We go into Fermat´s function II / Adentrándonos en la función de Fermat II

Sea f, el valor que satisfaga / Let f, the value that satisfies:  a^f + b^f = c^f.

La direfencia con Fermat es que no obligaremos a que f sea un número natural. En su lugar buscaremos números f reales.
The difference with Fermat is that we will not force "f" was a natural number. Instead look real numbers for "f".

Conjuntos de tres / Sets of three elements

Lo primero es buscar el orden en las ternas {a, b, c}. La forma más fácil ser buscar los conjuntos que cumplan que la suma de sus elementos sean un número determinado y que c>a || b>a.

The first is to find order in the triples {a, b, c}. The easiest way to find the sets be satisfying than the sum of its elements are a certain number and c> a || b> a.

C(3) = {Ø}, 0 elementos.
C(4) = {( 1 1 2)} 1  elementos.
C(5) = {( 1 1 3), ( 1 2 2)} 2  elementos.
C(6) = {( 1 1 4), ( 1 2 3), ( 2 2 2)} 3  elementos.
C(7) = {( 1 1 5), ( 1 2 4), ( 1 3 3), ( 2 2 3)}
C(8) = {( 1 1 6), ( 1 2 5), ( 1 3 4), ( 2 2 4), ( 2 3 3)}
...
C(20) = {( 1 1 18), ( 1 2 17), ( 1 3 16), ( 1 4 15), ( 1 5 14), ( 1 6 13), ( 1 7 12), ( 1 8 11), ( 1 9 10), ( 2 2 16), ( 2 3 15), ( 2 4 14), ( 2 5 13), ( 2 6 12), ( 2 7 11), ( 2 8 10), ( 2 9 9), ( 3 3 14), ( 3 4 13), ( 3 5 12), ( 3 6 11), ( 3 7 10), ( 3 8 9), ( 4 4 12), ( 4 5 11), ( 4 6 10), ( 4 7 9), ( 4 8 8), ( 5 5 10), ( 5 6 9), ( 5 7 8), ( 6 6 8), ( 6 7 7)}  33 elementos.

Por si alguno tiene curiosidad el número de elementos de cada conjunto es:
In case anyone was curious, the number of elements in each set is:

( 2 0), ( 3 0), ( 4 1), ( 5 1), ( 6 2), ( 7 3), ( 8 4), ( 9 5), ( 10 7), ( 11 8), ( 12 10), ( 13 12), ( 14 14), ( 15 16), ( 16 19), ( 17 21), ( 18 24), ( 19 27), ( 20 30), ( 21 33), ( 22 37), ( 23 40), ( 24 44), ( 25 48), ( 26 52), ( 27 56), ( 28 61), ( 29 65), ( 30 70), ( 31 75), ( 32 80), ( 33 85), ( 34 91), ( 35 96), ( 36 102), ( 37 108), ( 38 114), ( 39 120), ( 40 127), ( 41 133), ( 42 140), ( 43 147), ( 44 154), ( 45 161), ( 46 169), ( 47 176), ( 48 184), ( 49 192), ( 50 200), ( 51 208), ( 52 217), ( 53 225), ( 54 234), ( 55 243), ( 56 252), ( 57 261), ( 58 271), ( 59 280), ( 60 290), ( 61 300), ( 62 310), ( 63 320), ( 64 331), ( 65 341), ( 66 352), ( 67 363), ( 68 374), ( 69 385), ( 70 397), ( 71 408), ( 72 420), ( 73 432), ( 74 444), ( 75 456), ( 76 469), ( 77 481), ( 78 494), ( 79 507), ( 80 520), ( 81 533), ( 82 547), ( 83 560), ( 84 574), ( 85 588), ( 86 602), ( 87 616), ( 88 631), ( 89 645), ( 90 660), ( 91 675), ( 92 690), ( 93 705), ( 94 721), ( 95 736), ( 96 752), ( 97 768), ( 98 784), ( 99 800), ( 100 817), ( 101 833),

Satisface la ecuación: y = 0,0833x2 - 0,1663x - 0,0207
Pero esto nos despista de nuestro objetivo

La idea / The idea

Una vez ordenados los conjuntos C(n) buscar los f de cada elemento del conjunto y ver que pinta tienen.Para cumplir este objetivo lo primero que hay que hacer es programar una función para calcular f conociendo a, b y c.

Once ordered sets C (n), we have to find the f of each element of the set and see how they are. To meet this objective the first thing to do is to make an function to calculate b and c knowing f.


Function CalculaFa(ByVal a As Integer, 
                   ByVal b As Integer,
                   ByVal c As Integer) As Double
g = 0:p = 0.1:i1 = Sgn(a ^ g + b ^ g - c ^ g)
100:
Do
  i0 = i1: g = g + p
  h1 = a ^ g + b ^ g - c ^ g
  i1 = Sgn(h1)
Loop Until i0 <> i1
p = -p / 10
If Abs(p) > 0.0000001 Then GoTo 100
CalculaFa = g
End Function



Resultado numérico: / Numerical results:

C(5): (1 1 3)→0,630930 (1 2 2)→Inf.  C(6): (1 1 4)→0,500000 (1 2 3)→1,000000 (2 2 2)→Inf.  C(7): (1 1 5)→0,430677 (1 2 4)→0,694242 (1 3 3)→Inf. (2 2 3)→1,709511  C(8): (1 1 6)→0,386853 (1 2 5)→0,563896 (1 3 4)→1,000000 (2 2 4)→1,000000 (2 3 3)→Inf. C(9): (1 1 7)→0,356207 (1 2 6)→0,489536 (1 3 5)→0,727160 (1 4 4)→Inf. (2 2 5)→0,756471 (2 3 4)→1,507127 (3 3 3)→Inf.  C(10): (1 1 8)→0,333333 (1 2 7)→0,440660 (1 3 6)→0,600967 (1 4 5)→1,000000 (2 2 6)→0,630930 (2 3 5)→1,000000 (2 4 4)→Inf. (3 3 4)→2,409421  C(11): (1 1 9)→0,315465 (1 2 8)→0,405685 (1 3 7)→0,525764 (1 4 6)→0,748222 (1 5 5)→Inf. (2 2 7)→0,553295 (2 3 6)→0,787885 (2 4 5)→1,421586 (3 3 5)→1,356916 (3 4 4)→Inf.

C(12): (1 1 10)→0,301030 (1 2 9)→0,379195 (1 3 8)→0,474995 (1 4 7)→0,626250 (1 5 6)→1,000000 (2 2 8)→0,500000 (2 3 7)→0,668850 (2 4 6)→1,000000 (2 5 5)→Inf. (3 3 6)→1,000000 (3 4 5)→2,000000 (4 4 4)→Inf.  C(13): (1 1 11)→0,289065 (1 2 10)→0,358299 (1 3 9)→0,438018 (1 4 8)→0,551463 (1 5 7)→0,763203 (1 6 6)→Inf. (2 2 9)→0,460846 (2 3 8)→0,591710 (2 4 7)→0,807572 (2 5 6)→1,372508 (3 3 7)→0,818068 (3 4 6)→1,293174 (3 5 5)→Inf. (4 4 5)→3,106284  C(14): (1 1 12)→0,278943 (1 2 11)→0,341305 (1 3 10)→0,409664 (1 4 9)→0,500000 (1 5 8)→0,645009 (1 6 7)→1,000000 (2 2 10)→0,430677 (2 3 9)→0,537187 (2 4 8)→0,694242 (2 5 7)→1,000000 (2 6 6)→Inf. (3 3 8)→0,706695 (3 4 7)→1,000000 (3 5 6)→1,822550 (4 4 6)→1,709511 (4 5 5)→Inf.

C(15): (1 1 13)→0,270238 (1 2 12)→0,327153 (1 3 11)→0,387098 (1 4 10)→0,462006 (1 5 9)→0,571072 (1 6 8)→0,774576 (1 7 7)→Inf. (2 2 11)→0,406598 (2 3 10)→0,496338 (2 4 9)→0,618529 (2 5 8)→0,821352 (2 6 7)→1,339977 (3 3 9)→0,630930 (3 4 8)→0,836778 (3 5 7)→1,256656 (3 6 6)→Inf. (4 4 7)→1,238613 (4 5 6)→2,487939 (5 5 5)→Inf.  C(16): (1 1 14)→0,262650 (1 2 13)→0,315140 (1 3 12)→0,368624 (1 4 11)→0,432581 (1 5 10)→0,519463 (1 6 9)→0,659684 (1 7 8)→1,000000 (2 2 12)→0,386853 (2 3 11)→0,464425 (2 4 10)→0,563896 (2 5 9)→0,712809 (2 6 8)→1,000000 (2 7 7)→Inf. (3 3 10)→0,575717 (3 4 9)→0,731774 (3 5 8)→1,000000 (3 6 7)→1,719845 (4 4 8)→1,000000 (4 5 7)→1,581624 (4 6 6)→Inf. (5 5 6)→3,801784

C(17): (1 1 15)→0,255958 (1 2 14)→0,304786 (1 3 13)→0,353161 (1 4 12)→0,408985 (1 5 11)→0,480955 (1 6 10)→0,586741 (1 7 9)→0,783603 (1 8 8)→Inf. (2 2 13)→0,370310 (2 3 12)→0,438694 (2 4 11)→0,522352 (2 5 10)→0,638731 (2 6 9)→0,831677 (2 7 8)→1,316498 (3 3 11)→0,533484 (3 4 10)→0,658051 (3 5 9)→0,849761 (3 6 8)→1,232521 (3 7 7)→Inf. (4 4 9)→0,854756 (4 5 8)→1,207399 (4 6 7)→2,215951 (5 5 7)→2,060043 (5 6 6)→Inf.  C(18): (1 1 16)→0,250000 (1 2 15)→0,295743 (1 3 14)→0,339986 (1 4 13)→0,389554 (1 5 12)→0,450889 (1 6 11)→0,535259 (1 7 10)→0,671597 (1 8 9)→1,000000 (2 2 14)→0,356207 (2 3 13)→0,417433 (2 4 12)→0,489536 (2 5 11)→0,584449 (2 6 10)→0,727160 (2 7 9)→1,000000 (2 8 8)→Inf. (3 3 12)→0,500000 (3 4 11)→0,603151 (3 5 10)→0,749960 (3 6 9)→1,000000 (3 7 8)→1,651588 (4 4 10)→0,756471 (4 5 9)→1,000000 (4 6 8)→1,507127 (4 7 7)→Inf. (5 5 8)→1,474770 (5 6 7)→2,973549 (6 6 6)→Inf.

C(19) (1 1 17)→0,244651 (1 2 16)→0,287761 (1 3 15)→0,328594 (1 4 14)→0,373214 (1 5 13)→0,426624 (1 6 12)→0,496517 (1 7 11)→0,599669 (1 8 10)→0,791002 (1 9 9)→Inf. (2 2 15)→0,344010 (2 3 14)→0,399514 (2 4 13)→0,462854 (2 5 12)→0,542693 (2 6 11)→0,654699 (2 7 10)→0,839780 (2 8 9)→1,298569 (3 3 13)→0,472707 (3 4 12)→0,560499 (3 5 11)→0,678323 (3 6 10)→0,859418 (3 7 9)→1,215158 (3 8 8)→Inf. (4 4 11)→0,685198 (4 5 10)→0,867160 (4 6 9)→1,186814 (4 7 8)→2,057504 (5 5 9)→1,179250 (5 6 8)→1,867720 (5 7 7)→Inf. (6 6 7)→4,496556

C(20): (1 1 18)→0,239813 (1 2 17)→0,280648 (1 3 16)→0,318622 (1 4 15)→0,359240 (1 5 14)→0,406541 (1 6 13)→0,466065 (1 7 12)→0,548456 (1 8 11)→0,681533 (1 9 10)→1,000000 (2 2 16)→0,333333 (2 3 15)→0,384168 (2 4 14)→0,440660 (2 5 13)→0,509415 (2 6 12)→0,600967 (2 7 11)→0,738690 (2 8 10)→1,000000 (2 9 9)→Inf. (3 3 14)→0,449966 (3 4 13)→0,526285 (3 5 12)→0,624114 (3 6 11)→0,763919 (3 7 10)→1,000000 (3 8 9)→1,602326 (4 4 12)→0,630930 (4 5 11)→0,774279 (4 6 10)→1,000000 (4 7 9)→1,457470 (4 8 8)→Inf. (5 5 10)→1,000000 (5 6 9)→1,412357 (5 7 8)→2,605651 (6 6 8)→2,409421 (6 7 7)→Inf. ...

C(25): (1 1 23)→0,221065 (1 2 22)→0,253971 (1 3 21)→0,282600 (1 4 20)→0,311047 (1 5 19)→0,341358 (1 6 18)→0,375361 (1 7 17)→0,415359 (1 8 16)→0,464959 (1 9 15)→0,530840 (1 10 14)→0,628226 (1 11 13)→0,807162 (1 12 12)→Inf. (2 2 21)→0,294784 (2 3 20)→0,331166 (2 4 19)→0,368173 (2 5 18)→0,408588 (2 6 17)→0,455182 (2 7 16)→0,511801 (2 8 15)→0,584974 (2 9 14)→0,688046 (2 10 13)→0,856493 (2 11 12)→1,262874 (3 3 19)→0,375521 (3 4 18)→0,421736 (3 5 17)→0,473532 (3 6 16)→0,535062 (3 7 15)→0,612636 (3 8 14)→0,717976 (3 9 13)→0,878073 (3 10 12)→1,182930 (3 11 11)→Inf. (4 4 17)→0,479051 (4 5 16)→0,545214 (4 6 15)→0,626613 (4 7 14)→0,733936 (4 8 13)→0,889217 (4 9 12)→1,151962 (4 10 11)→1,817457 (5 5 15)→0,630930 (5 6 14)→0,741031 (5 7 13)→0,894776 (5 8 12)→1,137374 (5 9 11)→1,623440 (5 10 10)→Inf. (6 6 13)→0,896477 (6 7 12)→1,131270 (6 8 11)→1,552882 (6 9 10)→2,720194 (7 7 11)→1,533562 (7 8 10)→2,436346 (7 9 9)→Inf. (8 8 9)→5,884949

Resultado gráfico: / Graphic results:


Cada banda vertical es un número f, azul para C(1) y rojo para C(255) según arco iris
Each vertical strip is a number f, blue for C (1) and red for C (255) according rainbow scale.

Ídem, pero en lugar de formar una banda vertical, forma un punto y en ordenadas el valor de x de C(x).
Idem, but instead of forming a vertical band, it forms a neat point and the value of x in C (x).

Salvo ante valores de 0, 1 y 2 parece que los némeros "f" rehuyeran los demás números enteros, y quebrados y similares. Se puede concluir que: "x Î R, $ fÎ[x-e,x+e]Î R & a,b,c Î N | a^f+b^f=c^f . Es decir, dado un número real, x, siempre existe un número real en el intervalo [x-e,x+e] que satisface la ecuación a^f+b^f=c^f con valores enteros de a,b y c.

Pongamos un ejemplo. Sea x=3, vamos a ver cómo podemos aproximarnos:

 C(10); (3 3 4);       f = 2,409421;  e = 0,590579

 C(13); (4 4 5);       f = 3,106284;  e = 0,106283

 C(18); (5 6 7);       f = 2,973549;  e = 2,645111·10^-02

 C(23); (6 8 9);       f = 2,993245;  e = 6,755113·10^-03

 C(31); (9 10 12);     f = 3,002552;  e = 2,552032·10^-03

 C(82); (15 33 34);    f = 3,002088;  e = 2,088069·10^-03

 C(98); (17 40 41);    f = 2,998646;  e = 1,353979·10^-03

C(113); (23 44 46);    f = 3,001228;  e = 1,228094·10^-03

C(119); (36 37 46);    f = 2,998798;  e = 1,202106·10^-03

C(120); (24 47 49);    f = 2,999859;  e = 1,409053·10^-04

C(166); (43 58 65);    f = 2,999891;  e = 1,089572·10^-04

C(216); (41 86 89);    f = 3,000106;  e = 1,060962·10^-04

C(261); (64 94 103);   f = 3,000005;  e = 5,006790·10^-06

C(353); (71 138 144);  f = 2,999997;  e = 3,099441·10^-06

C(445); (135 138 172); f = 2,999999;  e = 9,536743·10^-06

La sucesión 10, 23, 28, 23, 31, 82, 98, 113, 119, 120, 166
, 216, 261, 353, 445,... ¿Tiene algún término general?

Para x=4

 C(13); (4 4 5);      f = 3,106284;  e = 0,8937160
 C(16); (5 5 6);      f = 3,801784;  e = 0,1982159
 C(24); (7 8 9);      f = 3,941391;  e = 5,860900·10^-02
 C(37); (10 13 14);   f = 4,026536;  e = 2,653598·10^-02
 C(59); (18 19 22);   f = 4,025973;  e = 2,597284·10^-02
 C(67); (16 25 26);   f = 3,982761;  e = 1,723909·10^-02
 C(72); (17 27 28);   f = 4,005021;  e = 5,021095·10^-03
 C(94); (21 36 37);   f = 3,999590;  e = 4,100799·10^-04
C(118); (37 37 44);   f = 4,000348;  e = 3,480911·10^-04
C(184); (53 62 69);   f = 3,999917;  e = 8,296966·10^-05
C(320); (64 127 129); f = 3,999935;  e = 6,508827·10^-05

Con e = 0,01 resulta:

C(23); (6 8 9); f = 2,993245
C(31); (9 10 12); f = 3,002552
C(46); (12 16 18); f = 2,993245
C(62); (18 20 24); f = 3,002552
C(69); (18 24 27); f = 2,993245
C(74); (19 26 29); f = 3,009558
C(75); (23 23 29); f = 2,990260
C(80); (23 26 31); f = 2,992867
C(82); (15 33 34); f = 3,002088
C(88); (27 27 34); f = 3,006838
C(91); (22 33 36); f = 2,991530
C(92); (24 32 36); f = 2,993245
[...]
C(171); (44 60 67); f = 3,007065

C(173); (50 56 67); f = 2,997841
[...]

Con e = 0,001 resulta:

C(120); (24 47 49); f = 2,999859
C(146); (31 56 59); f = 3,000988
C(163); (50 50 63); f = 2,999187
C(166); (43 58 65); f = 2,999891
C(179); (26 76 77); f = 3,000766
C(186); (40 71 75); f = 3,000602
[...]
C(319); (88 107 124); f = 2,999737

[...]

Con e = 0,0001 resulta:

C(261); (64  94 103); f = 3,000005
C(318); (94 101 123); f = 3,000042
C(328); (95 106 127); f = 3,000017
C(349);(103 111 135); f = 2,999970
C(353); (71 138 144); f = 2,999997
C(367); (73 144 150); f = 3,000003
C(374); (83 141 150); f = 3,000016
C(378); (54 161 163); f = 2,999991
C(380);(112 121 147); f = 2,999953
C(381); (44 168 169); f = 3,000049
C(400);(107 137 156); f = 2,999975
C(409);(102 146 161); f = 3,000080
C(411);(121 131 159); f = 2,999971
C(429); (59 184 186); f = 3,000089
C(432);(130 135 167); f = 2,999918
C(442);(130 141 171); f = 3,000009
C(445);(135 138 172); f = 2,999999

C(451);(120 155 176); f = 3,000087

[...]

De todo esto se puede deducir que si bien el teorema de Fermat se cumple para un matemático, sin embargo, se incumple para un ingeniero (si se admite un pequeño error). :-)

From all this, it can be deduced that although Fermat's theorem is true for a mathematician, however, fails to engineer (if a little error is supported). :-)

Queda ver que pasa cuando f forme parte de los números complejos.
It remains to see what happens when f was part of the complex numbers.