HOME plateWON | World!OfNumbers Palindromic Quasipronic Numbersof the form n(n+2) n(n+0) n(n+1) n(n+x) n^2+1 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 Products of Two Sequential Integers are defined and calculated by this extraordinary intricate and excruciatingly complex formula.
So, this line is for experts only

 base x ( base + n )
In you are interested in case n = 1 then visit this page about palindromic pronic numbers !

In case of n = 2 note that the formula n(n+2) can be written also as n2+2n.
Its always easier to refer something by name so I'll baptise these numbers Palindromic Quasipronic Numbers.
A repetitive infinite palindromic pattern can be made in the following manner :

9 x 11 = 99
99 x 101 = 9999
999 x 1001 = 999999
9999 x 10001 = 99999999

 PLAIN TEXT CONSEMOR

Palindromic Quasipronic Numbers

So far I compiled 130 Palindromic Quasipronic Numbers (coded as Q-PRONIC).
Here is the largest Sporadic Palindromic Quasipronic that Patrick De Geest
discovered, using CUDA code by Robert Xiao, on [ June 16, 2023 ].

 This nonpalindromic basenumber 2.869.583.606.927.752.572.852.824.676 has 28 digits yielding the following sporadic record palindromic quasipronic number 8.234.510.077.148.490.383.826.297.524.257.926.283.830.948.417.700.154.328 with a length of 55 digits.

Palindromic Quasipronic Numbers of the form x(x+2) can only start or end in one of the following digits : 0, 3, 4, 5, 8 or 9.
3 can only be followed by an even number : 30, 32, 34, 36 or 38
4 can only be followed by 2 : 42
5 can only be followed by an odd number : 51, 53, 55, 57, 59
8 can only be followed by an even number : 80, 82, 84, 86 or 88
9 can only be followed by 9 : 99
Alas, my palindromes may not have leading zero's! So the zero option must not be investigated.

There are no such palindromic quasipronic numbers of lengths 35, 43.

 Case Quapron Change of variables CUDApalin parameters Base Correction x(x+2) n.a. `A B C →  1 2 0 ` base = CUDAbase

Warut Roonguthai (email) 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. "

Proof that numbers of the form n^2–1 equal n(n+2) by substituting n with m+1 :

n^2 – 1 =
(m + 1)^2 – 1 =
m^2 + 2m + 1 – 1 =
m^2 + 2m =
m(m + 2)  QED

A finite palindromic pattern hides in the list in the following manner :

2 x 4 = 8
2966 x 2968 = 8803088
2967032 x 2967034 = 8803284823088
2967032966 x 2967032968 = 8803284627264823088
2967032967032 x 2967032967034 = 8803284627460647264823088

Note that the lengths of the resulting palindromes increases with steps of 6 !

Sources Revealed

 Neil Sloane's “Integer Sequences” Encyclopedia can be consulted online : Neil Sloane's Integer Sequences The regular quasipronic numbers are categorised as follows : %N n(n+2) under A005563. Check out the following two entries about Palindromic Quasipronic Numbers %N n(n+2) is a palindrome under A028503. %N Palindromes of form n(n+2) under A028504. %N Both n and its reverse are one less than a square under A066619. Click here to view some 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 Product of Sequence n(n+2)Length

[QP2] Twofold sequence (n) x (n+2)
Palindromic Quasipronic Numbers
Subsequence of A066619

One can find the regular numbers of the form n(n+2) at
%N n(n+2). under A005563.
The palindromic numbers of the form n(n+2) are categorised as follows :
%N n(n+2) is a palindrome. under A028503.
%N Palindromes of the form n(n+2). under A028504.
1302.869.583.606.927.752.572.852.824.67628
8.234.510.077.148.490.383.826.297.524.257.926.283.830.948.417.700.154.32855
1292.839.367.463.533.956.334.281.036.64628
8.062.007.592.975.252.853.725.763.273.723.675.273.582.525.792.957.002.60855
1281.858.533.249.512.946.375.265.289.38728
3.454.145.839.545.151.788.032.257.030.307.522.308.871.515.459.385.414.54355
127999.999.999.999.999.999.999.999.99927
999.999.999.999.999.999.999.999.999.999.999.999.999.999.999.999.999.99954
126907.622.710.543.480.493.039.799.17627
823.778.984.694.294.576.145.208.380.083.802.541.675.492.496.489.877.32854
125566.022.075.640.119.255.127.325.53127
320.380.990.111.948.883.478.999.034.430.999.874.388.849.111.099.083.02354
124290.957.452.721.121.988.428.818.34227
84.656.239.293.963.937.188.861.859.095.816.888.173.936.939.293.265.64853
123186.129.556.909.321.188.613.986.76127
34.644.211.955.260.235.023.092.246.864.229.032.053.206.255.911.244.64353
12299.999.999.999.999.999.999.999.99926
9.999.999.999.999.999.999.999.999.999.999.999.999.999.999.999.999.99952
12123.047.415.763.882.533.671.706.55526
531.183.373.393.261.113.083.114.757.411.380.311.162.393.373.381.13551
120Prime!    18.025.751.880.318.906.522.533.13126
324.927.730.850.820.594.096.293.434.392.690.495.028.058.037.729.42351
1199.999.999.999.999.999.999.999.99925
99.999.999.999.999.999.999.999.999.999.999.999.999.999.999.999.99950
1189.281.263.339.178.014.301.067.23625
86.141.849.171.169.824.133.657.799.775.633.142.896.117.194.814.16850
1177.188.432.787.621.709.530.912.79525
51.673.565.942.154.821.721.394.388.349.312.712.845.124.956.537.61550
1162.971.324.071.216.956.073.167.01625
8.828.766.736.193.306.646.148.659.568.416.466.033.916.376.678.28849
1151.846.392.722.689.497.797.462.36125
3.409.166.086.400.736.714.614.477.744.164.176.370.046.806.619.04349
114Prime!    1.757.294.218.102.431.600.179.70125
3.088.082.968.976.236.441.483.048.403.841.446.326.798.692.808.80349
113999.999.999.999.999.999.999.99924
999.999.999.999.999.999.999.999.999.999.999.999.999.999.999.99948
112909.440.099.007.674.131.076.27624
827.081.293.683.088.126.051.555.555.150.621.880.386.392.180.72848
111769.915.043.713.125.205.809.46324
592.769.174.535.783.496.498.902.209.894.694.387.535.471.967.29548
110748.178.082.592.971.093.142.36524
559.770.443.272.494.674.649.012.210.946.476.494.272.344.077.95548
109572.434.277.607.587.933.995.08124
327.681.002.180.121.048.789.960.069.987.840.121.081.200.186.72348
108179.952.055.392.509.877.201.98124
32.382.742.239.988.943.174.652.625.647.134.988.993.224.728.32347
10799.999.999.999.999.999.999.99923
9.999.999.999.999.999.999.999.999.999.999.999.999.999.999.99946
10664.825.429.061.399.630.637.04423
4.202.336.252.994.555.796.745.885.476.975.554.992.526.332.02446
10531.621.295.118.444.413.001.89923
999.906.304.967.756.463.298.909.892.364.657.769.403.609.99945
10420.566.244.031.577.809.646.31423
422.970.393.566.409.877.340.494.043.778.904.665.393.079.22445
1039.999.999.999.999.999.999.99922
99.999.999.999.999.999.999.999.999.999.999.999.999.999.99944
1026.483.218.910.752.681.076.69422
42.032.127.444.741.180.479.822.897.408.114.744.472.123.02444
1016.224.291.959.855.223.043.47122
38.741.810.401.518.373.506.999.960.537.381.510.401.814.78344
1006.041.636.672.175.825.109.25722
36.501.373.678.579.778.439.933.993.487.797.587.637.310.56344
995.693.758.099.575.300.973.86722
32.418.881.296.479.342.959.977.995.924.397.469.218.881.42344
98999.999.999.999.999.999.99921
999.999.999.999.999.999.999.999.999.999.999.999.999.99942
97554.092.910.028.082.027.05121
307.018.952.943.388.204.176.671.402.883.349.259.810.70342
96290.877.276.146.529.419.15621
84.609.589.778.424.332.617.971.623.342.487.798.590.64841
9599.999.999.999.999.999.99920
9.999.999.999.999.999.999.999.999.999.999.999.999.99940
9490.836.843.323.216.523.52220
8.251.332.104.926.586.308.998.036.856.294.012.331.52840
9389.703.862.578.415.825.65220
8.046.782.961.487.311.140.550.411.137.841.692.876.40840
9289.463.014.832.484.033.24620
8.003.631.022.917.258.135.335.318.527.192.201.363.00840
9173.279.502.387.125.583.30520
5.369.885.470.104.744.062.112.604.474.010.745.889.63540
9064.970.657.724.865.410.71420
4.221.186.365.201.613.466.886.643.161.025.636.811.22440
8924.367.639.093.196.434.91320
593.781.834.976.275.172.828.271.572.679.438.187.39539
8818.705.004.727.117.121.28720
349.877.201.841.473.853.020.358.374.148.102.778.94339
8717.594.739.560.626.811.64720
309.574.860.206.286.168.999.861.682.602.068.475.90339
869.999.999.999.999.999.99919
99.999.999.999.999.999.999.999.999.999.999.999.99938
859.087.624.822.702.230.72619
82.584.924.918.193.750.455.405.739.181.942.948.52838
846.503.375.380.654.101.68419
42.293.891.341.697.881.988.918.879.614.319.839.22438
833.159.884.456.126.887.92919
9.984.869.776.072.318.331.338.132.706.779.684.89937
823.153.133.622.994.482.34919
9.942.251.644.458.310.353.530.138.544.461.522.49937
81999.999.999.999.999.99918
999.999.999.999.999.999.999.999.999.999.999.99936
80918.101.589.661.512.80618
842.910.528.938.996.839.938.699.839.825.019.24836
79910.786.809.752.004.92218
829.532.612.818.234.809.908.432.818.216.235.92836
78906.394.740.346.503.62618
821.551.425.327.805.729.927.508.723.524.155.12836
77547.914.424.400.753.00118
300.210.216.466.408.476.674.804.664.612.012.00336
7699.999.999.999.999.99917
9.999.999.999.999.999.999.999.999.999.999.99934
7592.087.949.336.643.45617
8.480.190.413.028.212.112.128.203.140.910.84834
7476.110.360.069.816.32317
5.792.786.909.957.091.111.907.599.096.872.97534
7372.909.735.507.526.44317
5.315.829.531.777.462.332.647.771.359.285.13534
7228.296.773.213.909.50217
800.707.374.319.426.343.624.913.473.707.00833
7120.507.633.917.530.25417
420.563.048.895.437.313.734.598.840.365.02433
709.999.999.999.999.99916
99.999.999.999.999.999.999.999.999.999.99932
699.999.711.145.441.49916
99.994.222.992.266.955.966.229.922.249.99932
682.408.106.333.065.82316
5.798.976.111.351.729.271.531.116.798.97531
67999.999.999.999.99915
999.999.999.999.999.999.999.999.999.99930
66243.024.346.003.43515
59.060.832.750.397.779.305.723.806.09529
65185.008.189.318.33715
34.228.030.114.849.994.841.103.082.24329
6499.999.999.999.99914
9.999.999.999.999.999.999.999.999.99928
6364.860.859.637.35414
4.206.931.112.896.666.982.111.396.02428
6224.366.788.929.91314
593.740.402.754.979.457.204.047.39527
6117.556.561.248.20114
308.232.842.861.868.168.248.232.80327
609.999.999.999.99913
99.999.999.999.999.999.999.999.99926
596.035.200.088.90713
36.423.640.113.155.131.104.632.46326
583.153.099.043.04913
9.942.033.575.282.825.753.302.49925
572.967.841.885.46613
8.808.085.457.132.317.545.808.08825
562.967.032.967.03213
8.803.284.627.460.647.264.823.08825
552.939.136.124.28613
8.638.521.157.088.807.511.258.36825
542.839.531.502.89613
8.062.939.155.944.495.519.392.60825
532.050.695.871.49413
4.205.353.557.366.637.553.535.02425
52999.999.999.99912
999.999.999.999.999.999.999.99924
51186.125.268.23712
34.642.615.476.667.451.624.64323
50178.923.207.53112
32.013.514.193.539.141.531.02323
4999.999.999.99911
9.999.999.999.999.999.999.99922
4864.819.595.89411
4.201.580.011.991.100.851.02422
4754.809.426.00111
3.004.073.178.668.713.704.00322
4631.519.410.87911
993.473.262.222.262.374.39921
4524.064.405.17511
579.095.596.474.695.590.97521
4418.511.102.33711
342.660.909.767.909.066.24321
4318.163.818.61711
329.924.306.787.603.429.92321
429.999.999.99910
99.999.999.999.999.999.99920
413.146.471.48910
9.900.282.837.382.820.09919
402.967.032.96610
8.803.284.627.264.823.08819
392.441.052.18510
5.958.735.774.775.378.59519
38999.999.9999
999.999.999.999.999.99918
37185.812.3879
34.526.243.534.262.54317
3699.999.9998
9.999.999.999.999.99916
3593.809.7168
8.800.263.003.620.08816
3431.552.6598
995.570.353.075.59915
3322.765.8958
518.286.020.682.81515
329.999.9997
99.999.999.999.99914
319.200.1567
84.642.888.824.64814
302.967.0327
8.803.284.823.08813
29Prime!    1.868.2877
3.490.500.050.94313
28999.9996
999.999.999.99912
27552.1016
304.816.618.40312
26293.7866
86.310.801.36811
25243.0636
59.080.108.09511
24Prime!    179.3176
32.154.945.12311
23174.7476
30.536.863.50311
22174.6016
30.485.858.40311
2199.9995
9.999.999.99910
2020.5645
422.919.2249
1918.9915
360.696.0639
189.9994
99.999.9998
175.7314
32.855.8238
162.9664
8.803.0887
152.3654
5.597.9557
14Prime!    1.9074
3.640.4637
13Prime!    1.7474
3.055.5037
129993
999.9996
112043
42.0245
10Prime!    1913
36.8635
9992
9.9994
8752
5.7754
7642
4.2244
6Prime!    232
5753
5Prime!    172
3233
491
992
3Prime!    21
81
211
Prime!    31
101
01

Contributions

PDG's program completed the search up to length 26 inclusive.

[ January 15, 2008 ]
From length 27 on these quasipronic numbers were submitted by Feng Yuan.

[ November 4, 2022 ]
Robert Xiao (email) added two missing entries [87] and [88].

[ November 11, 2022 ]
Robert Xiao (email) added one missing entry [102].

[ November 13, 2022 ]
Robert Xiao (email) added one missing entry [102]
and eight more new entries [109] up to [116].

[ December 6, 2022 ]
Robert Xiao (email) added six new entries [117] up to [122].

[ June 16, 2023 ]
Patrick De Geest (email) added eight new entries [123] up to [130].

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