More about Palindromic Pronic Numbers

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Palindromic Pronic Numbers of the form n(n+1)
n(n+2) n(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 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 !
Palindromic Pronic Numbers can only begin (and end!) with the digits 2 and 6

and if it terminates in 6, it terminates in 06 or 56.
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.

So far I compiled more than 60 palindromic pronic numbers.

Here is the largest one which was submitted to me by Warut Roonguthai

from Bangkok (Thailand) on [ August 31, 1997 ].

This (palindromic!) basenumber
255.455.445.544.554.455.445.544.554.552
has 30 digits
yielding the following extraordinary palindromic pronic record number (ppn)
65.257.484.658.366.826.781.688.069.846.664.896.088.618.762.866.385.648.475.256
with a length of (hold your breath) 59 digits.

Warut Roonguthai 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 :

[25] 2554554552
[41] 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 upto 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 31 and using core numbers 4455 and 5544 gave birth to more palindromic pronic numbers :
[31] 255455445447
255455|445447
25545544|55445447
2554554455|4455445447
255455445544|554455445447
25545544554455|44554455445447

Warut 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 :

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

a curious pair of consecutives

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)
16 ³ + 17 ³ + ... + 25 ³ + 26 ³ = 108801 (#11)
16 ³ + 17 ³ + ... + 407 ³ + 408 ³ = 6961551696 (#393)

or summing the #311 squares from 16 up to 326

16 ² + 17 ² + ... + 325 ² + 326 ² = 11600611

See 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 = 9051509
173736 ² + 173737 ² + 173738 ² = 90553635509

The three consecutive numbers 16, 17 and 18 generate primes when applied to the formula n² + n + 5.
Another such triplet is {191,192,193}, can you find more ?

16 ² + 16 + 5 = 277 is also a prime with multiplicative persistence of value 4 - See A034051.
17 ² + 17 + 5 = 311
18 ² + 18 + 5 = 347

Last one in the queue are these two polynomials whereby 16 and 17 embrace the number 87

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 = 33
1706 + 1707 = 3413
170706 + 170707 = 341413
17070706 + 17070707 = 34141413

Chen Shuwen of the People's Republic of China made a profound search for solutions of the Diophantine System.
In his website about Equal Sums of Like Powers I came across the following :
( k = 1,2 )
[ 0, 16, 17 ] = [ 1, 12, 20 ] = [ 2, 10, 21 ] = [ 5, 6, 22 ]
meaning :
33 = 0 + 16 + 17 = 1 + 12 + 20 = 2 + 10 + 21 = 5 + 6 + 22
545 = 0² + 16² + 17² = 1² + 12² + 20² = 2² + 10² + 21² = 5² + 6² + 22²

The equation n² + n + 17 is known to produce 16 primes for consecutive values of n from 0 to 15.
The integer sequence whereby %N n² + n + 17 is composite. (Sloane's A007636) begins as follows :

%S 16,17,20,25,32,33,34,36,39,41,43,48,50,51,...

Making operations with the concatenation of our two consecutive numbers 16 and 17
gives rise to more palindromic outcome :

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.

or play around with their reversals :

16 + 61 = 77
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.

A small discovery made using Pi-Search

The string 1771 was first found at position 8448 counting from the first digit after the decimal point.
The 3. is not counted.

Let me tell you a story about, by now, two wellknown consecutive numbers :

The smallest number one must multiply with 16 to arrive at a palindrome is 17.

16 x 17 = 272
The smallest number one must multiply with 17 to arrive at a palindrome is 16.

17 x 16 = 272

Let me retell the above story but now with the 'concatenated' numbers 16 and 17 :

The smallest number one must multiply with 1617 to arrive at a palindrome is 123.

1617 x 123 = 198891
The smallest number one must multiply with 1716 to arrive at a palindrome is 147.

1716 x 147 = 252252
Note that when we multiply these two multipliers together we find again a palindrome !

123 x 147 = 18081

On [ September 1st, 1997 ] Bill Taylor 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,
with just 16 & 17 movable. (Bill doubts there's any real significance in this though.)

While skimming through the integer sequences in Sloane's table I came across entry number A006488.
The description is as follows : n! (n factorial) has a square number of digits.
The term 1617 pops up on the second line.
I asked Warut Roonguthai to calculate this square number for me.
He responded concise but to the point :

Since n! is asymptotically equal to (n/e)^n*sqrt(2*pi*n) (Stirling's formula),

we can find that log(1617!) is about 4.488,2.

So 1617! has 4489 = 67² digits.
All along the answer was hidden in the proper digits of the term itself !

The least integer such that the decimal part of its square root starts with
the ninedigits number 123456789 is, lo and behold [ Source: Solution to Problem #14 ]

1661201661617^(1/2) = 1288876.12345678900003617...
After careful observation we noticed the following facts :
1. the number begins with 16 and ends with 17.

1661201661617
2. we find twice the occurence of 1661.

1661201661617
3. and once the occurence of 1617.

1661201661617
4. the inverse reversal 6116 also occurs in a separated way.

1661201661617

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'.

"... and scientists were reticent to employ the base 10 system

because it had no simple way to denote fractions. But it became

popular when merchants adopted it, since it proved so

invaluable in their work and record keeping. Later, decimal

fractions made their appearance in the 16^th century, and the

decimal point was introduced in 1617 by John Napier."

Here is a prime starting with 1716 having the interesting property
that it will remain prime when any of its digits is deleted.

1716336911

[ Source: Prime Curios! 1716336911 ]

A second interesting equation I found is this one.
Note the presence of '17' and '61' in the prime result.

17¹⁷ – 16¹⁶ = 808793517812627212561

[ Source: Prime Curios! 808793517812627212561 ]

We knew already that 16 + 17 = 33
Did you know, a nice pattern exists, taken from WONplate 152, i.e.
Near_tautonymic numbers of the form (T)_(T + 1) as SQUARES

(4_n)5(3_n)2(8_n)8_(4_n)5(3_n)2(8_n)9 = (6_n)7(3_n)2(6_n)7²
Among the first 33333 values of n only 0, 16, 17, 33, 2738, 3096 yield PRIMES for the
right part of the equation ! Note the succession of the same three values 16, 17 and 33.
n = 16
prime ^ 2 = 666666666666666673333333333333333266666666666666667 ^ 2 =

444444444444444453333333333333333288888888888888888_444444444444444453333333333333333288888888888888889
n = 17
prime ^ 2 = 666666666666666667333333333333333332666666666666666667 ^ 2 =

444444444444444445333333333333333332888888888888888888_444444444444444445333333333333333332888888888888888889

1617 occurs many times as displacement to powers of ten
so that this number forms a (probable) prime nearest to those axes.

10²⁹³ + 1617 is the smallest prime of length 294.
10³⁸⁸ + 1617 is the smallest prime of length 389.
10¹²⁷⁶ + 1617 is the smallest prime of length 1277.
10¹⁶⁵¹ + 1617 is the smallest prime of length 1652.
10²⁴¹⁶ + 1617 is the smallest prime of length 2417.
10⁶⁶⁸⁹ + 1617 is the smallest prime of length 6690.

A prime gap of 1716 occurs twice
between the following nearest primes around powers of ten :
Between 10¹⁶⁵³ — 927 and 10¹⁶⁵³ + 789 (difference is 1716)
Between 10¹⁸⁵⁷ — 137 and 10¹⁸⁵⁷ + 1579 (difference is 1716)

We already knew that

16 ² + 17 ² = 545

Now Jean-Claude Rosa found an impressive extension of the expression of the square of two consecutives
[ Source : sumsquare.htm ]

1650889370330016 ² + 1650889370330017 ² = 5450871426137276727316241780545

Note that 16 and 17 appear also in the decimal expansion of the palindrome !

another curious pair of consecutives

Just a twofold construction this time starting with primenumber 1621 but with larger palindromic outcome !

1621 x 1622 = 2629262
1621 ² + 1622 ² = 5258525

Sources Revealed

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.

Neil Sloane's "Integer Sequences" Encyclopedia can be consulted online :
Neil Sloane's Integer Sequences
The normal pronic numbers are already categorised.
%N The pronic numbers: n(n+1) under A002378.
Check out the following two entries about Palindromic Pronic Numbers

%N n(n+1) is a palindrome under A028336.
%N Palindromes of form n(n+1) under A028337.
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.

There are no palindromic pronic numbers of lengths 2, 5, 9, 12, 18, 20, 30, 34 (Sloane's A034307).

My program (PDG) completed the search for ppn's upto length 34 inclusive.

Full Listing

Index Nr	Info	Basenumber	Length
Index Nr	Palindromic Pronic Numbers [Formula n(n+1)]		Length

	Two consecutives (n) x (n+1) Palindromic Pronic Numbers
?	?		?
?	?		?
p?	( wr ) 255.455.445.544.554.455.445.544.554.552		30
p?	65.257.484.658.366.826.781.688.069.846.664.896.088.618.762.866.385.648.475.256		59
p?	2.554.554.455.445.544.554.455.445.447		28
p?	6.525.748.465.836.682.678.168.806.483.846.088.618.762.866.385.648.475.256		55
p?	25.545.544.554.455.445.544.554.552		26
p?	652.574.846.583.668.267.816.885.656.588.618.762.866.385.648.475.256		51
p?	255.455.445.544.554.455.445.447		24
p?	65.257.484.658.366.826.781.638.483.618.762.866.385.648.475.256		47
p?	255.445.544.554.455.445.544.552		24
p?	65.252.426.232.722.282.129.202.020.292.128.222.723.262.425.256		47
p?	25.545.544.554.445.544.554.552		23
p?	652.574.846.583.162.415.454.424.454.514.261.385.648.475.256		45
p?	25.544.554.554.445.545.544.552		23
p?	652.524.267.385.044.663.854.424.458.366.440.583.762.425.256		45
p?	2.554.554.455.445.544.554.552		22
p?	6.525.748.465.836.682.678.664.668.762.866.385.648.475.256		43
p?	25.545.544.554.455.445.447		20
p?	652.574.846.583.668.262.858.262.866.385.648.475.256		39
p?	( wr ) 255.455.445.544.554.552		18
p?	65.257.484.658.366.876.367.866.385.648.475.256		35


58	( pdg ) 16.894.240.027.247.213		17
58	285.415.346.098.241.929.142.890.643.514.582		33
57	( pdg ) 16.527.595.524.408.908		17
57	273.161.413.818.461.383.164.818.314.161.372		33
56	( pdg ) 16.481.958.009.966.691		17
56	271.654.939.842.305.181.503.248.939.456.172		33
55	( pdg ) 5.177.562.474.527.178		16
55	26.807.153.177.631.999.913.677.135.170.862		32
54	( pdg ) 4.972.663.340.772.093		16
54	24.727.380.700.658.677.685.600.708.372.742		32
53	( pdg ) 4.570.386.636.562.273		16
53	20.888.434.007.667.011.076.670.043.488.802		32
52	( pdg ) 4.526.244.131.170.926		16
52	20.486.885.934.959.255.295.943.958.868.402		32
51	( wr ) 2.554.554.455.445.447		16
51	6.525.748.465.836.186.816.385.648.475.256		31
50	( pdg ) 2.450.326.711.358.322		16
50	6.004.100.992.396.091.906.932.990.014.006		31
49	( pdg ) 1.651.452.340.119.016		16
49	2.727.294.831.684.575.754.861.384.927.272		31
48	( pdg ) 1.578.831.273.783.306		16
48	2.492.708.191.076.218.126.701.918.072.942		31
47	( wr ) 255.455.444.554.552		15
47	65.257.484.152.564.046.525.148.475.256		29
46	( wr ) 255.445.545.544.552		15
46	65.252.426.738.554.045.583.762.425.256		29
45	( pdg ) 244.952.812.962.002		15
45	60.001.880.577.997.779.977.508.810.006		29
44	77.845.215.132.222		14
44	6.059.877.518.982.002.898.157.789.506		28
43	48.677.117.352.863		14
43	2.369.461.753.784.444.873.571.649.632		28
42	46.378.652.817.871		14
42	2.150.979.437.200.660.027.349.790.512		28
41	25.545.544.554.552		14
41	652.574.846.588.626.885.648.475.256		27
40	25.509.264.105.807		14
40	650.722.555.219.838.912.555.227.056		27
39	17.103.596.748.243		14
39	292.533.021.726.525.627.120.335.292		27
38	16.764.499.079.986		14
38	281.048.429.402.868.204.924.840.182		27
37	16.590.433.647.491		14
37	275.242.488.611.818.116.884.242.572		27
36	8.107.780.745.372		13
36	65.736.108.615.033.051.680.163.756		26
35	5.065.320.693.323		13
35	25.657.473.726.211.262.737.475.652		26
34	1.559.870.854.106		13
34	2.433.197.081.490.941.807.913.342		25
33	1.423.698.345.401		13
33	2.026.916.978.698.968.796.196.202		25
32	546.176.761.668 x 546.176.761.669		12
32	298.309.054.986.689.450.903.892		24
31	255.455.445.447 x 255.455.445.448		12
31	65.257.484.608.780.648.475.256		23
30	255.126.517.207 x 255.126.517.208		12
30	65.089.539.782.428.793.598.056		23
29	155.482.156.418 x 155.482.156.419		12
29	24.174.700.964.546.900.747.142		23
28	77.714.915.542 x 77.714.915.543		11
28	6.039.608.097.777.908.069.306		22
27	47.878.213.558 x 47.878.213.559		11
27	2.292.323.333.553.333.232.922		22
26	25.541.432.472 x 25.541.432.473		11
26	652.364.772.747.277.463.256		21
25	2.554.554.552 x 2.554.554.553		10
25	6.525.748.961.698.475.256		19
24	1.694.576.061 x 1.694.576.062		10
24	2.871.588.028.208.851.782		19
23	166.691.108 x 166.691.109		9
23	27.785.925.652.958.772		17
22	142.594.226 x 142.594.227		9
22	20.333.113.431.133.302		17
21	50.702.751 x 50.702.752		8
21	2.570.769.009.670.752		16
20	47.370.058 x 47.370.059		8
20	2.243.922.442.293.422		16
19	15.814.148 x 15.814.149		8
19	250.087.292.780.052		15
18	15.282.411 x 15.282.412		8
18	233.552.101.255.332		15
17	15.137.566 x 15.137.567		8
17	229.145.919.541.922		15
16	5.218.983 x 5.218.984		7
16	27.237.788.773.272		14
15	4.670.028 x 4.670.029		7
15	21.809.166.190.812		14
14	1.713.543 x 1.713.544		7
14	2.936.231.326.392		13
13	1.610.151 x 1.610.152		7
13	2.592.587.852.952		13
12	143.498 x 143.499		6
12	20.591.819.502		11
11	142.401 x 142.402		6
11	20.278.187.202		11
10	52.008 x 52.009		5
10	2.704.884.072		10
9	5.313 x 5.314		4
9	28.233.282		8
8	( cd ) 5.291 x 5.292		4
8	27.999.972		8
7	2.457 x 2.458		4
7	6.039.306		7
6	1.621 x 1.622		4
6	2.629.262		7
5	538 x 539		3
5	289.982		6
4	77 x 78		2
4	6.006		4
3	16 + 17	16 x 17	2
3	33	272	3
2	2 + 3	2 x 3	1
2	5	6	1
1	1 + 2	1 x 2	1
1	3	2	1

Contributions

Chad Davis is the first contributor to this topic. In fact he is the inspirator by sending me the first eight consecutives.
Much appreciated, thanks !

Warut Roonguthai got inspired by these palindromes and wrote and ran ' a very simple UBASIC program to search exhaustively for palindromic pronic numbers of length 18, 19 and 20.'
He discovered two new ones with index numbers 24 and 25 respectively.
Two days later he improved his program so it ran 100 times faster.
It produced the pronic numbers with index starting from 26 !

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