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Palindromic Pronic Numbers of the form n(n+1)
n(n+0) n(n+2) n(n+x) n^2+1 n^2+x n^2–x n^2+(n+1) n^2+(n+x)

Palindromic numbers are numbers which read the same from

  left to right (forwards) as from the right to left (backwards)

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).

 So far this compilation counts 110 palindromic pronic numbers.

Here is the largest Sporadic Pronic Palindromic that Patrick De Geest,

discovered, using CUDA code by Robert Xiao, on [ June 24, 2023 ]

There are no palindromic pronic numbers of lengths 2, 5, 9, 12, 18, 20, 30, 34, 41, 42, 48, 49.

A B C →  1 1 0 

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] 2554554552
[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!|

Admire for a moment this extraordinary manyfold construction with Two Consecutive Integers 16 and 17.

16 x 17 = 272   [ or 16 + 16² ]

16¹ + 17¹ = 33
16² + 17² = 545
16³ + 17³ = 9009

17¹ – 16¹ = 1
17² – 16² = 33

and this bellying construction (finite, alas) where 16 and 17 embrace the numbers 70 is quite phenomenal !

16² + 17² = 545
1706² + 1707² = 5824285
170706² + 170707² = 58281418285
17070706² + 17070707² = 582818040818285

and that is not all because when we add more consecutives to the expression we see the following :

16³ + 17³ + 18³ = 14841 (#3)
16³ + 17³ + ... + 25³ + 26³ = 108801 (#11)
16³ + 17³ + ... + 407³ + 408³ = 6961551696 (#393)

or summing the #311 squares from 16 up to 326

16² + 17² + ... + 325² + 326² = 11600611

Changing the cube to squares in the above expression allows us to stuff the three consecutive numbers with 73

1736² + 1737² + 1738² = 9051509
173736² + 173737² + 173738² = 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 ?

16² + 16 + 5 = 277 is also a prime with multiplicative persistence of value 4 - see A034051.
17² + 17 + 5 = 311
18² + 18+ 5 = 347

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

1876² + 1877¹ = 3521253
187876² + 187877¹ = 35297579253

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 (email) 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

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 :

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
*****************

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 ]

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

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

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

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 !

From wonplate 197 there is this nice FEP (First Encountered Prime) duo :
We have not only 16 + 1 = 17 but also 1945 + 1 = 1946 !

[1945][1_16] = prime!
[1946][1_17] = prime!

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

1621 x 1622 = 2629262
1621² + 1622² = 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.

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 ).

[ November 23, 2022 ]
David Griffeath (email) completed the 59-digit collection by adding the entries [103] up to [106].

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Patrick De Geest - Belgium - Short Bio - Some Pictures
E-mail address : pdg@worldofnumbers.com