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Palindromic Quasi_Over_Squares
of the form n^2+(n+1)
rood The Details
rood n^2+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 : 7, 3113, 44611644

Quasi_Over_Square numbers are defined and calculated by this extraordinary intricate and excruciatingly complex formula.
So, this line is for experts only

base2 + ( base + 1 )1



Palindromic Quasi_Over_Squares

Let me name the numbers of the form n + (n+1)^2 or (n-1) + n^2 as follows : Quasi_Under_Squares.
Numbers of the form n^2 + (n+1) then become the Quasi_Over_Squares.
Readers with more original suggestions for naming these numbers can always send their proposals to me at my e-mail address.


Just like Warut Roonguthai ( See Palindromic Pronic Numbers n(n+1) ) I also recognized
a similar finite growing pattern in these Quasi_Over_Squares basenumbers.
26 (5445)n 62, for n = 2 to 4
26545 (5445)n 437, for n = 0 to 4
|Astonishing!| the largest known Quasi_Over_Square now has a length of 47 digits !


flash So far I compiled 127 Palindromic Quasi_Over_Squares.

Here is the largest one that Feng Yuan discovered during [ January 19-25, 2008 ]

This basenumber
1.866.690.185.312.364.850.600.393

has 25 digits
yielding the following Palindromic Quasi_Over_Square
3.484.532.247.941.511.026.607.063.607.066.201.151.497.422.354.843
with a length of 49 digits.

Here is the largest one that I discovered on [ October 25, 1997 ].

This basenumber
265.455.445.544.554.455.445.437

has 24 digits
yielding the following Palindromic Quasi_Over_Square
70.466.593.569.257.915.890.542.124.509.851.975.296.539.566.407
with a length of 47 digits.


bu17 All palindromic numbers of the form n^2 + (n+1) can only end with 1, 3 or 7


It is no coincidence that I didn't find Palindromic Quasi_Over_Squares of even length.

Actually the following conjecture was proven for me by Neo Chee Beng and Warut Roonguthai almost simultaneously :
Every number of the form n^2+(n+1) is NOT divisible by 11.
Proof : by using mod 11
        n mod 11 :   0   1   2   3   4   5   6   7   8   9   10
(n^2+n+1) mod 11 :   1   3   7   2  10   9  10   2   7   3    1
[ if zero appeared in the second line then it would be divisible by 11 ]
Because Palindromic Quasi_Over_Squares of EVEN length are always divisible by 11 we immediately see that they cannot occur here.

And also that :
Every number of the form n^2+(n+1) is ODD.
Proof :
Either n or n+1 is even, so their product n(n+1) is even.
Therefore n(n+1)+1 is odd !!
Simple isn't it.



Sources Revealed


Neil Sloane's "Integer Sequences" Encyclopedia can be consulted online :
Neil Sloane's Integer Sequences
One can find the regular numbers of the form n2+(n+1) at
%N Central polygonal numbers: n^2 – n + 1 under A002061.
The palindromic numbers of the form n2+(n+1) are categorised as follows :
%N n^2 + (n+1) is a palindrome under A028413.
%N Palindromes of the form n^2 + (n+1) under A028414.
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.


If one searches the database for the short sequence {1,3,7,13} you get a positive reply and one of the entries shown is :
%N Primes of form n^2 + n + 1 under A002383.
(%N n^2 + n + 1 is prime. under A002384).
I will give a proof that every number of the form n(n – 1) + 1 is also a member of the numbers of the form n^2 + (n+1).
Substitute n with m+1 and we derive :
n(n – 1) + 1 =
(m+1) ( (m+1) – 1) + 1 =
(m+1) (m) + 1 =
m^2 + m + 1 =
m^2 + (m+1) QED


Two infinite expanding patterns are detected :
The first one starts with :
102 + 11 = 101
1002 + 101 = 10101
10002 + 1001 = 1001001
100002 + 10001 = 100010001
The stepvalue of the length of palindromes = + 2 and starts with length 3
The second pattern is less obvious :
11202 + 1121 = 1255521
10102002 + 1010201 = 1020505050201
10010020002 + 1001002001 = 1002005005005002001
10001000200002 + 1000100020001 = 1000200050005000500020001
The stepvalue of the length of palindromes = + 6 and starts with length 7.

And the following phenomenon totally surprised me !
All the 'least significant' terms of the above pattern when multiplied with their reversals produce palindromes :
1121 x 1211 = 1357531
1010201 x 1020101 = 1030507050301
1001002001 x 1002001001 = 1003005007005003001
1000100020001 x 1000200010001 = 1000300050007000500030001


A lot of the palindromes are semiprimes. Is there a logical explanation for this phenomenon ?



back [8]
F_qs  30.103 =
      PRIME

Comm 30103 is the only known multi-digit palindromic prime of the form sigma(n)/d(n), when n is composite. ( McCranie and Honaker ) sigma(n) = the sum of all the factors of n, including 1 and n itself. d(n) = the number of factors of n, including 1 and n itself. 30103 is also nice because it comes up for n = 149645 (an odd number) and once for n = 179574 (an even number). n sigma(n) d(n) sigma(n)/d(n)=palprime 149645 180618 6 30103 179574 361236 12 30103 30103 is still the only "known" palprime with the property.


The Table

My program searched for palindromes exhaustively upto length 31.

From length 33 on these numbers were discovered by Feng Yuan during [ January 19-25, 2008 ].


Index NrInfo BasenumberLength
Palindromic Quasi_Over_Squares of form n^2+(n+1)Length
   
Palindromic Quasi_Over_Squares
127 1.866.690.185.312.364.850.600.39325
3.484.532.247.941.511.026.607.063.607.066.201.151.497.422.354.84349
126 1.203.507.603.075.463.896.117.15925
1.448.430.550.660.448.354.463.702.073.644.538.440.660.550.348.44149
125 1.000.000.010.000.000.200.000.00025
1.000.000.020.000.000.500.000.005.000.000.050.000.000.200.000.00149
124 1.000.000.000.000.000.000.000.00025
1.000.000.000.000.000.000.000.001.000.000.000.000.000.000.000.00149
123 265.455.445.544.554.455.445.43724
70.466.593.569.257.915.890.542.124.509.851.975.296.539.566.40747
122 174.868.688.603.991.484.485.62624
30.579.058.254.079.741.275.153.435.157.214.797.045.285.097.50347
121 126.050.968.010.335.988.738.22524
15.888.846.536.342.746.771.487.678.417.764.724.363.564.888.85147
120 109.068.284.003.373.554.072.41424
Prime!   11.895.890.575.440.551.507.330.303.370.515.504.457.509.859.81147
119 107.122.509.064.061.532.741.31024
11.475.231.948.179.945.239.373.537.393.254.997.184.913.257.41147
118 100.000.000.000.000.000.000.00024
10.000.000.000.000.000.000.000.100.000.000.000.000.000.000.00147
117 19.311.068.891.689.189.856.98323
372.917.381.739.565.955.495.848.594.559.565.937.183.719.27345
116 18.448.546.164.376.797.088.99323
340.348.855.579.141.831.877.656.778.138.141.975.558.843.04345
115 17.988.729.019.015.444.317.36623
323.594.371.719.568.349.640.979.046.943.865.917.173.495.32345
114 11.442.728.977.039.858.149.00523
130.936.046.441.987.638.522.191.225.836.789.144.640.639.03145
113 11.134.188.044.349.820.552.92023
123.970.143.406.942.481.571.868.175.184.249.604.341.079.32145
112 10.000.000.000.000.000.000.00023
100.000.000.000.000.000.000.010.000.000.000.000.000.000.00145
111 2.646.181.864.462.346.192.40222
7.002.278.459.809.418.714.663.664.178.149.089.548.722.00743
110 1.866.239.981.123.248.444.39322
3.482.851.667.142.902.710.061.600.172.092.417.661.582.84343
109 1.257.643.369.684.401.760.64922
1.581.666.845.311.136.833.493.943.386.311.135.486.661.85143
108 1.240.734.862.303.538.673.14922
1.539.422.998.535.381.071.575.751.701.835.358.992.249.35143
107 1.042.776.663.690.308.997.80022
1.087.383.170.337.091.795.807.085.971.907.330.713.837.80143
106 1.007.779.342.176.626.844.89922
1.015.619.202.517.954.735.095.905.374.597.152.029.165.10143
105 1.000.000.100.000.020.000.00022
1.000.000.200.000.050.000.005.000.000.500.000.020.000.00143
104 1.000.000.000.000.000.000.00022
1.000.000.000.000.000.000.001.000.000.000.000.000.000.00143
103 188.608.385.819.602.861.64821
35.573.123.201.476.169.825.552.896.167.410.232.137.55341
102 173.770.124.996.346.340.29821
30.196.056.341.245.831.193.939.113.854.214.365.069.10341
101 131.077.218.569.828.232.85421
17.181.237.228.002.523.325.552.332.520.082.273.218.17141
100 121.350.673.633.339.893.14021
Prime!   14.725.985.991.265.373.941.814.937.356.219.958.952.74141
99 100.000.000.000.000.000.00021
10.000.000.000.000.000.000.100.000.000.000.000.000.00141
98 26.545.544.554.455.445.43720
704.665.935.692.579.153.222.351.975.296.539.566.40739
97 26.544.554.455.445.544.56220
704.613.371.238.113.910.828.019.311.832.173.316.40739
96 10.000.000.000.000.000.00020
100.000.000.000.000.000.010.000.000.000.000.000.00139
95 1.940.017.649.366.890.18319
3.763.668.479.855.034.063.604.305.589.748.663.67337
94 1.165.124.977.484.160.63019
1.357.516.213.157.465.816.185.647.513.126.157.53137
93 1.211.056.769.126.025.58419
1.466.658.498.045.967.635.367.695.408.948.566.64137
92 1.000.001.000.002.000.00019
1.000.002.000.005.000.005.000.005.000.002.000.00137
91 100.000.000.000.000.00018
10.000.000.000.000.000.100.000.000.000.000.00135
90 18.516.595.403.268.59317
342.864.305.328.347.606.743.823.503.468.24333
89 10.920.545.563.373.68517
119.258.315.401.720.686.027.104.513.852.91133
88 10.000.000.000.000.00017
100.000.000.000.000.010.000.000.000.000.00133
87 2.654.554.455.445.43716
7.046.659.356.925.223.225.296.539.566.40731
86 2.654.455.445.544.56216
7.046.133.712.381.181.811.832.173.316.40731
85 1.740.233.076.507.52616
3.028.411.160.570.850.580.750.611.148.20331
84 1.320.273.129.165.44516
1.743.121.135.596.317.136.955.311.213.47131
83 1.157.309.182.414.43016
1.339.364.543.700.757.570.073.454.639.33131
82 1.081.750.100.278.61016
1.170.183.279.452.783.872.549.723.810.71131
81 1.068.301.518.025.31016
1.141.268.133.415.182.815.143.318.621.41131
80 1.000.010.000.200.00016
1.000.020.000.500.005.000.050.000.200.00131
79 1.000.000.000.000.00016
1.000.000.000.000.001.000.000.000.000.00131
78 199.104.171.660.06815
39.642.471.172.442.024.427.117.424.69329
77 192.656.562.917.31615
37.116.551.235.113.931.153.215.561.17329
76 179.734.099.782.36615
32.304.346.624.577.677.542.664.340.32329
75 179.298.014.097.15815
32.147.777.859.184.848.195.877.774.12329
74 139.549.925.104.60915
19.474.181.596.702.120.769.518.147.49129
73 109.959.345.125.37915
12.091.057.580.402.320.408.575.019.02129
72 100.000.000.000.00015
10.000.000.000.000.100.000.000.000.00129
71 17.760.716.904.12814
315.443.064.948.595.849.460.344.51327
70 17.518.506.513.79814
306.898.070.474.000.474.070.898.60327
69 10.000.000.000.00014
100.000.000.000.010.000.000.000.00127
68 1.809.037.732.95813
3.272.617.519.267.629.157.162.72325
67 1.289.218.480.93913
1.662.084.291.595.951.924.802.66125
66 1.143.395.132.99413
1.307.352.430.155.510.342.537.03125
65 1.049.768.613.91013
1.102.014.142.751.572.414.102.01125
64 1.001.708.756.99913
1.003.420.433.849.483.340.243.00125
63 1.000.100.020.00013
1.000.200.050.005.000.500.020.00125
62 1.000.000.000.00013
1.000.000.000.001.000.000.000.00125
61 265.455.445.43712
70.466.593.512.421.539.566.40723
60 265.445.544.56212
70.461.337.128.082.173.316.40723
59 188.582.871.35112
35.563.499.367.176.399.436.55323
58 187.343.418.85112
35.097.556.586.968.565.579.05323
57 134.717.724.21912
18.148.865.218.881.256.884.18123
56 100.000.000.00012
10.000.000.000.100.000.000.00123
55 27.509.795.11211
756.788.827.131.728.887.65721
54 26.564.008.69711
705.646.558.080.855.646.50721
53 18.219.868.63611
331.963.613.131.316.369.13321
52 18.099.156.16611
327.579.453.939.354.975.72321
51 17.536.911.90111
307.543.279.040.972.345.70321
50 12.395.279.27411
153.642.948.292.849.246.35121
49 12.369.915.27411
153.014.803.898.308.410.35121
48 11.380.055.10411
129.505.654.181.456.505.92121
47 11.027.977.09511
121.616.278.818.872.616.12121
46 10.798.301.61411
116.603.317.757.713.306.61121
45 10.000.000.00011
100.000.000.010.000.000.00121
44 1.945.980.21610
3.786.839.003.009.386.87319
43 1.874.554.17610
3.513.953.360.633.593.15319
42 1.001.002.0002 + 1.001.002.00110
1.002.005.005.005.002.00119
41 1.000.000.0002 + 1.000.000.00110
1.000.000.001.000.000.00119
40 179.427.8412 + 179.427.8429
32.194.350.305.349.12317
39 100.000.0002 + 100.000.0019
10.000.000.100.000.00117
38 27.494.9322 + 27.494.9338
755.971.313.179.55715
37 26.545.4372 + 26.545.4388
704.660.252.066.40715
36 18.016.0582 + 18.016.0598
324.578.363.875.42315
35 12.783.2392 + 12.783.2408
163.411.212.114.36115
34 12.643.5492 + 12.643.5508
159.859.343.958.95115
33 11.842.1842 + 11.842.1858
140.237.333.732.04115
32 10.000.0002 + 10.000.0018
100.000.010.000.00115
31 1.975.3562 + 1.975.3577
3.902.033.302.09313
30 1.821.6362 + 1.821.6377
3.318.359.538.13313
29 1.780.7282 + 1.780.7297
3.170.993.990.71313
28 1.228.4252 + 1.228.4267
1.509.029.209.05113
27 1.190.9152 + 1.190.9167
1.418.279.728.14113
26 1.010.2002 + 1.010.2017
1.020.505.050.20113
25 1.000.0002 + 1.000.0017
1.000.001.000.00113
24 196.8132 + 196.8146
38.735.553.78311
23 192.3832 + 192.3846
37.011.411.07311
22 188.3482 + 188.3496
35.475.157.45311
21 187.8762 + 187.8776
35.297.579.25311
20 126.9352 + 126.9366
16.112.621.16111
19 119.0842 + 119.0856
14.181.118.14111
18 100.0002 + 100.0016
10.000.100.00111
17 11.9152 + 11.9165
141.979.1419
16 10.0002 + 10.0015
100.010.0019
15 1.9612 + 1.9624
3.847.4837
14 1.8762 + 1.8774
3.521.2537
13 1.8362 + 1.8374
3.372.7337
12 1.3552 + 1.3564
1.837.3817
11 1.1842 + 1.1854
1.403.0417
10 1.1202 + 1.1214
1.255.5217
9 1.0002 + 1.0014
1.001.0017
8 INFO 1732 + 1743
Prime!          30.1035
7 1252 + 1263
15.7515
6 1002 + 1013
10.1015
5 272 + 282
Prime!          7573
4 182 + 192
3433
3 102 + 112
1113
2 22 + 31
Prime!          71
1 12 + 21
Prime!          31
0 02 + 11
11



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Patrick De Geest - Belgium flag - Short Bio - Some Pictures
E-mail address : pdg@worldofnumbers.com