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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
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 |
| 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 |
| 5) (6n) * 9 + 1 = 5(9n–1)5 |
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
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