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WON plate
153 |

[ 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

666n * 9 + 1 = 5(93.n–1)5

666[6k] = 66[W]  siendo : W = (2(10k –1)(2.10k +1))/9
equivalent to : 666[4k+2k] = 66[4k.10k+2k] , 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 abax = ccy
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[nk] = aa[(n(n(10k –1)+9)(10k –1))/81]  if 'nk > a' and 'n > a' , no zeros

Operaciones Aritméticas Básicas con Palindrómicos Generalizados

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

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

 c) (2(1n))2 – ((1n)2)2(9n) = 3(2n–1)3

 d) (1n)0(6n)0(1n)1(0n)6(0n)1 = 1n

 e)   1n + 2n + 3n = 6n

 f)   (9n) * (1(0n)1) = (9n)0(9n)

 g)   (2(0n)2)2 = 4(0n)8(0n)4

 h) (5(0n)5)2 + 494 + 15 + 5 = 25(0n–1)5(0n–1)52

 i) (3n) * (6n) – 83 * 6 – 8 – 5n–1 = (2n–1)1(2n–1)

 j) (9n) * (3n) – 79 ÷ 3 + 7 – 3n–1 = (3n–1)2(3n–1)

 k) (6n) * (9n) – 45 + 5 + 3n–1 = (6n–1)5(6n–1)

 l) (3n) * (6n) + 1n – 95 + 5 – 6n–1 = (2n–1)1(2n–1)

 m) (7n) * (9n) – 35 + 5 + 5n–1 = (7n–1)6(7n–1)

 n) (3n)2 – 95 + 5 – 7n–1 = (1n–1)0(1n–1)

 o) (9n)2 – 15 + 5 + 9n–1 = (9n–1)8(9n–1)

 p) (6n)2 – 65 + 5 – 1n–1 = (4n–1)3(4n–1)

 q) (3n) * (6n) * (9n) – 25 + 5 – 5.10n–1 = (2n–1)1(5n–1)1(2n–1)

 r) (2n) * (5n)2 * 5 = (10n – 1)2(10 – 1)2 = (9n)292

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 ! )

 s) (3n)2 + (6n)2 – 55 + 5 + 1n = 52.n–1

 t) (6n)2 + (9n)2 – (1n) – 5 = 1(4n–1)1(4n–1)1

 u) (3n)2 + (6n)2 + (9n)2 + (1n) – 6 = 1(5n–1)2(5n–1)1

 v) (3n)3 + (6n)3 – 35 + 5 – 2(0n–2)2 + 2 = (3n–1)2(3n–1)2(3n–1)

 w) (9n)3 + (9n)2 + (9n) – 95 + 5 + 7(0n–2)7 – 7 = (9n–1)8(0n–1)8(9n–1)

 x) (9n)3 + 5(0n–1)5 – 9 – 55 + 5 = (9n–1)7(0n–1)7(9n–1)

 y) (6n) * (6n) – 65 + 5 – 1n–1 = (4n–1)3(4n–1)

 z) (3n) * (3n) – 95 + 5 – 7n–1 = (1n–1)0(1n–1)

Continuación de "Operaciones Aritméticas...

 aa) (3n) * (3n) – 95 + 5 – 7n–1 + 1(0n–2)1 – 1 = 12.n–1

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

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

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

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

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

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

 hh)   (6n) * (9n) + 1(3n–1)1 + 1 = 62.n

 ii)   (3n) * (6n) + 4n = 22.n

 jj)  (3n) * (9n) + 6n = 32.n

 kk)   (3n) * (6n) * (9n) + (6n) * (9n + 1) = 23.n

 ll) (9(0n)9)2 –19 + 1 – 9n + 9 = 81(0n–2)161(0n–2)18

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

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

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

Miscelanea de Patrones

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

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

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

 4)   (6n) * 9 = (9n) * 6

 5)   (6n) * 9 + 1 = 5(9n–1)5

 6) nSj=1 9j + 1 + n = 1n+1

 7)    | [ A(0n)B ]2 – [ XY(92.n)YX ] – [ B(0n)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) (8n) * (9n) + 7n – 95 + 5 = (8n–1)7(8n–1)

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

 I.3) (9n * 3n + 6n)2 – 72.n – 25 + 5 = (12.n–1)0(12.n–1)

 I.4) 3n * 6n * 9n – 5(0n–1)5 + 35 + 5 = (2n–1)1(5n–1)1(2n–1)

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

 I.6) (9n + 6n)2 + 5n+1 – 55 – 55 + 5 = 2(72(n–1))2

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

II. Patrones Similares:

 II.1) (3n)2 – 7n – 25 + 5 = (1n–1)0(1n–1)

 II.2) (3n + 3n)2 – 1n+1 – 55 + 5 = (4n–1)3(4n–1)

 II.3) (3n + 3n + 3n)2 + 9n – 9 – 15 + 5 = (9n–1)8(9n–1)

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

 II.5) (3n + 3n + 3n + 3n + 3n)2 – 55 + 5 + 5n – 5 = 2(72(n–1))2

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

 II.7) (3n + 3n + 3n + 3n + 3n + 3n + 3n)2 – 15 + 5 – 1n–1 = 5(4n–2)33(4n–2)5

 II.8) (6n)2 – 65 + 5 – 1n–1 = (4n–1)3(4n–1)

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

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

 II.11) (9n)2 + 9n – 9 – 15 + 5 = (9n–1)8(9n–1)

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

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

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

 II.15) (9n + 9n + 9n + 9n + 9n)2 – 55 + 5 + 9n – 55 – 4 = 24(9n–2)5(9n–2)42

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

A000153 Prime Curios! Prime Puzzle
Wikipedia 153 Le nombre 153 Curious Properties of 153
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