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WON plate
153 |


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


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 + 1
5 + 5
= 25(0n–1)5(0n–1)52

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

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

k)   (6n) * (9n) – 4
5 + 5
+ 3n–1= (6n–1)5(6n–1)

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

m)   (7n) * (9n) – 3
5 + 5
+ 5n–1= (7n–1)6(7n–1)

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

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

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

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

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

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

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

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

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

z)   (3n) * (3n) – 9
5 + 5
– 7n–1= (1n–1)0(1n–1)


Continuación de "Operaciones Aritméticas...

aa)   (3n) * (3n) – 9
5 + 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 –1
9 + 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)   n
S
j=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 – 9
5 + 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 – 2
5 + 5
= (12.n–1)0(12.n–1)

I.4)   3n * 6n * 9n – 5(0n–1)5 + 3
5 + 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 – 5
5 + 5
= 2(72(n–1))2

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

II. Patrones Similares:

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

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

II.3)   (3n + 3n + 3n)2 + 9n – 9 – 1
5 + 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 – 5
5 + 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 – 1
5 + 5
– 1n–1= 5(4n–2)33(4n–2)5

II.8)   (6n)2 – 6
5 + 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 – 1
5 + 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 – 5
5 + 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
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