Palindromic numbers are numbers which read the same from
Palindromic Pronic Numbers are defined and calculated by this extraordinary intricate and excruciatingly complex formula.
So, this line is for experts only _{}
base x ( base + n )
In case of n = 1 we speak about our 'pronic' numbers a.k.a. oblong numbers!
Every pronic number is the double of a triangular. A quick look at both formulae reveals this fact n(n+1)/2 versus n(n+1).
To know more about the case whereby n = 2 visit this page named Palindromic Quasipronic Numbers.
The case whereby n > 2 please go to this page Palindromic Quasipronics of the form n(n+x).
PLAIN TEXT CONSEMOR |
This basenumber 542.552.107.088.554.950.844.257.896.843 has 30 digits yielding the following extraordinary palindromic pronic record number (ppn) 294.362.788.906.230.799.339.509.080.886.688.080.905.933.997.032.609.887.263.492 with a staggering length of 60 digits. |
This world record was achieved using CUDA code written by Robert Xiao and no longer on Rust. Recently he generalized the program to handle arbitrary quadratics. CUDA is a programming language, or more properly a programming toolkit, for writing software to run on GPUs rather than CPUs. It runs about 50 times faster on our GPUs though the logic of the code follows the Rust version closely. I asked Robert now that his CUDA is running at warp speed how far it would reach. He answered that as for 70 digits the time estimate on that is around ~400 days on one of our GPUs. 60 digits is about two days of GPU time, and it’ll go up by a factor of 10 every 4 digits. Doable but it’ll be a pretty decent power bill :) “Maybe we could get some palindrome enthusiasts together”, as David Griffeath put it, “and get a distributed computation going.” The program is very amenable to divide-and-conquer approaches. |
Case Pronic | Change of variables | CUDApalin parameters | Base Correction |
x(x+1) | n.a. | A B C → 1 1 0 | base = CUDAbase |
Warut Roonguthai (website) found this record number not by letting his UBASIC program perform an exhaustive search but by 'computer-assisted construction'.
He probably saw the following pattern emerging after finding these two ppn's :
[26] 2554554552Warut emailed me after reading all these assumptions and wrote that the way he recognized the patterns of the two ppn series is quite different from what I guessed above ! Actually he sees them as follows :
[42] 25545544554552
Could it be that he noticed that the second basenumber equals the first one with a core number 5445 added in between ?
25545|54552
2554554|4554552
Excited by this initial discovery maybe he tried to find an expansion by setting up a grow-pattern.
Perhaps he succeeded after realising he had to use two alternate core numbers 5445 and 4554 !
That insight paves the way to construct ppn's up to the record one :
255455445|544554552
25545544554|45544554552
2554554455445|5445544554552
255455445544554|455445544554552
It's a pity that this remarkable expansion is finite.
Note that the last basenumber has 30 digits and leads sofar to the largest constructed palindromic number of my whole website nl. 59 digits. Quite an achievement !A similar procedure applied on the non-palindromic basenumber [32] and using core numbers 4455 and 5544 gave birth to more palindromic pronic numbers :
[32] 255455445447
255455|445447
25545544|55445447
2554554455|4455445447
255455445544|554455445447
25545544554455|44554455445447
255 (4554)_{n} 552, for n = 1 to 6
255 (4554)_{n} 45447, for n = 1 to 5
Simplicity is the hall-mark of truth.
|Astonishing!|
16 & 17 |
---|
Admire for a moment this extraordinary manyfold construction with Two Consecutive Integers 16 and 17.
16 x 17 = 272 [ or 16 + 16^{2} ]
16^{1} + 17^{1} = 33
16^{2} + 17^{2} = 545
16^{3} + 17^{3} = 9009
17^{1} – 16^{1} = 1
17^{2} – 16^{2} = 33
and this bellying construction (finite, alas) where 16 and 17 embrace the numbers 70 is quite phenomenal !
16^{2} + 17^{2} = 545
1706^{2} + 1707^{2} = 5824285
170706^{2} + 170707^{2} = 58281418285
17070706^{2} + 17070707^{2} = 582818040818285
and that is not all because when we add more consecutives to the expression we see the following :
16^{3} + 17^{3} + 18^{3} = 14841 (#3)or summing the #311 squares from 16 up to 326
16^{3} + 17^{3} + ... + 25^{3} + 26^{3} = 108801 (#11)
16^{3} + 17^{3} + ... + 407^{3} + 408^{3} = 6961551696 (#393)
16^{2} + 17^{2} + ... + 325^{2} + 326^{2} = 11600611See Powered consecutives summing to palindromes for the database.
Changing the cube to squares in the above expression allows us to stuff the three consecutive numbers with 73
1736^{2} + 1737^{2} + 1738^{2} = 9051509The three consecutive numbers 16, 17 and 18 generate primes when applied to the formula n^{2} + n + 5.
173736^{2} + 173737^{2} + 173738^{2} = 90553635509
16^{2} + 16 + 5 = 277 is also a prime with multiplicative persistence of value 4 - see A034051.Last one in the queue are these two polynomials whereby 16 and 17 embrace the number 87
17^{2} + 17 + 5 = 311
18^{2} + 18+ 5 = 347
1876^{2} + 1877^{1} = 3521253
187876^{2} + 187877^{1} = 35297579253
All above operations yield a palindromic result!
Note that 9009 can be expressed as the product of four consecutives i.e. the sequence (n) x (n+2) x (n+4) x (n+6)
or 9009 = 7 x 9 x 11 x 13
A remarkable similar repetitive infinite nonpalindromic pattern can be made with 16 and 17 in the following manner :
16 + 17 = 33Chen Shuwen (email) of the People's Republic of China made a profound search for solutions of the Diophantine System.
1706 + 1707 = 3413
170706 + 170707 = 341413
17070706 + 17070707 = 34141413
Making operations with the concatenation of our two consecutive numbers 16 and 17
gives rise to more palindromic outcome :
or play around with their reversals :1716 – 1617 = 99
1716 + 1617 = 3333
1716 x 1617 = 2774772
GCD ( 1716 , 1617 ) = 33
1617 = the 33rd Pentagonal Number !Note 1617 is also 7 times the 21^{st} triangular number (7 x 231) and 231 is the 16^{th} partition number.
16 + 61 = 77A small discovery made using Pi-Search
17 + 71 = 88
1617 + 7161 = 8778
1716 + 6171 = 7887
7117 – 6116 = 1001
1717 – 1616 = 101
1661 + 1771 = 3432 = 2 x 1716 !!! Note that 1716 = 11 * 12 * 13 the product of three consecutive integers.
Let me tell you a story about, by now, two wellknown consecutive numbers :
Let me retell the above story but now with the 'concatenated' numbers 16 and 17 :
On [ September 1st, 1997 ] Bill Taylor (email) posted the following puzzle in sci.math involving the numbers 16 and 17 :
Partition the integers 1 to 23 into three sets, such that for no set are there three different numbers with two adding to the third.
There are three solutions... (23 was the largest number for which this could be done.)
***************** 1 2 4 8 11 16 22 3 5 6 7 19 21 23 9 10 12 13 14 15 17 18 20 ***************** 1 2 4 8 11 17 22 3 5 6 7 19 21 23 9 10 12 13 14 15 16 18 20 ***************** 1 2 4 8 11 22 3 5 6 7 19 21 23 9 10 12 13 14 15 16 17 18 20 *****************It's intriguing to note that most of the integers are fixed,
Let me quote a paragraph from Theoni Pappas' Book “The Joy of Mathematics”, Wide World Publishing/Tetra,
Edition 1998, page 3, from the very first chapter 'The evolution of Base Ten'.
Here is a prime starting with 1716 having the interesting property
that it will remain prime when any of its digits is deleted.
[ Source: Prime Curios! 1716336911 ]
A second interesting equation I found is this one.
Note the presence of '17' and '61' in the prime result.
[ Source: Prime Curios! 808793517812627212561 ]
1617 occurs many times as displacement to powers of ten
so that this number forms a (probable) prime nearest to those axes.
10^{9109} – 1617 |
10^{293} + 1617 10^{388} + 1617 10^{1276} + 1617 10^{1651} + 1617 10^{2416} + 1617 10^{6689} + 1617 10^{18448} + 1617 10^{25344} + 1617 10^{28725} + 1617 10^{41905} + 1617 |
A prime gap of 1716 occurs twice
between the following nearest primes around powers of ten :
Between 10^{1653} – 927 and 10^{1653} + 789 (difference is 1716)
Between 10^{1857} – 137 and 10^{1857} + 1579 (difference is 1716)
We already knew that
Now Jean Claude Rosa found an impressive extension of the expression of the square of two consecutives
[ Source : sumsquare.htm ]
Note that 16 and 17 appear also in the decimal expansion of the palindrome !
[1945][1_16] = prime!One needs to add at least 16 ones to 1945 to arrive at a prime.
[1946][1_17] = prime!
1621 & 1622 |
---|
Just a twofold construction this time starting with primenumber 1621 but with larger palindromic outcome !
1621 x 1622 = 2629262
1621^{2} + 1622^{2} = 5258525
John H. Conway and Richard K. Guy, in their work “The Book of Numbers”, speak about pronic numbers
when they refer to the product of two consecutive integers n(n + 1) or twice the triangular number T(n).
So, in fact this page tells you all there is about Palindromic Pronic Numbers.
Here is Eric Weisstein's definition of a Pronic Number.
Huen Y.K. from Singapore developed a general generating function for palindromic products
of consecutive integers using concise programcode written for Macsyma 2.2.1.
Global Generating Function For Palindromic Products of Consecutive Integers.
Click here to view some entries to the table about palindromes. |
Index Nr | Basenumber | Length |
---|---|---|
Palindromic Pronic Numbers [ Formula n(n+1) ] | Length | |
[QP1] Two consecutives (n) x (n+1) Palindromic Pronic Numbers | ||
| ||
110 | 542.552.107.088.554.950.844.257.896.843 | 30 |
294.362.788.906.230.799.339.509.080.886.688.080.905.933.997.032.609.887.263.492 | 60 | |
109 | 489.359.849.293.616.698.599.019.712.388 | 30 |
239.473.062.100.671.247.666.800.361.701.107.163.008.666.742.176.001.260.374.932 | 60 | |
108 | 256.375.288.386.585.303.594.683.736.052 | 30 |
65.728.288.495.304.781.240.757.153.154.145.135.175.704.218.740.359.488.282.756 | 59 | |
107 | ( wr ) 255.455.445.544.554.455.445.544.554.552 | 30 |
65.257.484.658.366.826.781.688.069.846.664.896.088.618.762.866.385.648.475.256 | 59 | |
106 | 166.335.325.122.303.670.171.268.680.191 | 30 |
27.667.440.383.542.466.468.219.750.901.310.905.791.286.466.424.538.304.476.672 | 59 | |
105 | 156.576.909.589.458.650.620.752.632.381 | 30 |
24.516.328.616.585.508.342.478.257.655.455.675.287.424.380.558.561.682.361.542 | 59 | |
104 | 147.717.705.191.208.602.318.386.620.403 | 30 |
21.820.520.426.956.816.858.020.231.337.973.313.202.085.861.865.962.402.502.812 | 59 | |
103 | 147.714.724.743.240.911.096.643.847.403 | 30 |
21.819.639.905.971.428.648.705.618.418.781.481.650.784.682.417.950.993.691.812 | 59 | |
102 | 47.133.925.696.199.327.011.587.850.333 | 29 |
2.221.606.951.534.839.213.522.814.668.448.664.182.253.129.384.351.596.061.222 | 58 | |
101 | 14.151.717.193.775.152.311.341.052.126 | 29 |
200.271.099.532.591.271.832.798.573.636.375.897.238.172.195.235.990.172.002 | 57 | |
100 | 8.100.996.822.390.891.944.117.676.512 | 28 |
65.626.149.516.387.328.478.238.194.555.549.183.287.482.378.361.594.162.656 | 56 | |
99 | 4.821.320.883.331.063.084.866.217.336 | 28 |
23.245.135.060.044.222.418.421.392.577.529.312.481.422.244.006.053.154.232 | 56 | |
98 | ( wr ) 2.554.554.455.445.544.554.455.445.447 | 28 |
6.525.748.465.836.682.678.168.806.483.846.088.618.762.866.385.648.475.256 | 55 | |
97 | 2.549.581.638.365.707.381.113.017.367 | 28 |
6.500.366.530.691.564.691.667.376.054.506.737.661.964.651.960.356.630.056 | 55 | |
96 | 1.720.455.397.703.583.339.735.582.856 | 28 |
2.959.966.775.487.395.117.669.765.672.765.679.667.115.937.845.776.699.592 | 55 | |
95 | 779.801.512.547.589.097.035.972.557 | 27 |
608.090.398.971.507.755.946.586.056.650.685.649.557.705.179.893.090.806 | 54 | |
94 | 478.301.650.243.599.779.631.219.866 | 27 |
228.772.468.625.750.853.133.838.559.955.838.331.358.057.526.864.277.822 | 54 | |
93 | 160.521.853.830.056.733.172.095.248 | 27 |
25.767.265.557.038.099.496.816.774.647.761.869.499.083.075.556.276.752 | 53 | |
92 | 53.385.378.822.409.047.765.025.961 | 26 |
2.849.998.672.012.120.447.468.574.114.758.647.440.212.102.768.999.482 | 52 | |
91 | 47.474.640.248.485.781.264.512.766 | 26 |
2.253.841.466.723.146.083.048.160.220.618.403.806.413.276.641.483.522 | 52 | |
90 | ( wr ) 25.545.544.554.455.445.544.554.552 | 26 |
652.574.846.583.668.267.816.885.656.588.618.762.866.385.648.475.256 | 51 | |
89 | 24.664.219.622.898.547.441.653.882 | 26 |
608.323.729.606.573.765.768.282.565.282.867.567.375.606.927.323.806 | 51 | |
88 | 16.077.772.292.644.112.713.002.548 | 26 |
258.494.761.894.114.728.324.489.525.984.423.827.411.498.167.494.852 | 51 | |
87 | 15.898.002.337.555.893.441.543.876 | 26 |
252.746.478.324.932.652.034.884.060.488.430.256.239.423.874.647.252 | 51 | |
86 | 4.996.103.555.848.566.189.128.693 | 25 |
24.961.050.740.762.687.134.037.400.473.043.178.626.704.705.016.942 | 50 | |
85 | ( wr ) 255.455.445.544.554.455.445.447 | 24 |
65.257.484.658.366.826.781.638.483.618.762.866.385.648.475.256 | 47 | |
84 | ( wr ) 255.445.544.554.455.445.544.552 | 24 |
65.252.426.232.722.282.129.202.020.292.128.222.723.262.425.256 | 47 | |
83 | 255.026.231.105.195.553.803.792 | 24 |
65.038.378.551.720.612.220.143.934.102.221.602.715.587.383.056 | 47 | |
82 | 160.710.793.885.413.017.976.548 | 24 |
25.827.959.271.279.706.287.024.942.078.260.797.217.295.972.852 | 47 | |
81 | 142.318.630.031.242.794.328.401 | 24 |
20.254.592.453.969.763.373.355.855.337.336.796.935.429.545.202 | 47 | |
80 | 50.041.660.886.233.890.263.851 | 23 |
2.504.167.824.252.830.819.532.442.359.180.382.524.287.614.052 | 46 | |
79 | ( wr ) 25.545.544.554.445.544.554.552 | 23 |
652.574.846.583.162.415.454.424.454.514.261.385.648.475.256 | 45 | |
78 | ( wr ) 25.544.554.554.445.545.544.552 | 23 |
652.524.267.385.044.663.854.424.458.366.440.583.762.425.256 | 45 | |
77 | 25.515.127.911.702.848.979.212 | 23 |
651.021.752.350.557.787.134.939.431.787.755.053.257.120.156 | 45 | |
76 | 4.535.567.920.553.179.774.373 | 22 |
20.571.376.361.951.095.277.588.577.259.015.916.367.317.502 | 44 | |
75 | ( wr ) 2.554.554.455.445.544.554.552 | 22 |
6.525.748.465.836.682.678.664.668.762.866.385.648.475.256 | 43 | |
74 | 2.461.611.964.203.752.530.097 | 22 |
6.059.533.462.311.056.627.608.067.266.501.132.643.359.506 | 43 | |
73 | 1.650.614.605.673.801.852.608 | 22 |
2.724.528.576.463.680.383.037.303.830.863.646.758.254.272 | 43 | |
72 | 1.489.721.139.127.855.915.966 | 22 |
2.219.269.072.364.396.642.539.352.466.934.632.709.629.122 | 43 | |
71 | 1.488.064.682.926.106.222.841 | 22 |
2.214.336.500.571.973.048.839.388.403.791.750.056.334.122 | 43 | |
70 | 77.983.954.983.470.964.042 | 20 |
6.081.497.234.864.025.807.667.085.204.684.327.941.806 | 40 | |
69 | ( wr ) 25.545.544.554.455.445.447 | 20 |
652.574.846.583.668.262.858.262.866.385.648.475.256 | 39 | |
68 | 15.251.279.843.549.799.288 | 20 |
232.601.536.866.268.390.262.093.862.668.635.106.232 | 39 | |
67 | 5.398.911.022.504.737.056 | 19 |
29.148.240.228.923.145.399.354.132.982.204.284.192 | 38 | |
66 | 5.381.424.298.612.875.413 | 19 |
28.959.727.481.701.078.088.087.010.718.472.795.982 | 38 | |
65 | 2.457.845.237.463.156.837 | 19 |
6.041.003.211.320.321.823.281.230.231.123.001.406 | 37 | |
64 | 1.505.031.457.201.893.533 | 19 |
2.265.119.687.167.255.086.805.527.617.869.115.622 | 37 | |
63 | 777.568.874.404.375.437 | 18 |
604.613.354.442.487.383.383.784.244.453.316.406 | 36 | |
62 | 452.092.594.175.225.926 | 18 |
204.387.713.708.085.523.325.580.807.317.783.402 | 36 | |
61 | ( wr ) 255.455.445.544.554.552 | 18 |
65.257.484.658.366.876.367.866.385.648.475.256 | 35 | |
60 | 164.091.059.979.032.478 | 18 |
26.925.875.965.042.434.343.424.056.957.852.962 | 35 | |
59 | 16.894.240.027.247.213 | 17 |
285.415.346.098.241.929.142.890.643.514.582 | 33 | |
58 | 16.527.595.524.408.908 | 17 |
273.161.413.818.461.383.164.818.314.161.372 | 33 | |
57 | 16.481.958.009.966.691 | 17 |
271.654.939.842.305.181.503.248.939.456.172 | 33 | |
56 | 5.177.562.474.527.178 | 16 |
26.807.153.177.631.999.913.677.135.170.862 | 32 | |
55 | 4.972.663.340.772.093 | 16 |
24.727.380.700.658.677.685.600.708.372.742 | 32 | |
54 | 4.570.386.636.562.273 | 16 |
20.888.434.007.667.011.076.670.043.488.802 | 32 | |
53 | 4.526.244.131.170.926 | 16 |
20.486.885.934.959.255.295.943.958.868.402 | 32 | |
52 | ( wr ) 2.554.554.455.445.447 | 16 |
6.525.748.465.836.186.816.385.648.475.256 | 31 | |
51 | 2.450.326.711.358.322 | 16 |
6.004.100.992.396.091.906.932.990.014.006 | 31 | |
50 | 1.651.452.340.119.016 | 16 |
2.727.294.831.684.575.754.861.384.927.272 | 31 | |
49 | 1.578.831.273.783.306 | 16 |
2.492.708.191.076.218.126.701.918.072.942 | 31 | |
48 | ( wr ) 255.455.444.554.552 | 15 |
65.257.484.152.564.046.525.148.475.256 | 29 | |
47 | ( wr ) 255.445.545.544.552 | 15 |
65.252.426.738.554.045.583.762.425.256 | 29 | |
46 | 244.952.812.962.002 | 15 |
60.001.880.577.997.779.977.508.810.006 | 29 | |
45 | 77.845.215.132.222 | 14 |
6.059.877.518.982.002.898.157.789.506 | 28 | |
44 | 48.677.117.352.863 | 14 |
2.369.461.753.784.444.873.571.649.632 | 28 | |
43 | 46.378.652.817.871 | 14 |
2.150.979.437.200.660.027.349.790.512 | 28 | |
42 | 25.545.544.554.552 | 14 |
652.574.846.588.626.885.648.475.256 | 27 | |
41 | 25.509.264.105.807 | 14 |
650.722.555.219.838.912.555.227.056 | 27 | |
40 | 17.103.596.748.243 | 14 |
292.533.021.726.525.627.120.335.292 | 27 | |
39 | 16.764.499.079.986 | 14 |
281.048.429.402.868.204.924.840.182 | 27 | |
38 | 16.590.433.647.491 | 14 |
275.242.488.611.818.116.884.242.572 | 27 | |
37 | 8.107.780.745.372 | 13 |
65.736.108.615.033.051.680.163.756 | 26 | |
36 | 5.065.320.693.323 | 13 |
25.657.473.726.211.262.737.475.652 | 26 | |
35 | 1.559.870.854.106 | 13 |
2.433.197.081.490.941.807.913.342 | 25 | |
34 | Prime! 1.423.698.345.401 | 13 |
2.026.916.978.698.968.796.196.202 | 25 | |
33 | 546.176.761.668 | 12 |
298.309.054.986.689.450.903.892 | 24 | |
32 | 255.455.445.447 | 12 |
65.257.484.608.780.648.475.256 | 23 | |
31 | 255.126.517.207 | 12 |
65.089.539.782.428.793.598.056 | 23 | |
30 | 155.482.156.418 | 12 |
24.174.700.964.546.900.747.142 | 23 | |
29 | 77.714.915.542 | 11 |
6.039.608.097.777.908.069.306 | 22 | |
28 | 47.878.213.558 | 11 |
2.292.323.333.553.333.232.922 | 22 | |
27 | 25.541.432.472 | 11 |
652.364.772.747.277.463.256 | 21 | |
26 | 2.554.554.552 | 10 |
6.525.748.961.698.475.256 | 19 | |
25 | 1.694.576.061 | 10 |
2.871.588.028.208.851.782 | 19 | |
24 | 166.691.108 | 9 |
27.785.925.652.958.772 | 17 | |
23 | 142.594.226 | 9 |
20.333.113.431.133.302 | 17 | |
22 | 50.702.751 | 8 |
2.570.769.009.670.752 | 16 | |
21 | 47.370.058 | 8 |
2.243.922.442.293.422 | 16 | |
20 | 15.814.148 | 8 |
250.087.292.780.052 | 15 | |
19 | 15.282.411 | 8 |
233.552.101.255.332 | 15 | |
18 | 15.137.566 | 8 |
229.145.919.541.922 | 15 | |
17 | 5.218.983 | 7 |
27.237.788.773.272 | 14 | |
16 | 4.670.028 | 7 |
21.809.166.190.812 | 14 | |
15 | 1.713.543 | 7 |
2.936.231.326.392 | 13 | |
14 | 1.610.151 | 7 |
2.592.587.852.952 | 13 | |
13 | 143.498 | 6 |
20.591.819.502 | 11 | |
12 | 142.401 | 6 |
20.278.187.202 | 11 | |
11 | 52.008 | 5 |
2.704.884.072 | 10 | |
10 | 5.313 | 4 |
28.233.282 | 8 | |
9 | ( cd ) 5.291 | 4 |
27.999.972 | 8 | |
8 | 2.457 | 4 |
6.039.306 | 7 | |
7 | Prime! 1.621 | 4 |
2.629.262 | 7 | |
6 | 538 | 3 |
289.982 | 6 | |
5 | 77 | 2 |
6.006 | 4 | |
4 | 16 | 2 |
272 | 3 | |
3 | Prime! 2 | 1 |
6 | 1 | |
2 | 1 | 1 |
2 | 1 | |
1 | 0 | 1 |
0 | 1 |
Chad Davis (email) is the first contributor to this topic. In fact he is the inspirator by sending me the first nine consecutives.
Much appreciated, thanks !
PDG's program completed the search for ppn's up to length 43 inclusive.
More terms from Warut Roonguthai, Index Nrs [46], [47], [51], [60] and nine higher ppn's in the table indicated with ( wr ).
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