WON TOPIC 153

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[ November 3, 2003 ]
Patrones Palindromicos con Multiples Operaciones Aritmeticas
Hugo Sanchez (email)

This plate will display a goodly number of palindromic patterns
obtained by various arithmetic operations. These patterns are very unique
and they are not like those that I have sent to you before.
Some contain correction factors in order to achieve palindrome-ness.

Palindromic schemes of the form aba_[n] = aa_[m]
Example : 363_[7] = 192_[10] = (3*63+3)_[10] = 33_[63]
363_[7] = 33_[63]. Siendo 63 = 7(7+6/3).
In general : aba_[n] = aa_[n(n+b/a)] if 'a divides b' and
'n > a' and 'n > b' and 'm = [n(n+b/a)]'.

Palindromic schemes of the form aba_[n] = bb_[m]
Example : 636_[7] = 321_[10] = (3*106+3)_[10] = 33_[106]
636_[7] = 33_[106]. Siendo 106 = (6/3)*(7² +1)+(7-1).
In general : aba_[n] = bb_{[(a/b)(n² +1)+(n-1)]} if 'b divides a'.

Palindromes and Numerical Bases
General note : the following convention will be used : a "numerical base"
will be a number with subindex to the right put between square brackets.
Example : number 13 from "base 5" is written as (13)_[5]
while a "numerical repetition" number is accompanied by a subindex
but this time without brackets.
Example : the repetition of five times the number 13 is 1313131313 = (13)₅.

The Number of the Beast

666_[n] = 66_[n(n+1)] , n > 6

666_n * 9 + 1 = 5(9_3.n–1)5

666_{[6_k]} = 66_[W] siendo : W = (2(10^k –1)(2.10^k +1))/9
equivalent to : 666_{[4_k+2_k]} = 66_{[4_k.10^k+2_k]} , k > 1
Example : 666_[6₅] = 666_[66666] = 26666533338_[10] =
(6 * 4444422222 + 6)_[10] = 66_[4444422222] =
66_{[4₅.10⁵+2₅]} = 66_{[(2(10⁵ –1)(2.10⁵ +1))/9]}

Palindromes of the form aba_x = cc_y
1) aba_[n] = aa_[n(n+b/a)] if 'a divides b' and 'n > a,b' , no zeros
2) aba_[n] = bb_{[(a/b)(n²+1)+(n+1)]} if 'b divides a' and 'n > a,b' , no zeros
3) aaa_{[n_k]} = aa_{[(n(n(10^k –1)+9)(10^k –1))/81]} if 'n_k > a' and 'n > a' , no zeros

Operaciones Aritméticas Básicas con Palindrómicos Generalizados

a) (3_n) * (2(0_n)2) = (6_n)0(6_n)
Example with n = 5 : 33333 * 2000002 = 66666066666

b) (3_n) * (2(0_n)2(0_n)2) = (6_n)0(6_n)0(6_n)
Example with n = 4 : 3333 * 20000200002 = 66660666606666

(2(1_n))² – ((1_n)2)²

(9_n)

= 3(2_n–1)3

(1_n)0(6_n)0(1_n)

1(0_n)6(0_n)1

= 1_n

e) 1_n + 2_n + 3_n = 6_n

f) (9_n) * (1(0_n)1) = (9_n)0(9_n)

g) (2(0_n)2)² = 4(0_n)8(0_n)4

(5(0_n)5)² + 494 + 1

5 + 5

= 25(0_n–1)5(0_n–1)52

(3_n) * (6_n) – 8

3 * 6 – 8

– 5_n–1

= (2_n–1)1(2_n–1)

(9_n) * (3_n) – 7

9 ÷ 3 + 7

– 3_n–1

= (3_n–1)2(3_n–1)

(6_n) * (9_n) – 4

5 + 5

+ 3_n–1

= (6_n–1)5(6_n–1)

(3_n) * (6_n) + 1_n – 9

5 + 5

– 6_n–1

= (2_n–1)1(2_n–1)

(7_n) * (9_n) – 3

5 + 5

+ 5_n–1

= (7_n–1)6(7_n–1)

(3_n)² – 9

5 + 5

– 7_n–1

= (1_n–1)0(1_n–1)

(9_n)² – 1

5 + 5

+ 9_n–1

= (9_n–1)8(9_n–1)

(6_n)² – 6

5 + 5

– 1_n–1

= (4_n–1)3(4_n–1)

(3_n) * (6_n) * (9_n) – 2

5 + 5

– 5.10^n–1

= (2_n–1)1(5_n–1)1(2_n–1)

(2_n) * (5_n)

2 * 5

(10ⁿ – 1)²

(10 – 1)²

(9_n)²

9²

n = 1 1
n = 2 121
n = 3 12321
n = 4 1234321
n = 5 123454321
n = 6 12345654321
n = 7 1234567654321
n = 8 123456787654321
n = 9 12345678987654321
( end of pattern ! )

(3_n)² + (6_n)² – 5

5 + 5

+ 1_n

= 5_2.n–1

(6_n)² + (9_n)² – (1_n) – 5

= 1(4_n–1)1(4_n–1)1

(3_n)² + (6_n)² + (9_n)² + (1_n) – 6

= 1(5_n–1)2(5_n–1)1

(3_n)³ + (6_n)³ – 3

5 + 5

– 2(0_n–2)2 + 2

= (3_n–1)2(3_n–1)2(3_n–1)

(9_n)³ + (9_n)² + (9_n) – 9

5 + 5

+ 7(0_n–2)7 – 7

= (9_n–1)8(0_n–1)8(9_n–1)

(9_n)³ + 5(0_n–1)5 – 9 – 5

5 + 5

= (9_n–1)7(0_n–1)7(9_n–1)

(6_n) * (6_n) – 6

5 + 5

– 1_n–1

= (4_n–1)3(4_n–1)

(3_n) * (3_n) – 9

5 + 5

– 7_n–1

= (1_n–1)0(1_n–1)

Continuación de "Operaciones Aritméticas...

aa)

(3_n) * (3_n) – 9

5 + 5

– 7_n–1 + 1(0_n–2)1 – 1

= 1_2.n–1

bb) (9_n)² + 8(9_n–2)8 = (9_n–1)88(9_n–1)

cc) (3_n)² – (1_n–1)00(1_n–1) = 8(7_n–2)8

dd) (6_n)² – (4_n–1)33(4_n–1) = 2(1_n–2)2

ee) (3_n) * (6_n) – 6(5_n–2)6 = (2_n–1)11(2_n–1)

ff) (3_n) * (9_n) – 4(3_n–2)4 = (3_n–1)22(3_n–1)

gg) (6_n) * (9_n) + 2(3_n–2)2 = (6_n–1)55(6_n–1)

hh) (6_n) * (9_n) + 1(3_n–1)1 + 1 = 6_2.n

ii) (3_n) * (6_n) + 4_n = 2_2.n

jj) (3_n) * (9_n) + 6_n = 3_2.n

kk) (3_n) * (6_n) * (9_n) + (6_n) * (9_n + 1) = 2_3.n

ll)

(9(0_n)9)² –1

9 + 1

– 9_n + 9

= 81(0_n–2)161(0_n–2)18

mm) (3(0_n)3)² + 9_n + 1 = 9(0_n–1)181(0_n–1)9

nn) (3(0_n)3)² – 9_n+2 – 1 = 9(0_n)8(0_n)9

oo)   (2(0_n)2) * (2(0_n+1)2) * (2(0_n+2)2) =
         2 * (2(0_n)222(0_n–1)222(0_n)2) * 2 =
         8(0_n)888(0_n–1)888(0_n)8

Miscelanea de Patrones

1) (6_n) * (9_n) + (6_n) + (9_n) = (6_n) * (9_n + 1) + 9_n = (6_n)(9_n)

2) (9_n) * (9_n) + (6_n) + (9_n) = (9_n) * (9_n + 1) + 6_n = (9_n)(6_n)

3) (6_n) * 9 + (9_n) * 6 – 77 = 11(9_n–2)11 , n >= 2

4) (6_n) * 9 = (9_n) * 6

5) (6_n) * 9 + 1 = 5(9_n–1)5

n
S
j=1

9_j + 1 + n

= 1_n+1

7) | [ A(0_n)B ]² – [ XY(9_2.n)YX ] – [ B(0_n)A ]² | = Palindrome

Palindrome equal to D

Palindromico Monodigital

Palindrome equal to EE

Palindromico Bidigital

"|" = absolute value

A, B : digits with A<>B , B < A

XY, YX : 'two distinct digit' numbers obtained from XY = A^2 – B^2 – 1

YX the reverse of XY
(ps. XY = '10' then YX = '01' or XY = '04' and YX = '40').

Examples A=6, B=5, n=3, XY=6^2 – 5^2 – 1=10, YX=01

60005² – 1099999901 – 50006² =

ps. note the above beautiful palindromic expression itself !!

3600600025 – 1099999901 – 2500600036 = EE

EE equal to 88

I. Operaciones Aritméticas con Factores de Corrección, que
generan Patrones Palindrómicos Infinitos. Continuación:

I.1)

(8_n) * (9_n) + 7_n – 9

5 + 5

= (8_n–1)7(8_n–1)

I.2) (9_n + 9_n)² + 9_n – 9 – 1 = (3_n + 6_n + 9_n)² + 9_n – 9 – 1 =
3(9_n–1)2(9_n–1)3

I.3)

(9_n * 3_n + 6_n)² – 7_2.n – 2

5 + 5

= (1_2.n–1)0(1_2.n–1)

I.4)

3_n * 6_n * 9_n – 5(0_n–1)5 + 3

5 + 5

= (2_n–1)1(5_n–1)1(2_n–1)

I.5) (9_n + 3_n)² + 5_n – 8 = (6_n + 6_n)² + 5_n – 8 = 1(7_n–1)4(7_n–1)1

I.6)

(9_n + 6_n)² + 5_n+1 – 55 – 5

5 + 5

= 2(7_2(n–1))2

I.7) (3_n)² + (3_n)² + 3_n + 1_n = (22)_n

II. Patrones Similares:

II.1)

(3_n)² – 7_n – 2

5 + 5

= (1_n–1)0(1_n–1)

II.2)

(3_n + 3_n)² – 1_n+1 – 5

5 + 5

= (4_n–1)3(4_n–1)

II.3)

(3_n + 3_n + 3_n)² + 9_n – 9 – 1

5 + 5

= (9_n–1)8(9_n–1)

II.4) (3_n + 3_n + 3_n + 3_n)² + 5_n – 5 – 3 = 1(7_n–1)9(7_n–1)1

II.5)

(3_n + 3_n + 3_n + 3_n + 3_n)² – 5

5 + 5

+ 5_n – 5

= 2(7_2(n–1))2

II.6) (3_n + 3_n + 3_n + 3_n + 3_n + 3_n)² + 9_n – 9 – 1 = 3(9_n–1)2(9_n–1)3

II.7)

(3_n + 3_n + 3_n + 3_n + 3_n + 3_n + 3_n)² – 1

5 + 5

– 1_n–1

= 5(4_n–2)33(4_n–2)5

II.8)

(6_n)² – 6

5 + 5

– 1_n–1

= (4_n–1)3(4_n–1)

II.9) (6_n + 6_n)² + 5_n – 8 = 1(7_n–1)4(7_n–1)1

II.10) (6_n + 6_n + 6_n)² – 9_n – 9 – 3 = 3(9_n–1)0(9_n–1)3

II.11)

(9_n)² + 9_n – 9 – 1

5 + 5

= (9_n–1)8(9_n–1)

II.12) (9_n + 9_n)² + 9_n – 9 –1 = 3(9_n–1)2(9_n–1)3

II.13) (9_n + 9_n + 9_n)² + 9_n+1 – 9 –1 = 8(9_n–1)1(9_n–1)8

II.14) (9_n + 9_n + 9_n + 9_n)² + 9_n+1 – 66 + 2 = 15(9_n–2)77(9_n–2)51

II.15)

(9_n + 9_n + 9_n + 9_n + 9_n)² – 5

5 + 5

+ 9_n – 55 – 4

= 24(9_n–2)5(9_n–2)42

Other Hugo Sánchez WONplates
won84.htm won106.htm won162.htm

A000153

Curious Properties of 153

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