Square numbers are defined and calculated by this extraordinary intricate and excruciatingly complex formula.
So, this line is for experts only
( base ) x ( base ) or base 2
|square 1||square 4||square 9|
|PLAIN TEXT SQUARES||PLAIN TEXT SSP|
This basenumber |
has 33 digits
yielding the following palindromic square record number with even number of digits
having a staggering length of 66 digits.
This world record was achieved using CUDA code written by Robert Xiao and no longer|
on Rust. Recently he generalized the program to handle arbitrary quadratics.
CUDA is a programming language, or more properly a programming toolkit,
for writing software to run on GPUs rather than CPUs. It runs about 50 times faster on our
GPUs though the logic of the code follows the Rust version closely.
I asked Robert now that his CUDA is running at warp speed how far it would reach.
He answered that as for 70 digits the time estimate on that is around ~400 days on one of
our GPUs. 60 digits is about two days of GPU time, and it’ll go up by a factor of 10 every 4 digits.
Doable but it’ll be a pretty decent power bill :) “Maybe we could get some palindrome enthusiasts
together”, as David Griffeath put it, “and get a distributed computation going.”
The program is very amenable to divide-and-conquer approaches.
Mike Keith made an original study of the palindromic squares.
He was able to classify and enumerate them in a logical manner
and went even further as he tried also to create a general formula for
calculating the number of certain classes of palindromic squares,
except for the 'sporadic' solutions. Wonderful!
[Please refer to the “JRM” source detailed in section 'Sources Revealed'].
Mike had most trouble with the Asymmetric Root Family but thanks to
Dave Wilson's “lemma” he made significant progress.
Unlike Palindromic Triangulars where it is impossible to
predict a next higher one, whether its basenumber is palindromic or not,
with the Palindromic Squares (and Cubes) we have an opposite situation. Finding a next higher number is very easy.
Start e.g. with the number 11. Then repeatedly add a zero between the two 'ones' and square them.
A pattern emerges that can go on forever.
There are other numbers with the same properties (e.g. 10101). The repunit numbers like '111' looks also a good candidate
for finding palindromic squares (not for cubes though) but the expansion doesn't go on forever. Nine 'ones' still produce
a palindrome when squared, but not ten 'ones'. It's a near miss as only the number 8 doesn't show up in the left part of the square.
This anomaly stays with longer repunits as well, so we must abandon this candidate.
If we take resort to higher basenumbers only then we can extend the pattern (for a while)!
|1111111111||1234567900987654321 = NOT palindromic in base 10 !|
Here is how Natalie Dickendasher [ November 4, 2005 ] (email) figures them out quickly in her head and impresses friends or boss :
The highest digit of the sequence will always be the quantity of 1's in the number your are squaring.
As an example lets start with my salary for next year (yah right): what is 111111 squared ?
EASY !! This number has a total of 6 digits. Therefore the highest number in the palindrome will be 6
so count to 6 and then back down. The answer is 12345654321 .
Finding large palindromic squares is easy if you start with some palindromic basenumbers as explained above.
So, let's concentrate on the difficult ones namely those with a non palindromic basenumber. The first six are
(26 ) (264 ) (307 ) (836 ) (2285 ) and (2636 ) and can be found without too much trouble with a spreadsheet or calculator.
|# odd_length SSP's||--||1||2||2||3||0||5||4||1||2||6||4||4||1||3||5||5||4|
|# even_length SSP's||0||0||1||0||0||1||0||1||0||0||1||0||1||2||0||1||1|
|# odd_length SSP's||2||1||4||4||4||1||2||1||5||4||2||3||6||1||3||-||-|
|# even_length SSP's||3||0||0||1||1||0||2||1||1||0||0||1||1||0||0||2||-||-|
|A263616 | A263617||# BN's giving SP's||4||3||8||5||11||6||19||14||25||18||49||31||71||46||105||71||154||101||209||132||292||182||384||236||497||302||636||383||799||475||981||578||1201||-|
|# BN's giving odd_length SP's||--||3||7||5||11||5||19||13||25||18||48||31||70||44||105||70||153||98||209||132||291||181||384||234||496||301||636||383||798||474||981||578||1199||-|
|# BN's giving even_length (S)SP's||--||0||1||0||0||1||0||1||0||0||1||0||1||2||0||1||1||3||0||0||1||1||0||2||1||1||0||0||1||1||0||0||2||-|
|# BN's giving sporadics (↑)+(↓)||--||1||3||2||3||1||5||5||1||2||7||4||5||3||3||6||6||7||2||1||5||5||4||3||3||2||5||4||3||4||6||1||5||-|
|# BN's giving odd_length SSP's||--||1||2||2||3||0||5||4||1||2||6||4||4||1||3||5||5||4||2||1||4||4||4||1||2||1||5||4||2||3||6||1||3||-|
|# BN's giving NSSP's||--||2||5||3||8||5||14||9||24||16||42||27||66||43||102||65||148||94||207||131||287||177||380||233||494||300||631||379||796||471||975||577||1196||698|
|A263614||# BN's binaries → |B|||--||1||2||2||4||4||8||8||16||15||30||26||52||42||84||64||128||93||186||130||260||176||352||232||454||299||598||378||756||470||940||576||1152||697|
|A142150||# BN's ternaries → |T|||--||0||1||0||2||0||3||0||4||0||5||0||6||0||7||0||8||0||9||0||10||0||11||0||12||0||13||0||14||0||15||0||16||0|
|A007573||# BN's asymmetrics → |A|||--||0||0||0||0||0||1||0||2||0||5||0||6||0||9||0||10||0||10||0||15||0||15||0||16||0||18||0||24||0||18||0||26||0|
|A000034||# BN's even_root → |E|||--||1||2||1||2||1||2||1||2||1||2||1||2||1||2||1||2||1||2||1||2||1||2||1||2||1||2||1||2||1||2||1||2||1|
Can we find formulas for all or some of these sequences ?
“I've been feeling remiss about not showing you how that messy formula for the number of binaries doesn't|
involve fancy math, but rather just a bunch of unpleasant, unenlightening algebra. So I've attached a
handwritten sketch of the computations.
The first step, in (1) and (2) is to change variables and express the formulas for the even and odd cases
Thanks David for this proper answer, much appreciated!"
Click here to view some entries to the table about palindromes.
|Squares||Acquaint yourself with SQUARES - by Richard Phillips|
|Square Number||From Eric Weisstein's Math Encyclopedia|
|Ask Dr. Math||Someone gave me different numbers to find the square root of, including 14641.|
My answer was 121. Something special about this is that both numbers are read
the same left to right as right to left. Do you know any other numbers in which
both the number and its square root are this way ?
A very interesting webpage by William Rex Marshall (1930-2010) is titled “Palindromic Squares”
and gives an overview of the first 67 palindromic squares known up to May 28, 2001.
The classification of these squares as set out by Michael Keith in the JRM from 1990 is also revealed.
The second highest one (3069306930693 ) is copied from Martin Gardner's book
“The Ambidextrous Universe”
see page 40.
Another source “Curious and Interesting Numbers” by David Wells, page 185, provided me the basenumber 798644 .
While surfing the Internet I came across the work of Keith Devlin namely
All the Math that's Fit to Print.
In chapters 17, 75 and 110 he printed some palindromic squares he got from other readers.
The highest one that originates from his publication is 6360832925898 .
Gustavus J. Simmons published an article about palindromic squares in the “Journal of Recreational Mathematics”,
J. Rec. Math., 3 (No. 2, 1970), 93-98, titled “Palindromic Powers”.
OEIS Source (annotated scanned copy) → https://oeis.org/A002778/a002778_2.pdf
Gustavus J. Simmons published an article about palindromic squares in the “Journal of Recreational Mathematics”, Mike Keith (email) (website) published an article about palindromic squares in the “Journal of Recreational Mathematics”,
Volume 5, No. 1, 1972, pp. 11-19, titled “On Palindromic Squares of Non-Palindromic Numbers”.
OEIS Source (annotated scanned copy) → https://oeis.org/A002778/a002778.pdf
Volume 22, Number 2 - 1990, pp. 124-132, titled “Classification and enumeration of palindromic squares”,
and was followed by an article by Charles Ashbacher, pp. 133-135, “More on palindromic squares”.
OEIS Source (annotated scanned copy) → https://oeis.org/A002778/a002778_1.pdf
Mike Keith (email) (website) published an article about palindromic squares in the “Journal of Recreational Mathematics”,
I found Feng Yuan's record palindromic square of 55 digits  in the following archived blog
Careful observation of some numbers in the list lead me to the discovery of ever greater non-palindromic basenumbers whose squares are palindromic.
Specially the numbers with indexes , ,  and  attracted my attention because of the CORE number 091 in them.
A second promising CORE number exists. It is 09901 and it shows up from , , , ....
I think a new third CORE number is emerging... It is 0999001
Watch out from
Quite interesting. Let's put these three CORE numbers in a row :
David W. Wilson took up the thread where I left off.
Read what he posted to me about these core 'pseudopalindromes'
[ February 16, 1998 ].
If we allow a digit "n" with value "1", then 091 is equal to the palindrome 1n1,
while 09901 = 10n01, 0999001 = 100n001, etc. Thus
10110n01101 x 10110n01101 ----------- 10110n01101 10110n01101.. 10110n01101... n0nn010nn0n..... 10110n01101....... 10110n01101........ 10110n01101.......... --------------------- 102210100272001012201
And what about this repeat pattern of the item 3069
and that is finalized alternatively with a chopped off and rounded version of it namely a 3 or 307...
 3Each time the length increases with 2
 3069 3
 3069 307
 3069 3069 3
 3069 3069 307
 3069 3069 3069 3
3069 occurs as displacements to powers of ten so that this number forms the nearest (probable) prime to those axes :
101001 3069 is the largest prime of 1001 digits
101555 + 3069|
102399 + 3069
105016 + 3069
105063 + 3069
109896 + 3069
NEVER ODD OR EVEN Read title backwards please !
It looks as though the decent sufficiently-random nonpatternlike unpredictable nonpalindromic basenumbers become very rare.
In the table (see Full Listing or the Subsets) I highlighted them using a lightblue background in the side-cells.
And even then a further distinction can be made. Consider the even and odd length values for a minute.
Sporadic Palindromic Squares with an even length are just a plain minority. Until now I can only list twenty of them.
These are highlighted in the table by means of a aquamarine background color in the side-cells.
Score 'subset E' = 20
I 'almost' solved a very ancient problem namely : squaring the circle (pi = 3,1415....)
MORELAND PI welcomes PALINDROMES
This basenumber starts with the the number 2201 which is (to me) the only known nonpalindromic basenumber
of a palindromic cube ! The cube of 2201 equalling 10662526601.
|10101010101010101||102030405060708090807060504030201 (° see conjecture below)|
|1010101010101010101||10203040506070809100908070605004030201 = QUITE UNpalindromic !|
|10101010101010101 is indeed the smallest integer that generates|
a palindromic square 102030405060708090807060504030201
that is also pandigital (contains all the ten digits).
Here is a good place to talk about a famous conjecture that nobody
could prove or disprove for more than 80 years.
I must thank Mike Keith for letting me know this fact through an email message.
Mike has heard that this conjecture originates from
L. E. Dickson's famous “History of the Theory of Numbers”.
Can someone with access to this source confirm this ?
My table lists exhaustively all palindromic squares up to lengths 31.
I also listed all the palindromic squares of length 32.
Palindromic squares of 'even' nature seem to be very popular among the record
breaking submitters. They told me there are no gaps for the 'even' length 32 !
So, in fact I had to finish checking only length 33...
Of which I checked all squares beginning with 1, 5 and 9 !
( and 50 % of those beginning with digits 4 and 6 ).
I collected a total of 151 palindromic squares beginning with digit 1.
(Un)lucky for me Dickson's number 102030405060708090807060504030201
( 101010101010101012 )  is the 86th one. Anyway after checking these 86 squares
I could proudly announce that the conjecture was finally proven [ May 3, 1998 ].
The second smallest number  that answers to these rules is this one (also of length 33) :
118431915157648212 = 140261185279083838380972581162041.
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