More about Palindromic Products of Sequential Integers

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Palindromic Quasipronic Numbers of the form n(n+2)
n(n+1) 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 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

Palindromic Quasipronic Numbers

So far I compiled 59 Palindromic Quasipronic Numbers.

Here is the largest non trivial one that I discovered on [ May 6, 2000 ].

This nonpalindromic basenumber
6.035.200.088.907
has 13 digits
yielding the following palindromic quasipronic number
36.423.640.113.155.131.104.632.463
with a length of 26 digits.

Palindromic Quasipronic Numbers can only end in one of the following digits : 3, 4, 5, 8 or 9.

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.
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

My program completed the search upto length 26 inclusive.

Index Nr	Info	Length
Index Nr	Palindromic Product of Sequence n(n+2)	Length

	Twofold sequence (n) x (n+2) Palindromic Quasipronic Numbers
59	9.999.999.999.999	13
59	99.999.999.999.999.999.999.999.999	26
58	6.035.200.088.907	13
58	36.423.640.113.155.131.104.632.463	26
57	3.153.099.043.049	13
57	9.942.033.575.282.825.753.302.499	25
56	2.967.841.885.466	13
56	8.808.085.457.132.317.545.808.088	25
55	2.967.032.967.032	13
55	8.803.284.627.460.647.264.823.088	25
54	2.939.136.124.286	13
54	8.638.521.157.088.807.511.258.368	25
53	2.839.531.502.896	13
53	8.062.939.155.944.495.519.392.608	25
52	2.050.695.871.494	13
52	4.205.353.557.366.637.553.535.024	25
51	999.999.999.999	12
51	999.999.999.999.999.999.999.999	24
50	186.125.268.237	12
50	34.642.615.476.667.451.624.643	23
49	178.923.207.531	12
49	32.013.514.193.539.141.531.023	23
48	99.999.999.999	11
48	9.999.999.999.999.999.999.999	22
47	64.819.595.894	11
47	4.201.580.011.991.100.851.024	22
46	54.809.426.001	11
46	3.004.073.178.668.713.704.003	22
45	31.519.410.879	11
45	993.473.262.222.262.374.399	21
44	24.064.405.175	11
44	579.095.596.474.695.590.975	21
43	18.511.102.337	11
43	342.660.909.767.909.066.243	21
42	18.163.818.617	11
42	329.924.306.787.603.429.923	21
41	9.999.999.999	10
41	99.999.999.999.999.999.999	20
40	3.146.471.489	10
40	9.900.282.837.382.820.099	19
39	2.967.032.966	10
39	8.803.284.627.264.823.088	19
38	2.441.052.185	10
38	5.958.735.774.775.378.595	19
37	999.999.999	9
37	999.999.999.999.999.999	18
36	185.812.387	9
36	34.526.243.534.262.543	17
35	99.999.999	8
35	9.999.999.999.999.999	16
34	93.809.716	8
34	8.800.263.003.620.088	16
33	31.552.659	8
33	995.570.353.075.599	15
32	22.765.895	8
32	518.286.020.682.815	15
31	9.999.999	7
31	99.999.999.999.999	14
30	9.200.156	7
30	84.642.888.824.648	14
29	2.967.032	7
29	8.803.284.823.088	13
28	1.868.287	7
28	3.490.500.050.943	13
27	999.999 x 1.000.001	6-7
27	999.999.999.999	12
26	552.101 x 552.103	6
26	304.816.618.403	12
25	293.786 x 293.788	6
25	86.310.801.368	11
24	243.063 x 243.065	6
24	59.080.108.095	11
23	179.317 x 179.319	6
23	32.154.945.123	11
22	174.747 x 174.749	6
22	30.536.863.503	11
21	174.601 x 174.603	6
21	30.485.858.403	11
20	99.999 x 100.001	5-6
20	9.999.999.999	10
19	20.564 x 20.566	5
19	422.919.224	9
18	18.991 x 18.993	5
18	360.696.063	9
17	9.999 x 10.001	4-5
17	99.999.999	8
16	5.731 x 5.733	4
16	32.855.823	8
15	2.966 x 2.968	4
15	8.803.088	7
14	2.365 x 2.367	4
14	5.597.955	7
13	1.907 x 1.909	4
13	3.640.463	7
12	1.747 x 1.749	4
12	3.055.503	7
11	999 x 1.001	3-4
11	999.999	6
10	204 x 206	3
10	42.024	5
9	191 x 193	3
9	36.863	5
8	99 x 101	2-3
8	9.999	4
7	75 x 77	2
7	5.775	4
6	64 x 66	2
6	4.224	4
5	23 x 25	2
5	575	3
4	17 x 19	2
4	323	3
3	9 x 11	1-2
3	99	2
2	2 x 4	1
2	8	1
1	1 x 3	1
1	3	1

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