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Palindromic Incremented Squares
of Form n^2+1
rood n^2+(n+1) rood n^2+(n+x) rood comments



Introduction

Palindromic numbers are numbers which read the same from
 p_right left to right (forwards) as from the right to left (backwards) p_left
Here are a few random examples : 535, 3773, 246191642

Palindromic Incremented Squares are defined and calculated by this extraordinary intricate and excruciatingly complex formula.
So, this line is for experts only

base2 + 1


Palindromic Incremented Squares

flash So far I compiled 91 Palindromic Incremented Squares.

Here is the largest sporadic one that Feng Yuan from Washington State, USA,
discovered on [ January 25, 2008 ].

This basenumber
279.060.176.858.911.581.141.126
has 24 digits
yielding the following palindromic incremented square
77.874.582.308.527.010.741.090.909.014.701.072.580.328.547.877
with a length of 47 digits.
Here is the largest sporadic one that Feng Yuan from Washington State, USA,
discovered on [ April 4, 2002 ].

This basenumber
23.702.356.087.344.642
has 17 digits
yielding the following palindromic incremented square
561.801.684.091.283.606.382.190.486.108.165
with a length of 33 digits.


bu17 Palindromic Incremented Squares can only end in one of the following digits : 1, 2, 5, 6 or 7.


Here is Warut Roonguthai's synopsis of his basenumbers investigation :
No less than six patterns are detected !
Four infinite patterns :
10n » n = 0, 1 , 2, ... » 1, 10, 100, 1000, ...
103n+2*10n » n = 1, 2, 3, ... » 1020, 1000200, 1000002000, ...
103n+1+9*10n » n = 1, 2, 3, ... » 10090, 10000900, 10000009000, ...
105n+2*103n+2*10n » n = 1, 2, 3, ... » 102020, 10002000200, 1000002000002000, ...
Two finite patterns :
10 (09)n 10, for n = 1 to 4
100 (90)n, for n = 0 to 5

[By Patrick De Geest] Yet, this second finite pattern gives rise to infinite extensions in the following way :

100
1000
10000
100000 10n+1 n = 1, 2, 3, ...

10090
10000900
10000009000
10000000090000 103n+1+9*10n n = 1, 2, 3, ...

1009090
100009000900
10000009000009000
1000000009000000090000 105n+1+9*103n+9*10n n = 1, 2, 3, ...

100909090
1000090009000900
10000009000009000009000
100000000900000009000000090000 107n+1+9*105n+9*103n+9*10n n = 1, 2, 3, ...

10090909090
10000900090009000900
10000009000009000009000009000
10000000090000000900000009000000090000 109n+1+9*107n+9*105n+9*103n+9*10n n = 1, 2, 3, ...

1009090909090
100009000900090009000900
10000009000009000009000009000009000
1000000009000000090000000900000009000000090000 1011n+1+9*109n+9*107n+9*105n+9*103n+9*10n n = 1, 2, 3, ...


Warut Roonguthai from Bangkok Thailand informed me that every (palindromic) number of the form n(n+2)
is also of the form n^2–1.

" It's just one step away from being a palindromic square. And
that is why I think that it is interesting to investigate palindromes
of the form
n^2+1, another near miss, as well. "


It is no coincidence that there aren't any Palindromic Incremented Squares of even length.
Statement :
Every number of the form n^2+1 is NOT divisible by 11.
Proof there is no even_length palindromic number of the form n^2+1
( proof that -1 is a non-quadratic residue modulo 11 ) :
      n mod 11 :   0   1   2   3   4   5   6   7   8   9   10
(n^2+1) mod 11 :   1   2   5  10   6   4   4   6  10   5    2
[ if zero appeared in the second line then it would be divisible by 11 ]
Because Palindromic Numbers of EVEN length are always divisible by 11 ( for a general proof of this refer to the palindromic primes page ),
we immediately see that we can safely skip searching for them.


Sources Revealed


Neil Sloane's "Integer Sequences" Encyclopedia can be consulted online :
Neil Sloane's Integer Sequences
The regular incremented squares are categorised as follows :
%N n^2 + 1 under A002522.
Soon the following two entries about Palindromic Incremented Squares will be present :
%N n^2 + 1 is a palindrome under A027719.
%N Palindromes of form n^2 + 1 under A027720
Palindromic entry number Thanks N. Sloane for providing the palindromic entry number 027720
Click here to view some of the author's [P. De Geest] entries to the table.
Click here to view some entries to the table about palindromes.




The Table

My search went exhaustively upto (only odd!) length 33.

From length 35 on these numbers were discovered by Feng Yuan during [ January 20-25, 2008 ].



Index Nr BasenumberLength
Palindromic Incremented squares of Form n^2 + 1Length
   
Palindromic Incremented Squares
91 279.060.176.858.911.581.141.12624
77.874.582.308.527.010.741.090.909.014.701.072.580.328.547.87747
90 100.009.000.900.090.009.000.90024
10.001.800.261.034.204.230.504.040.503.240.243.016.200.810.00147
89 100.000.000.000.000.000.000.00024
10.000.000.000.000.000.000.000.000.000.000.000.000.000.000.00147
88 26.808.352.876.763.477.422.10423
718.687.783.965.072.615.665.828.566.516.270.569.387.786.81745
87 14.894.364.910.663.458.809.73923
221.842.106.092.002.903.330.454.033.309.200.290.601.248.12245
86 14.164.462.066.619.017.431.99923
200.631.985.636.689.086.223.868.322.680.986.636.589.136.00245
85 10.003.490.889.533.741.533.60023
100.069.829.976.984.567.458.181.854.765.489.679.928.960.00145
84 10.000.009.000.009.000.009.00023
100.000.180.000.261.000.342.000.243.000.162.000.081.000.00145
83 10.000.000.000.000.090.000.00023
100.000.000.000.001.800.000.000.000.008.100.000.000.000.00145
82 10.000.000.000.000.000.000.00023
100.000.000.000.000.000.000.000.000.000.000.000.000.000.00145
81 1.000.000.009.000.000.090.00022
1.000.000.018.000.000.261.000.001.620.000.008.100.000.00143
80 1.000.000.000.000.020.000.00022
1.000.000.000.000.040.000.000.000.000.400.000.000.000.00143
79 1.000.000.000.000.000.000.00022
1.000.000.000.000.000.000.000.000.000.000.000.000.000.00143
78 Prime!  142.127.953.804.140.488.85121
20.200.355.252.551.892.856.265.829.815.525.255.300.20241
77 100.000.002.000.000.020.00021
10.000.000.400.000.008.000.000.080.000.000.400.000.00141
76 100.000.000.000.000.000.00021
10.000.000.000.000.000.000.000.000.000.000.000.000.00141
75 Prime!   16.348.455.263.549.970.18120
267.271.989.504.294.724.969.427.492.405.989.172.76239
74 10.451.418.805.724.483.23020
109.232.155.052.651.383.333.383.156.250.551.232.90139
73 10.003.360.644.429.332.60020
100.067.224.182.517.632.404.236.715.281.422.760.00139
72 10.000.900.090.009.000.90020
100.018.002.610.342.042.303.240.243.016.200.810.00139
71 10.000.000.000.009.000.00020
100.000.000.000.180.000.000.000.081.000.000.000.00139
70 10.000.000.000.000.000.00020
100.000.000.000.000.000.000.000.000.000.000.000.00139
69 2.422.996.639.944.927.52819
5.870.912.717.184.408.770.778.044.817.172.190.78537
68 1.000.210.940.682.331.80019
1.000.421.925.860.635.062.605.360.685.291.240.00137
67 1.000.000.000.002.000.00019
1.000.000.000.004.000.000.000.004.000.000.000.00137
66 1.000.000.000.000.000.00019
1.000.000.000.000.000.000.000.000.000.000.000.00137
65 100.000.000.000.000.00018
10.000.000.000.000.000.000.000.000.000.000.00135
64 23.702.356.087.344.64217
561.801.684.091.283.606.382.190.486.108.16533
63 10.313.381.620.663.06017
106.365.840.453.430.606.034.354.048.563.60133
62 10.000.009.000.009.00017
100.000.180.000.261.000.162.000.081.000.00133
61 10.000.000.000.900.00017
100.000.000.018.000.000.000.810.000.000.00133
60 10.000.000.000.000.00017
100.000.000.000.000.000.000.000.000.000.00133
59 2.416.653.284.019.97816
5.840.213.095.164.544.454.615.903.120.48531
58 1.687.257.100.543.55916
2.846.836.523.334.657.564.333.256.386.48231
57 1.000.090.009.000.90016
1.000.180.026.103.420.243.016.200.810.00131
56 1.000.002.000.002.00016
1.000.004.000.008.000.008.000.004.000.00131
55 1.000.000.000.200.00016
1.000.000.000.400.000.000.040.000.000.00131
54 1.000.000.000.000.00016
1.000.000.000.000.000.000.000.000.000.00131
53 142.360.550.071.85115
20.266.526.216.759.995.761.262.566.20229
52 100.000.000.000.00015
10.000.000.000.000.000.000.000.000.00129
51 16.168.393.768.63114
261.416.957.057.505.750.759.614.16227
50 10.000.000.090.00014
100.000.018.000.000.008.100.000.00127
49 10.000.000.000.00014
100.000.000.000.000.000.000.000.00127
48 1.009.090.909.09013
1.018.264.462.808.082.644.628.10125
47 1.000.000.020.00013
1.000.000.040.000.000.400.000.00125
46 1.000.000.000.00013
1.000.000.000.000.000.000.000.00125
45 234.253.293.56212
54.874.605.544.644.550.647.84523
44 100.909.090.91012
10.182.644.628.282.644.628.10123
43 100.009.000.90012
10.001.800.261.016.200.810.00123
42 100.000.000.00012
10.000.000.000.000.000.000.00123
41 16.353.780.06911
267.446.122.545.221.644.76221
40 15.577.088.67111
242.645.691.464.196.546.24221
39 10.462.738.43011
109.468.895.454.598.864.90121
38 10.212.242.32011
104.289.893.202.398.982.40121
37 10.090.909.09011
101.826.446.262.644.628.10121
36 10.062.715.39011
101.258.241.020.142.852.10121
35 10.002.000.20011
100.040.008.000.800.040.00121
34 10.000.009.00011
100.000.180.000.081.000.00121
33 10.000.000.00011
100.000.000.000.000.000.00121
32 1.577.033.47110
2.487.034.568.654.307.84219
31 1.009.090.91010
Prime Curios!          1.018.264.464.644.628.10119
30 1.000.002.00010
1.000.004.000.004.000.00119
29 1.000.000.00010
1.000.000.000.000.000.00119
28 271.867.4569
73.911.913.631.911.93717
27 103.226.6609
10.655.743.334.755.60117
26 100.909.0909
10.182.644.444.628.10117
25 100.000.0009
10.000.000.000.000.00117
24 24.917.1958
620.866.606.668.02615
23 10.090.9108
101.826.464.628.10115
22 10.000.9008
100.018.000.810.00115
21 10.000.0008
100.000.000.000.00115
20 2.744.9347
7.534.662.664.35713
19 1.009.0907
1.018.262.628.10113
18 1.000.2007
1.000.400.040.00113
17 1.000.0007
1.000.000.000.00113
16 167.4916
28.053.235.08211
15 102.0206
10.408.080.40111
14 100.9106
10.182.828.10111
13 100.0006
10.000.000.00111
12 10.0905
101.808.1019
11 10.0005
100.000.0019
10 2.2484
5.053.5057
9 1.4894
2.217.1227
8 1.0204
1.040.4017
7 1.0004
1.000.0017
6 1003
10.0015
5 252
6263
4 102
Prime!          1013
3 21
Prime!          51
2 11
Prime!          21
1 01
11


Contributions

Warut Roonguthai (email) from Bangkok (Thailand) helped searching for these Palindromic Incremented Squares.
He discovered those starting from index number 21 upto 41 and 48.
Number 48 was the largest sporadic one which he submitted to me on [ November 3, 1997 ].







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( © All rights reserved ) - Last modified : October 29, 2016.
Patrick De Geest - Belgium flag - Short Bio - Some Pictures
E-mail address : pdg@worldofnumbers.com