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 ! 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).
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 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 [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
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 + 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
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)
16^{2} + 17^{2} + ... + 325^{2} + 326^{2} = 11600611
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
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
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
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.
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.
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 ]
1617 occurs many times as displacement to powers of ten so that this number forms a (probable) prime nearest to those axes.
10^{293} + 1617 is the smallest prime of length 294. 10^{388} + 1617 is the smallest prime of length 389. 10^{1276} + 1617 is the smallest prime of length 1277. 10^{1651} + 1617 is the smallest prime of length 1652. 10^{2416} + 1617 is the smallest prime of length 2417. 10^{6689} + 1617 is the smallest prime of length 6690. 10^{9109} + 1617 is the smallest prime of length 9110. 10^{18448} + 1617 is the smallest prime of length 18449. 10^{25344} + 1617 is the smallest prime of length 25345. 10^{28725} + 1617 is the smallest prime of length 28725. 10^{41905} + 1617 is the smallest prime of length 41906.
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! [1946][1_17] = prime!
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.
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 39 inclusive.
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 !
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