More about Palindromic Incremented Squares

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Palindromic Incremented Squares of Form n^2+1
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 Incremented Squares are defined and calculated by this extraordinary intricate and excruciatingly complex formula.
So, this line is for experts only

base² + 1

Palindromic Incremented Squares

So far I compiled 64 Palindromic Incremented Squares.

Here is the largest sporadic one that Feng Yuan from Washington State, USA,

discovered on [ April 4, 2002 ].

This basenumber
23.702.356.087.344.642
has 17 digits
yielding the following palindromic incremented square
561.801.684.091.283.606.382.190.486.108.165
with a length of 33 digits.

Palindromic Incremented Squares can only end in one of the following digits : 1, 2, 5, 6 or 7.

Here is Warut Roonguthai's synopsis of his basenumbers investigation :
No less than six patterns are detected !
Four infinite patterns :

10ⁿ » n = 0, 1 , 2, ... » 1, 10, 100, 1000, ...

10³ⁿ+2*10ⁿ » n = 1, 2, 3, ... » 1020, 1000200, 1000002000, ...

10³ⁿ⁺¹+9*10ⁿ » n = 1, 2, 3, ... » 10090, 10000900, 10000009000, ...

10⁵ⁿ+2*10³ⁿ+2*10ⁿ » n = 1, 2, 3, ... » 102020, 10002000200, 1000002000002000, ...
Two finite patterns :

10 (09)_n 10, for n = 1 to 4

100 (90)_n, for n = 0 to 5

Warut Roonguthai 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. "

It is no coincidence that there aren't any Palindromic Incremented Squares of even length.
Statement :

Every number of the form n^2+1 is NOT divisible by 11.
Proof there is no even_length palindromic number of the form n^2+1

( proof that -1 is a non-quadratic residue modulo 11 ) :

      n mod 11 :   0   1   2   3   4   5   6   7   8   9   10
(n^2+1) mod 11 :   1   2   5  10   6   4   4   6  10   5    2
[ if zero appeared in the second line then it would be divisible by 11 ]

Because Palindromic Numbers of EVEN length are always divisible by 11 ( for a general proof of this refer to the palindromic primes page ),
we immediately see that we can safely skip searching for them.

Sources Revealed

Neil Sloane's "Integer Sequences" Encyclopedia can be consulted online :
Neil Sloane's Integer Sequences
The regular incremented squares are categorised as follows :
%N n^2 + 1 under A002522.
Soon the following two entries about Palindromic Incremented Squares will be present :
%N n^2 + 1 is a palindrome under A027719.
%N Palindromes of form n^2 + 1 under A027720

Thanks N. Sloane for providing the palindromic entry number 027720
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.

The Table

The search went exhaustively upto (only odd!) length 33.

Index Nr	Basenumber	Length
Index Nr	Palindromic Incremented squares of Form n^2 + 1	Length

	Palindromic Incremented Squares
64	23.702.356.087.344.642	17
64	561.801.684.091.283.606.382.190.486.108.165	33
63	10.313.381.620.663.060	17
63	106.365.840.453.430.606.034.354.048.563.601	33
62	10.000.009.000.009.000	17
62	100.000.180.000.261.000.162.000.081.000.001	33
61	10.000.000.000.900.000	17
61	100.000.000.018.000.000.000.810.000.000.001	33
60	10.000.000.000.000.000	17
60	100.000.000.000.000.000.000.000.000.000.001	33
59	2.416.653.284.019.978	16
59	5.840.213.095.164.544.454.615.903.120.485	31
58	1.687.257.100.543.559	16
58	2.846.836.523.334.657.564.333.256.386.482	31
57	1.000.090.009.000.900	16
57	1.000.180.026.103.420.243.016.200.810.001	31
56	1.000.002.000.002.000	16
56	1.000.004.000.008.000.008.000.004.000.001	31
55	1.000.000.000.200.000	16
55	1.000.000.000.400.000.000.040.000.000.001	31
54	1.000.000.000.000.000	16
54	1.000.000.000.000.000.000.000.000.000.001	31
53	142.360.550.071.851	15
53	20.266.526.216.759.995.761.262.566.202	29
52	100.000.000.000.000	15
52	10.000.000.000.000.000.000.000.000.001	29
51	16.168.393.768.631	14
51	261.416.957.057.505.750.759.614.162	27
50	10.000.000.090.000	14
50	100.000.018.000.000.008.100.000.001	27
49	10.000.000.000.000	14
49	100.000.000.000.000.000.000.000.001	27
48	1.009.090.909.090	13
48	1.018.264.462.808.082.644.628.101	25
47	1.000.000.020.000	13
47	1.000.000.040.000.000.400.000.001	25
46	1.000.000.000.000	13
46	1.000.000.000.000.000.000.000.001	25
45	234.253.293.562	12
45	54.874.605.544.644.550.647.845	23
44	100.909.090.910	12
44	10.182.644.628.282.644.628.101	23
43	100.009.000.900	12
43	10.001.800.261.016.200.810.001	23
42	100.000.000.000	12
42	10.000.000.000.000.000.000.001	23
41	16.353.780.069	11
41	267.446.122.545.221.644.762	21
40	15.577.088.671	11
40	242.645.691.464.196.546.242	21
39	10.462.738.430	11
39	109.468.895.454.598.864.901	21
38	10.212.242.320	11
38	104.289.893.202.398.982.401	21
37	10.090.909.090	11
37	101.826.446.262.644.628.101	21
36	10.062.715.390	11
36	101.258.241.020.142.852.101	21
35	10.002.000.200	11
35	100.040.008.000.800.040.001	21
34	10.000.009.000	11
34	100.000.180.000.081.000.001	21
33	10.000.000.000	11
33	100.000.000.000.000.000.001	21
32	1.577.033.471	10
32	2.487.034.568.654.307.842	19
31	1.009.090.910	10
31	Prime Curios! 1.018.264.464.644.628.101	19
30	1.000.002.000	10
30	1.000.004.000.004.000.001	19
29	1.000.000.000	10
29	1.000.000.000.000.000.001	19
28	271.867.456	9
28	73.911.913.631.911.937	17
27	103.226.660	9
27	10.655.743.334.755.601	17
26	100.909.090	9
26	10.182.644.444.628.101	17
25	100.000.000	9
25	10.000.000.000.000.001	17
24	24.917.195	8
24	620.866.606.668.026	15
23	10.090.910	8
23	101.826.464.628.101	15
22	10.000.900	8
22	100.018.000.810.001	15
21	10.000.000	8
21	100.000.000.000.001	15
20	2.744.934	7
20	7.534.662.664.357	13
19	1.009.090	7
19	1.018.262.628.101	13
18	1.000.200	7
18	1.000.400.040.001	13
17	1.000.000	7
17	1.000.000.000.001	13
16	167.491	6
16	28.053.235.082	11
15	102.020	6
15	10.408.080.401	11
14	100.910	6
14	10.182.828.101	11
13	100.000	6
13	10.000.000.001	11
12	10.090	5
12	101.808.101	9
11	10.000	5
11	100.000.001	9
10	2.248	4
10	5.053.505	7
9	1.489	4
9	2.217.122	7
8	1.020	4
8	1.040.401	7
7	1.000	4
7	1.000.001	7
6	100	3
6	10.001	5
5	25	2
5	626	3
4	10	2
4	Prime! 101	3
3	2	1
3	Prime! 5	1
2	1	1
2	Prime! 2	1
1	0	1
1	1	1

Contributions

Warut Roonguthai (email) from Bangkok (Thailand) helped searching for these Palindromic Incremented Squares.
He discovered those starting from index number 21 upto 41 and 48.
Number 48 was the largest sporadic one which he submitted to me on [ November 3, 1997 ].

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