)(x-11)(x-13)(x-19)(x-23)(x-29)(x-31)(x-37))*13/((17-2)(17-3)(17-5)(17-7)(17-11)(17-13)(17-19)(17-23)(17-29)(17-31)(17-37)) + ((x-2)(x-3)(x-5)(x-7)(x-11)(x-13)(x-17)(x-23)(x-29)(x-31)(x-37))*14/((19-2)(19-3)(19-5)(19-7)(19-11)(19-13)(19-17)(19-23)(19-29)(19-31)(19-37)) + ((x-2)(x-3)(x-5)(x-7)(x-11)(x-13)(x-17)(x-19)(x-29)(x-31)(x-37))*15/((23-2)(23-3)(23-5)(23-7)(23-11)(23-13)(23-17)(23-19)(23-29)(23-31)(23-37)) + ((x-2)(x-3)(x-5)(x-7)(x-11)(x-13)(x-17)(x-19)(x-23)(x-31)(x-37))*16/((29-2)(29-3)(29-5)(29-7)(29-11)(29-13)(29-17)(29-19)(29-23)(29-31)(29-37)) + ((x-2)(x-3)(x-5)(x-7)(x-11)(x-13)(x-17)(x-19)(x-23)(x-29)(x-37))*17/((31-2)(31-3)(31-5)(31-7)(31-11)(31-13)(31-17)(31-19)(31-23)(31-29)(31-37)) + ((x-2)(x-3)(x-5)(x-7)(x-11)(x-13)(x-17)(x-19)(x-23)(x-29)(x-31))*18/((37-2)(37-3)(37-5)(37-7)(37-11)(37-13)(37-17)(37-19)(37-23)(37-29)(37-31))
Reduciendo un poco quedaría:
F8(x)= (((x-3)*(x-5)*(x-7)*(x-11)*(x-13)*(x-17)*(x-19)*(x-23)*(x-29)*(x-31)*(x-37))*7/(135)) + (((x-2)*(x-5)*(x-7)*(x-11)*(x-13)*(x-17)*(x-19)*(x-23)*(x-29)*(x-31)*(x-37))*8/(-64)) + (((x-2)*(x-3)*(x-7)*(x-11)*(x-13)*(x-17)*(x-19)*(x-23)*(x-29)*(x-31)*(x-37))*9/(72)) + (((x-2)*(x-3)*(x-5)*(x-11)*(x-13)*(x-17)*(x-19)*(x-23)*(x-29)*(x-31)*(x-37))*10/(-160)) + (((x-2)*(x-3)*(x-5)*(x-7)*(x-13)*(x-17)*(x-19)*(x-23)*(x-29)*(x-31)*(x-37))*11/(1728)) + (((x-2)*(x-3)*(x-5)*(x-7)*(x-11)*(x-17)*(x-19)*(x-23)*(x-29)*(x-31)*(x-37))*12/(10560)) + (((x-2)*(x-3)*(x-5)*(x-7)*(x-11)*(x-13)*(x-19)*(x-23)*(x-29)*(x-31)*(x-37))*13/(604800)) + (((x-2)*(x-3)*(x-5)*(x-7)*(x-11)*(x-13)*(x-17)*(x-23)*(x-29)*(x-31)*(x-37))*14/(4386816)) + (((x-2)*(x-3)*(x-5)*(x-7)*(x-11)*(x-13)*(x-17)*(x-19)*(x-29)*(x-31)*(x-37))*15/(348364800)) + (((x-2)*(x-3)*(x-5)*(x-7)*(x-11)*(x-13)*(x-17)*(x-19)*(x-23)*(x-31)*(x-37))*16/(76859228160)) + (((x-2)*(x-3)*(x-5)*(x-7)*(x-11)*(x-13)*(x-17)*(x-19)*(x-23)*(x-29)*(x-37))*17/(490311843840)) + (((x-2)*(x-3)*(x-5)*(x-7)*(x-11)*(x-13)*(x-17)*(x-19)*(x-23)*(x-29)*(x-31))*18/(172454510592000))
F9(x)= ((x-3)(x-5)(x-7)(x-11)(x-13)(x-17)(x-19)(x-23)(x-29)(x-31)(x-37)(x-39))*7/((2-3)(2-5)(2-7)(2-11)(2-13)(2-17)(2-19)(2-23)(2-29)(2-31)(2-37)(2-39)) + ((x-2)(x-5)(x-7)(x-11)(x-13)(x-17)(x-19)(x-23)(x-29)(x-31)(x-37)(x-39))*8/((3-2)(3-5)(3-7)(3-11)(3-13)(3-17)(3-19)(3-23)(3-29)(3-31)(3-37)(3-39)) + ((x-2)(x-3)(x-7)(x-11)(x-13)(x-17)(x-19)(x-23)(x-29)(x-31)(x-37)(x-39))*9/((5-2)(5-3)(5-7)(5-11)(5-13)(5-17)(5-19)(5-23)(5-29)(5-31)(5-37)(5-39)) + ((x-2)(x-3)(x-5)(x-11)(x-13)(x-17)(x-19)(x-23)(x-29)(x-31)(x-37)(x-39))*10/((7-2)(7-3)(7-5)(7-11)(7-13)(7-17)(7-19)(7-23)(7-29)(7-31)(7-37)(7-39)) + ((x-2)(x-3)(x-5)(x-7)(x-13)(x-17)(x-19)(x-23)(x-29)(x-31)(x-37)(x-39))*11/((11-2)(11-3)(11-5)(11-7)(11-13)(11-17)(11-19)(11-23)(11-29)(11-31)(11-37)(11-39)) + ((x-2)(x-3)(x-5)(x-7)(x-11)(x-17)(x-19)(x-23)(x-29)(x-31)(x-37)(x-39))*12/((13-2)(13-3)(13-5)(13-7)(13-11)(13-17)(13-19)(13-23)(13-29)(13-31)(13-37)(13-39)) + ((x-2)(x-3)(x-5)(x-7)(x-11)(x-13)(x-19)(x-23)(x-29)(x-31)(x-37)(x-39))*13/((17-2)(17-3)(17-5)(17-7)(17-11)(17-13)(17-19)(17-23)(17-29)(17-31)(17-37)(17-39)) + ((x-2)(x-3)(x-5)(x-7)(x-11)(x-13)(x-17)(x-23)(x-29)(x-31)(x-37)(x-39))*14/((19-2)(19-3)(19-5)(19-7)(19-11)(19-13)(19-17)(19-23)(19-29)(19-31)(19-37)(19-39)) + ((x-2)(x-3)(x-5)(x-7)(x-11)(x-13)(x-17)(x-19)(x-29)(x-31)(x-37)(x-39))*15/((23-2)(23-3)(23-5)(23-7)(23-11)(23-13)(23-17)(23-19)(23-29)(23-31)(23-37)(23-39)) + ((x-2)(x-3)(x-5)(x-7)(x-11)(x-13)(x-17)(x-19)(x-23)(x-31)(x-37)(x-39))*16/((29-2)(29-3)(29-5)(29-7)(29-11)(29-13)(29-17)(29-19)(29-23)(29-31)(29-37)(29-39)) + ((x-2)(x-3)(x-5)(x-7)(x-11)(x-13)(x-17)(x-19)(x-23)(x-29)(x-37)(x-39))*17/((31-2)(31-3)(31-5)(31-7)(31-11)(31-13)(31-17)(31-19)(31-23)(31-29)(31-37)(31-39)) + ((x-2)(x-3)(x-5)(x-7)(x-11)(x-13)(x-17)(x-19)(x-23)(x-29)(x-31)(x-39))*18/((37-2)(37-3)(37-5)(37-7)(37-11)(37-13)(37-17)(37-19)(37-23)(37-29)(37-31)(37-39)) + ((x-2)(x-3)(x-5)(x-7)(x-11)(x-13)(x-17)(x-19)(x-23)(x-29)(x-31)(x-37))*19/((39-2)(39-3)(39-5)(39-7)(39-11)(39-13)(39-17)(39-19)(39-23)(39-29)(39-31)(39-37))
Reduciendo un poco quedaría:
F9(x)= (((x-3)*(x-5)*(x-7)*(x-11)*(x-13)*(x-17)*(x-19)*(x-23)*(x-29)*(x-31)*(x-37)*(x-39))*7/(135)) + (((x-2)*(x-5)*(x-7)*(x-11)*(x-13)*(x-17)*(x-19)*(x-23)*(x-29)*(x-31)*(x-37)*(x-39))*8/(-64)) + (((x-2)*(x-3)*(x-7)*(x-11)*(x-13)*(x-17)*(x-19)*(x-23)*(x-29)*(x-31)*(x-37)*(x-39))*9/(72)) + (((x-2)*(x-3)*(x-5)*(x-11)*(x-13)*(x-17)*(x-19)*(x-23)*(x-29)*(x-31)*(x-37)*(x-39))*10/(-160)) + (((x-2)*(x-3)*(x-5)*(x-7)*(x-13)*(x-17)*(x-19)*(x-23)*(x-29)*(x-31)*(x-37)*(x-39))*11/(1728)) + (((x-2)*(x-3)*(x-5)*(x-7)*(x-11)*(x-17)*(x-19)*(x-23)*(x-29)*(x-31)*(x-37)*(x-39))*12/(10560)) + (((x-2)*(x-3)*(x-5)*(x-7)*(x-11)*(x-13)*(x-19)*(x-23)*(x-29)*(x-31)*(x-37)*(x-39))*13/(604800)) + (((x-2)*(x-3)*(x-5)*(x-7)*(x-11)*(x-13)*(x-17)*(x-23)*(x-29)*(x-31)*(x-37)*(x-39))*14/(4386816)) + (((x-2)*(x-3)*(x-5)*(x-7)*(x-11)*(x-13)*(x-17)*(x-19)*(x-29)*(x-31)*(x-37)*(x-39))*15/(348364800)) + (((x-2)*(x-3)*(x-5)*(x-7)*(x-11)*(x-13)*(x-17)*(x-19)*(x-23)*(x-31)*(x-37)*(x-39))*16/(76859228160)) + (((x-2)*(x-3)*(x-5)*(x-7)*(x-11)*(x-13)*(x-17)*(x-19)*(x-23)*(x-29)*(x-37)*(x-39))*17/(490311843840)) + (((x-2)*(x-3)*(x-5)*(x-7)*(x-11)*(x-13)*(x-17)*(x-19)*(x-23)*(x-29)*(x-31)*(x-39))*18/(172454510592000)) + (((x-2)*(x-3)*(x-5)*(x-7)*(x-11)*(x-13)*(x-17)*(x-19)*(x-23)*(x-29)*(x-31)*(x-37))*19/(1188384944947200))
F10(x)= ((x-3)(x-5)(x-7)(x-11)(x-13)(x-17)(x-19)(x-23)(x-29)(x-31)(x-37)(x-39)(x-41))*7/((2-3)(2-5)(2-7)(2-11)(2-13)(2-17)(2-19)(2-23)(2-29)(2-31)(2-37)(2-39)(2-41)) + ((x-2)(x-5)(x-7)(x-11)(x-13)(x-17)(x-19)(x-23)(x-29)(x-31)(x-37)(x-39)(x-41))*8/((3-2)(3-5)(3-7)(3-11)(3-13)(3-17)(3-19)(3-23)(3-29)(3-31)(3-37)(3-39)(3-41)) + ((x-2)(x-3)(x-7)(x-11)(x-13)(x-17)(x-19)(x-23)(x-29)(x-31)(x-37)(x-39)(x-41))*9/((5-2)(5-3)(5-7)(5-11)(5-13)(5-17)(5-19)(5-23)(5-29)(5-31)(5-37)(5-39)(5-41)) + ((x-2)(x-3)(x-5)(x-11)(x-13)(x-17)(x-19)(x-23)(x-29)(x-31)(x-37)(x-39)(x-41))*10/((7-2)(7-3)(7-5)(7-11)(7-13)(7-17)(7-19)(7-23)(7-29)(7-31)(7-37)(7-39)(7-41)) + ((x-2)(x-3)(x-5)(x-7)(x-13)(x-17)(x-19)(x-23)(x-29)(x-31)(x-37)(x-39)(x-41))*11/((11-2)(11-3)(11-5)(11-7)(11-13)(11-17)(11-19)(11-23)(11-29)(11-31)(11-37)(11-39)(11-41)) + ((x-2)(x-3)(x-5)(x-7)(x-11)(x-17)(x-19)(x-23)(x-29)(x-31)(x-37)(x-39)(x-41))*12/((13-2)(13-3)(13-5)(13-7)(13-11)(13-17)(13-19)(13-23)(13-29)(13-31)(13-37)(13-39)(13-41)) + ((x-2)(x-3)(x-5)(x-7)(x-11)(x-13)(x-19)(x-23)(x-29)(x-31)(x-37)(x-39)(x-41))*13/((17-2)(17-3)(17-5)(17-7)(17-11)(17-13)(17-19)(17-23)(17-29)(17-31)(17-37)(17-39)(17-41)) + ((x-2)(x-3)(x-5)(x-7)(x-11)(x-13)(x-17)(x-23)(x-29)(x-31)(x-37)(x-39)(x-41))*14/((19-2)(19-3)(19-5)(19-7)(19-11)(19-13)(19-17)(19-23)(19-29)(19-31)(19-37)(19-39)(19-41)) + ((x-2)(x-3)(x-5)(x-7)(x-11)(x-13)(x-17)(x-19)(x-29)(x-31)(x-37)(x-39)(x-41))*15/((23-2)(23-3)(23-5)(23-7)(23-11)(23-13)(23-17)(23-19)(23-29)(23-31)(23-37)(23-39)(23-41)) + ((x-2)(x-3)(x-5)(x-7)(x-11)(x-13)(x-17)(x-19)(x-23)(x-31)(x-37)(x-39)(x-41))*16/((29-2)(29-3)(29-5)(29-7)(29-11)(29-13)(29-17)(29-19)(29-23)(29-31)(29-37)(29-39)(29-41)) + ((x-2)(x-3)(x-5)(x-7)(x-11)(x-13)(x-17)(x-19)(x-23)(x-29)(x-37)(x-39)(x-41))*17/((31-2)(31-3)(31-5)(31-7)(31-11)(31-13)(31-17)(31-19)(31-23)(31-29)(31-37)(31-39)(31-41)) + ((x-2)(x-3)(x-5)(x-7)(x-11)(x-13)(x-17)(x-19)(x-23)(x-29)(x-31)(x-39)(x-41))*18/((37-2)(37-3)(37-5)(37-7)(37-11)(37-13)(37-17)(37-19)(37-23)(37-29)(37-31)(37-39)(37-41)) + ((x-2)(x-3)(x-5)(x-7)(x-11)(x-13)(x-17)(x-19)(x-23)(x-29)(x-31)(x-37)(x-41))*19/((39-2)(39-3)(39-5)(39-7)(39-11)(39-13)(39-17)(39-19)(39-23)(39-29)(39-31)(39-37)(39-41)) + ((x-2)(x-3)(x-5)(x-7)(x-11)(x-13)(x-17)(x-19)(x-23)(x-29)(x-31)(x-37)(x-39))*20/((41-2)(41-3)(41-5)(41-7)(41-11)(41-13)(41-17)(41-19)(41-23)(41-29)(41-31)(41-37)(41-39))
Reduciendo un poco quedaría:
F10(x)= (((x-3)*(x-5)*(x-7)*(x-11)*(x-13)*(x-17)*(x-19)*(x-23)*(x-29)*(x-31)*(x-37)*(x-39)*(x-41))*7/(135)) + (((x-2)*(x-5)*(x-7)*(x-11)*(x-13)*(x-17)*(x-19)*(x-23)*(x-29)*(x-31)*(x-37)*(x-39)*(x-41))*8/(-64)) + (((x-2)*(x-3)*(x-7)*(x-11)*(x-13)*(x-17)*(x-19)*(x-23)*(x-29)*(x-31)*(x-37)*(x-39)*(x-41))*9/(72)) + (((x-2)*(x-3)*(x-5)*(x-11)*(x-13)*(x-17)*(x-19)*(x-23)*(x-29)*(x-31)*(x-37)*(x-39)*(x-41))*10/(-160)) + (((x-2)*(x-3)*(x-5)*(x-7)*(x-13)*(x-17)*(x-19)*(x-23)*(x-29)*(x-31)*(x-37)*(x-39)*(x-41))*11/(1728)) + (((x-2)*(x-3)*(x-5)*(x-7)*(x-11)*(x-17)*(x-19)*(x-23)*(x-29)*(x-31)*(x-37)*(x-39)*(x-41))*12/(10560)) + (((x-2)*(x-3)*(x-5)*(x-7)*(x-11)*(x-13)*(x-19)*(x-23)*(x-29)*(x-31)*(x-37)*(x-39)*(x-41))*13/(604800)) + (((x-2)*(x-3)*(x-5)*(x-7)*(x-11)*(x-13)*(x-17)*(x-23)*(x-29)*(x-31)*(x-37)*(x-39)*(x-41))*14/(4386816)) + (((x-2)*(x-3)*(x-5)*(x-7)*(x-11)*(x-13)*(x-17)*(x-19)*(x-29)*(x-31)*(x-37)*(x-39)*(x-41))*15/(348364800)) + (((x-2)*(x-3)*(x-5)*(x-7)*(x-11)*(x-13)*(x-17)*(x-19)*(x-23)*(x-31)*(x-37)*(x-39)*(x-41))*16/(76859228160)) + (((x-2)*(x-3)*(x-5)*(x-7)*(x-11)*(x-13)*(x-17)*(x-19)*(x-23)*(x-29)*(x-37)*(x-39)*(x-41))*17/(490311843840)) + (((x-2)*(x-3)*(x-5)*(x-7)*(x-11)*(x-13)*(x-17)*(x-19)*(x-23)*(x-29)*(x-31)*(x-39)*(x-41))*18/(172454510592000)) + (((x-2)*(x-3)*(x-5)*(x-7)*(x-11)*(x-13)*(x-17)*(x-19)*(x-23)*(x-29)*(x-31)*(x-37)*(x-41))*19/(1188384944947200)) + (((x-2)*(x-3)*(x-5)*(x-7)*(x-11)*(x-13)*(x-17)*(x-19)*(x-23)*(x-29)*(x-31)*(x-37)*(x-39))*20/(13902297189580800))
Todas estas funciones pasan por los 5 puntos dados, pero además la última pasa por los primeros 14 números primos
Ven que fácil es interpolar una función que pase por n puntos.
Si seguimos agregando puntos sigo encontrando funciones…
Así que culmino como terminé:
“Qué pasa, este tipo de soluciones fuerzan a que el ejercicio tome la solución que yo quiera y no la que «sugiere o muestra» el ejercicio. Por eso no me gusta aplicarla. Pero es válida cuando no se tiene la posibilidad de encontrar la lógica de lo que se quiere expresar con el ejercicio”
Disculpen por no saberme explicar, hace años que pasé por la matemática y no estoy al tanto de todo el vocabulario ni de todo el conocimiento actual para realizar una demostración más sencilla.
ya vi que para los primos tienes un xE+23 y otro xE+22 para nada divertido lidiar con semejantes números, si dejas este post abierto no creo que a Nestor le guste, tener un side-site donde se desarrolle una discusión paralela pienso que debería consultarlo con él, en cualquier caso mi opinión es que es un improvement con respecto a cd, jajajaja, si agregas una sección de reglas, para comentarios y algo para interpretar latex y código, los comentarios y tu respuesta se van a enriquecer esto se me parece wordpress, te puedo ayudar con esto, mi internet es bastante decente. me dices y te puedo enviar los plugins para que los instales.
saludos
Saludos, es verdad, el cálculo es enorme, es 24! = 620448401733239439360000, lo que pasa es que hice el cáculo en excel para andar más rápido que en una calculadora.
Je, jee
A Nestor le comuniqué con un SMS y le le hice un usuario, esperemos por su respuesta.
Además, nuca va a competir con su sección. No se aceptarán acertijos nuevos, solamente explicaciones a los ya publicados en Cubadebate o extensiones de los mismo después de terminada la semana de publicación o para exponer aluna tabla etc, que en Cubadebate no se pueda poner.
Las otras secciones como las de los programadores ayudaría a que los que resuelven el acertijo por esa vía compartan la solución.
Lo de los Pluggin, los pone el sitio, no ha incluío latex, no obstante regístrate y los dos (y otros de para pensar) podemos administrarlos y mejorarlo. Siempre con la idea de extender, no suplantar.
La secuencia de funciones la realicé en excel,(Hacer la función literal)