HOME plateWON | World!OfNumbers Palindromic Incremented Squaresof Form n^2+1 n(n+0) n(n+1) n(n+2) n(n+x) n^2+x n^2–x n^2+(n+1) n^2+(n+x)

Introduction

Palindromic numbers are numbers which read the same from
left to right (forwards) as from the right to left (backwards)
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

So far this compilation counts 117 Palindromic Incremented Squares (coded as SQUADD1).
Here is the largest sporadic Palindromic Incremented Square that Patrick De Geest,
discovered, using CUDA code from Robert Xiao, on [ June 17, 2023 ].

 This basenumber2.812.249.296.687.150.319.581.220.564 has 28 digits yielding the following palindromic incremented square 7.908.746.106.717.371.621.449.100.653.560.019.441.261.737.176.016.478.097 with a length of 55 digits.

Palindromic Incremented Squares can only start or end with digits : 0, 1, 2, 5, 6 or 7.
1 can only be followed by 0 : 10
2 can only be followed by an even number : 20, 22, 24, 26 or 28
5 can only be followed by an even number : 50, 52, 54, 56 or 58
6 can only be followed by 2 : 62
7 can only be followed by an uneven number : 71, 73, 75, 77 or 79
Alas, my palindromes may not have leading zero's! So the zero option must not be investigated.

There are no palindromic incremented squares of type [SQ1] of EVEN lengths.

 Case Squadd1 Change of variables CUDApalin parameters Base Correction base n.a. `A B C →  1 0 1 ` base = CUDAbase

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 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

Index NrBasenumberLength
Palindromic Incremented squares of Form n^2 + 1Length

[SQ1] Palindromic Incremented Squares
One can find the regular numbers of the form n2 + 1 at
%N n^2 + 1. under A002522.
The palindromic numbers of the form n2 + 1 are categorised as follows :
%N n^2 + 1 is a palindrome. under A027719.
%N Palindromes of the form n^2 + 1. under A027720.
1172.812.249.296.687.150.319.581.220.56428
7.908.746.106.717.371.621.449.100.653.560.019.441.261.737.176.016.478.09755
1162.754.261.449.396.633.799.499.196.68428
7.585.956.131.632.445.967.819.384.237.324.839.187.695.442.361.316.595.85755
1152.491.847.781.298.753.072.044.331.65528
6.209.305.365.163.518.320.370.455.334.335.540.730.238.153.615.635.039.02655
1141.551.296.962.488.030.712.756.915.57128
2.406.522.265.824.590.568.363.169.516.159.613.638.650.954.285.622.256.04255
1131.494.077.004.455.004.324.432.286.36128
2.232.266.095.241.239.011.916.697.749.477.966.191.109.321.425.906.622.32255
1121.000.000.000.000.000.002.000.000.00028
1.000.000.000.000.000.004.000.000.000.000.000.004.000.000.000.000.000.00155
1111.000.000.000.000.000.000.000.000.00028
1.000.000.000.000.000.000.000.000.000.000.000.000.000.000.000.000.000.00155
110101.992.801.384.131.628.756.386.98027
10.402.531.534.182.922.686.669.217.671.296.668.622.928.143.513.520.40153
109100.000.000.009.000.000.000.900.00027
10.000.000.001.800.000.000.261.000.000.016.200.000.000.810.000.000.00153
108100.000.000.000.000.000.000.000.00027
10.000.000.000.000.000.000.000.000.000.000.000.000.000.000.000.000.00153
10727.191.364.491.912.530.415.430.95626
739.370.302.932.041.583.352.046.262.640.253.385.140.239.203.073.93751
10623.735.687.120.580.638.190.017.55826
563.382.843.086.097.587.216.696.868.696.612.785.790.680.348.283.36551
10510.000.000.002.000.000.000.200.00026
100.000.000.040.000.000.008.000.000.000.800.000.000.040.000.000.00151
10410.000.000.000.000.000.900.000.00026
100.000.000.000.000.018.000.000.000.000.000.810.000.000.000.000.00151
10310.000.000.000.000.000.000.000.00026
100.000.000.000.000.000.000.000.000.000.000.000.000.000.000.000.00151
1021.618.624.351.248.784.060.217.38125
2.619.944.790.455.547.077.078.899.988.707.707.455.540.974.499.16249
1011.030.375.982.106.737.764.302.96025
1.061.674.664.502.424.381.431.686.861.341.834.242.054.664.761.60149
1001.000.000.000.000.000.200.000.00025
1.000.000.000.000.000.400.000.000.000.000.040.000.000.000.000.00149
991.000.000.000.000.000.000.000.00025
1.000.000.000.000.000.000.000.000.000.000.000.000.000.000.000.00149
981.618.624.351.248.784.060.217.38125
2.619.944.790.455.547.077.078.899.988.707.707.455.540.974.499.16249
971.030.375.982.106.737.764.302.96025
1.061.674.664.502.424.381.431.686.861.341.834.242.054.664.761.60149
961.000.000.000.000.000.200.000.00025
1.000.000.000.000.000.400.000.000.000.000.040.000.000.000.000.00149
951.000.000.000.000.000.000.000.00025
1.000.000.000.000.000.000.000.000.000.000.000.000.000.000.000.00149
94279.060.176.858.911.581.141.12624
77.874.582.308.527.010.741.090.909.014.701.072.580.328.547.87747
93100.009.000.900.090.009.000.90024
10.001.800.261.034.204.230.504.040.503.240.243.016.200.810.00147
92100.000.000.000.000.000.000.00024
10.000.000.000.000.000.000.000.000.000.000.000.000.000.000.00147
9126.808.352.876.763.477.422.10423
718.687.783.965.072.615.665.828.566.516.270.569.387.786.81745
9014.894.364.910.663.458.809.73923
221.842.106.092.002.903.330.454.033.309.200.290.601.248.12245
8914.164.462.066.619.017.431.99923
200.631.985.636.689.086.223.868.322.680.986.636.589.136.00245
8810.003.490.889.533.741.533.60023
100.069.829.976.984.567.458.181.854.765.489.679.928.960.00145
8710.000.009.000.009.000.009.00023
100.000.180.000.261.000.342.000.243.000.162.000.081.000.00145
8610.000.000.000.000.090.000.00023
100.000.000.000.001.800.000.000.000.008.100.000.000.000.00145
8510.000.000.000.000.000.000.00023
100.000.000.000.000.000.000.000.000.000.000.000.000.000.00145
841.000.000.009.000.000.090.00022
1.000.000.018.000.000.261.000.001.620.000.008.100.000.00143
831.000.000.000.000.020.000.00022
1.000.000.000.000.040.000.000.000.000.400.000.000.000.00143
821.000.000.000.000.000.000.00022
1.000.000.000.000.000.000.000.000.000.000.000.000.000.00143
81167.346.823.633.589.201.24121
28.004.959.380.251.609.231.013.290.615.208.395.940.08241
80Prime!    142.127.953.804.140.488.85121
20.200.355.252.551.892.856.265.829.815.525.255.300.20241
79100.000.002.000.000.020.00021
10.000.000.400.000.008.000.000.080.000.000.400.000.00141
78100.000.000.000.000.000.00021
10.000.000.000.000.000.000.000.000.000.000.000.000.00141
77Prime!    16.348.455.263.549.970.18120
267.271.989.504.294.724.969.427.492.405.989.172.76239
7610.451.418.805.724.483.23020
109.232.155.052.651.383.333.383.156.250.551.232.90139
7510.003.360.644.429.332.60020
100.067.224.182.517.632.404.236.715.281.422.760.00139
7410.000.900.090.009.000.90020
100.018.002.610.342.042.303.240.243.016.200.810.00139
7310.000.000.000.009.000.00020
100.000.000.000.180.000.000.000.081.000.000.000.00139
7210.000.000.000.000.000.00020
100.000.000.000.000.000.000.000.000.000.000.000.00139
712.422.996.639.944.927.52819
5.870.912.717.184.408.770.778.044.817.172.190.78537
701.692.363.327.880.989.70919
2.864.093.633.556.418.279.728.146.553.363.904.68237
691.007.384.510.708.540.29019
1.014.823.552.415.485.126.215.845.142.553.284.10137
681.000.210.940.682.331.80019
1.000.421.925.860.635.062.605.360.685.291.240.00137
671.000.000.000.002.000.00019
1.000.000.000.004.000.000.000.004.000.000.000.00137
661.000.000.000.000.000.00019
1.000.000.000.000.000.000.000.000.000.000.000.00137
65100.000.000.000.000.00018
10.000.000.000.000.000.000.000.000.000.000.00135
6423.702.356.087.344.64217
561.801.684.091.283.606.382.190.486.108.16533
6310.313.381.620.663.06017
106.365.840.453.430.606.034.354.048.563.60133
6210.000.009.000.009.00017
100.000.180.000.261.000.162.000.081.000.00133
6110.000.000.000.900.00017
100.000.000.018.000.000.000.810.000.000.00133
6010.000.000.000.000.00017
100.000.000.000.000.000.000.000.000.000.00133
592.416.653.284.019.97816
5.840.213.095.164.544.454.615.903.120.48531
581.687.257.100.543.55916
2.846.836.523.334.657.564.333.256.386.48231
571.000.090.009.000.90016
1.000.180.026.103.420.243.016.200.810.00131
561.000.002.000.002.00016
1.000.004.000.008.000.008.000.004.000.00131
551.000.000.000.200.00016
1.000.000.000.400.000.000.040.000.000.00131
541.000.000.000.000.00016
1.000.000.000.000.000.000.000.000.000.00131
53142.360.550.071.85115
20.266.526.216.759.995.761.262.566.20229
52100.000.000.000.00015
10.000.000.000.000.000.000.000.000.00129
5116.168.393.768.63114
261.416.957.057.505.750.759.614.16227
5010.000.000.090.00014
100.000.001.800.000.008.100.000.00127
4910.000.000.000.00014
100.000.000.000.000.000.000.000.00127
481.009.090.909.09013
1.018.264.462.808.082.644.628.10125
471.000.000.020.00013
1.000.000.040.000.000.400.000.00125
461.000.000.000.00013
1.000.000.000.000.000.000.000.00125
45234.253.293.56212
54.874.605.544.644.550.647.84523
44100.909.090.91012
10.182.644.628.282.644.628.10123
43100.009.000.90012
10.001.800.261.016.200.810.00123
42100.000.000.00012
10.000.000.000.000.000.000.00123
4116.353.780.06911
267.446.122.545.221.644.76221
4015.577.088.67111
242.645.691.464.196.546.24221
3910.462.738.43011
109.468.895.454.598.864.90121
3810.212.242.32011
104.289.893.202.398.982.40121
3710.090.909.09011
101.826.446.262.644.628.10121
3610.062.715.39011
101.258.241.020.142.852.10121
3510.002.000.20011
100.040.008.000.800.040.00121
3410.000.009.00011
100.000.180.000.081.000.00121
3310.000.000.00011
100.000.000.000.000.000.00121
321.577.033.47110
2.487.034.568.654.307.84219
311.009.090.91010
Prime Curios!    1.018.264.464.644.628.10119
301.000.002.00010
1.000.004.000.004.000.00119
291.000.000.00010
1.000.000.000.000.000.00119
28271.867.4569
73.911.913.631.911.93717
27103.226.6609
10.655.743.334.755.60117
26100.909.0909
10.182.644.444.628.10117
25100.000.0009
10.000.000.000.000.00117
2424.917.1958
620.866.606.668.02615
2310.090.9108
101.826.464.628.10115
2210.000.9008
100.018.000.810.00115
2110.000.0008
100.000.000.000.00115
202.744.9347
7.534.662.664.35713
191.009.0907
1.018.262.628.10113
181.000.2007
1.000.400.040.00113
171.000.0007
1.000.000.000.00113
16Prime!    167.4916
28.053.235.08211
15102.0206
10.408.080.40111
14100.9106
10.182.828.10111
13100.0006
10.000.000.00111
1210.0905
101.808.1019
1110.0005
100.000.0019
102.2484
5.053.5057
9Prime!    1.4894
2.217.1227
81.0204
1.040.4017
71.0004
1.000.0017
61003
10.0015
5252
6263
4102
Prime!    1013
3Prime!    21
Prime!    51
211
Prime!    21
101
11

Contributions

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

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

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

[ November 4, 2022 ]
Robert Xiao (email) spotted an error in entry [50] and has now been fixed.
Furthermore he added two missing entries [69] and [70].

[ November 11, 2022 ]
Robert Xiao (email) added missing entry [81].

[ November 13, 2022 ]
Robert Xiao (email) added four new entries [95] up to [98].

[ December 6, 2022 ]
Robert Xiao (email) added nine new entries [99] up to [107].

[ June 17, 2023 ]
Patrick De Geest (email) added nine new entries [108] up to [117].

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