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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 more than 70 Palindromic Quasi_Over_Squares.

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 ?


The Table

My program searched for palindromes exhaustively upto length 31.


Index NrInfo BasenumberLength
Palindromic Quasi_Over_Squares of form n^2+(n+1)Length
   
 
? Info 265.455.445.544.554.455.445.43724
70.466.593.569.257.915.890.542.124.509.851.975.296.539.566.40747
? Info 26.545.544.554.455.445.43720
704.665.935.692.579.153.222.351.975.296.539.566.40739
? Info 26.544.554.455.445.544.56220
704.613.371.238.113.910.828.019.311.832.173.316.40739
   
 
88 Info 10.000.000.000.000.00017
100.000.000.000.000.010.000.000.000.000.00133
87 Info 2.654.554.455.445.43716
7.046.659.356.925.223.225.296.539.566.40731
86 Info 2.654.455.445.544.56216
7.046.133.712.381.181.811.832.173.316.40731
85 Info 1.740.233.076.507.52616
3.028.411.160.570.850.580.750.611.148.20331
84 Info 1.320.273.129.165.44516
1.743.121.135.596.317.136.955.311.213.47131
83 Info 1.157.309.182.414.43016
1.339.364.543.700.757.570.073.454.639.33131
82 Info 1.081.750.100.278.61016
1.170.183.279.452.783.872.549.723.810.71131
81 Info 1.068.301.518.025.31016
1.141.268.133.415.182.815.143.318.621.41131
80 Info 1.000.010.000.200.00016
1.000.020.000.500.005.000.050.000.200.00131
79 Info 1.000.000.000.000.00016
1.000.000.000.000.001.000.000.000.000.00131
78 Info 199.104.171.660.06815
39.642.471.172.442.024.427.117.424.69329
77 Info 192.656.562.917.31615
37.116.551.235.113.931.153.215.561.17329
76 Info 179.734.099.782.36615
32.304.346.624.577.677.542.664.340.32329
75 Info 179.298.014.097.15815
32.147.777.859.184.848.195.877.774.12329
74 Info 139.549.925.104.60915
19.474.181.596.702.120.769.518.147.49129
73 Info 109.959.345.125.37915
12.091.057.580.402.320.408.575.019.02129
72 Info 100.000.000.000.00015
10.000.000.000.000.100.000.000.000.00129
71 Info 17.760.716.904.12814
315.443.064.948.595.849.460.344.51327
70 Info 17.518.506.513.79814
306.898.070.474.000.474.070.898.60327
69 Info 10.000.000.000.00014
100.000.000.000.010.000.000.000.00127
68 Info 1.809.037.732.95813
3.272.617.519.267.629.157.162.72325
67 Info 1.289.218.480.93913
1.662.084.291.595.951.924.802.66125
66 Info 1.143.395.132.99413
1.307.352.430.155.510.342.537.03125
65 Info 1.049.768.613.91013
1.102.014.142.751.572.414.102.01125
64 Info 1.001.708.756.99913
1.003.420.433.849.483.340.243.00125
63 Info 1.000.100.020.00013
1.000.200.050.005.000.500.020.00125
62 Info 1.000.000.000.00013
1.000.000.000.001.000.000.000.00125
61 Info 265.455.445.43712
70.466.593.512.421.539.566.40723
60 Info 265.445.544.56212
70.461.337.128.082.173.316.40723
59 Info 188.582.871.35112
35.563.499.367.176.399.436.55323
58 Info 187.343.418.85112
35.097.556.586.968.565.579.05323
57 Info 134.717.724.21912
18.148.865.218.881.256.884.18123
56 Info 100.000.000.00012
10.000.000.000.100.000.000.00123
55 Info 27.509.795.11211
756.788.827.131.728.887.65721
54 Info 26.564.008.69711
705.646.558.080.855.646.50721
53 Info 18.219.868.63611
331.963.613.131.316.369.13321
52 Info 18.099.156.16611
327.579.453.939.354.975.72321
51 Info 17.536.911.90111
307.543.279.040.972.345.70321
50 Info 12.395.279.27411
153.642.948.292.849.246.35121
49 Info 12.369.915.27411
153.014.803.898.308.410.35121
48 Info 11.380.055.10411
129.505.654.181.456.505.92121
47 Info 11.027.977.09511
121.616.278.818.872.616.12121
46 Info 10.798.301.61411
116.603.317.757.713.306.61121
45 Info 10.000.000.00011
100.000.000.010.000.000.00121
44 Info 1.945.980.21610
3.786.839.003.009.386.87319
43 Info 1.874.554.17610
3.513.953.360.633.593.15319
42 Info 1.001.002.0002 + 1.001.002.00110
1.002.005.005.005.002.00119
41 Info 1.000.000.0002 + 1.000.000.00110
1.000.000.001.000.000.00119
40 Info 179.427.8412 + 179.427.8429
32.194.350.305.349.12317
39 Info 100.000.0002 + 100.000.0019
10.000.000.100.000.00117
38 Info 27.494.9322 + 27.494.9338
755.971.313.179.55715
37 Info 26.545.4372 + 26.545.4388
704.660.252.066.40715
36 Info 18.016.0582 + 18.016.0598
324.578.363.875.42315
35 Info 12.783.2392 + 12.783.2408
163.411.212.114.36115
34 Info 12.643.5492 + 12.643.5508
159.859.343.958.95115
33 Info 11.842.1842 + 11.842.1858
140.237.333.732.04115
32 Info 10.000.0002 + 10.000.0018
100.000.010.000.00115
31 Info 1.975.3562 + 1.975.3577
3.902.033.302.09313
30 Info 1.821.6362 + 1.821.6377
3.318.359.538.13313
29 Info 1.780.7282 + 1.780.7297
3.170.993.990.71313
28 Info 1.228.4252 + 1.228.4267
1.509.029.209.05113
27 Info 1.190.9152 + 1.190.9167
1.418.279.728.14113
26 Info 1.010.2002 + 1.010.2017
1.020.505.050.20113
25 Info 1.000.0002 + 1.000.0017
1.000.001.000.00113
24 Info 196.8132 + 196.8146
38.735.553.78311
23 Info 192.3832 + 192.3846
37.011.411.07311
22 Info 188.3482 + 188.3496
35.475.157.45311
21 Info 187.8762 + 187.8776
35.297.579.25311
20 Info 126.9352 + 126.9366
16.112.621.16111
19 Info 119.0842 + 119.0856
14.181.118.14111
18 Info 100.0002 + 100.0016
10.000.100.00111
17 Info 11.9152 + 11.9165
141.979.1419
16 Info 10.0002 + 10.0015
100.010.0019
15 Info 1.9612 + 1.9624
3.847.4837
14 Info 1.8762 + 1.8774
3.521.2537
13 Info 1.8362 + 1.8374
3.372.7337
12 Info 1.3552 + 1.3564
1.837.3817
11 Info 1.1842 + 1.1854
1.403.0417
10 Info 1.1202 + 1.1214
1.255.5217
9 Info 1.0002 + 1.0014
1.001.0017
8 INFO 1732 + 1743
Prime!          30.1035
7 Info 1252 + 1263
15.7515
6 Info 1002 + 1013
10.1015
5 Info 272 + 282
Prime!          7573
4 Info 182 + 192
3433
3 Info 102 + 112
1113
2 Info 22 + 31
Prime!          71
1 Info 12 + 21
Prime!          31
0 Info 02 + 11
11



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(All rights reserved) - Last modified : November 23, 2011.
Patrick De Geest - Belgium flag - Short Bio - Some Pictures
E-mail address : pdg@worldofnumbers.com