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Palindromic Polygonal Sporadic Number Record Table

General formula for n-gonal is x/2[(n–2)x–(n–4)]

Patrick De Geest
& R. Xiao's CUDApalin
September 18, 2023
TRIANGLE3-gonalbase = 1160435589776671713726420981475132
palindrome6733053790101659583793681919514415919186397385956101097350337662
Patrick De Geest
& R. Xiao's CUDApalin
October 18, 2023
SQUARE4-gonalbase = 310988584438015288876391088595036334
palindrome9671389965076056510789702463424611164243642079870156506705699831769 67
Patrick De Geest
& R. Xiao's CUDApalin
June 16, 2023
PENTA5-gonalbase = 82941261649088231606387478627
palindrome103188793259137714210547979997974501241773195239788130155
Patrick De Geest
& R. Xiao's CUDApalin
September 18, 2023
HEXA6-gonalbase = 580217794888335856863210490737631
palindrome6733053790101659583793681919514415919186397385956101097350337662
Patrick De Geest
& R. Xiao's CUDApalin
June 16, 2023
HEPTA7-gonalbase = 187372056476546961149644176328
palindrome877707188706257600235472803330827453200675260788170777855
Patrick De Geest
& R. Xiao's CUDApalin
June 16, 2023
OCTA8-gonalbase = 150518903431627398987636455828
palindrome679678208707787231610807339793370801613278770780287697655
Patrick De Geest
& R. Xiao's CUDApalin
June 16, 2023
NONA9-gonalbase = 167918150634033580179178099328
palindrome986877685932389721579448174247184497512798323958677868955


Various Palindromic Sporadic Numbers Record Table


Patrick De Geest
& R. Xiao's CUDApalin
June 24, 2023
PRONICn(n+1)base = 54255210708855495084425789684330
palindrome29436278890623079933950908088668808090593399703260988726349260
Patrick De Geest
& R. Xiao's CUDApalin
June 16, 2023
Q-PRONICn(n+2)base = 286958360692775257285282467628
palindrome823451007714849038382629752425792628383094841770015432855
Patrick De Geest
& R. Xiao's CUDApalin
Jul 26, 2023
QU-SQRn+(n+1)^2base = 302045497563515132545037480328
palindrome912314825983914258707227459395472270785241938952841321955
Patrick De Geest
& R. Xiao's CUDApalin
July 26, 2023
QO-SQRn^2+(n+1)base = 274044690604774196849092961228
palindrome751004924486664149567691438683419676594146668442940015755
Patrick De Geest
& R. Xiao's CUDApalin
June 17, 2023
SQUADD1n^2+1base = 281224929668715031958122056428
palindrome790874610671737162144910065356001944126173717601647809755
Patrick De Geest
& R. Xiao's CUDApalin
July 9, 2023
SQUDEC2n^2–2base = 299252852952377703754971016028
palindrome895522700001373929667956620402665976692937310000722559855


Various Non Quadratic Palindromic Numbers Record Table


Patrick De Geest
-- -- ----
TETRA(n)(n+1)(n+2)/6base = 3363
palindrome63787367
Charles Trigg
-- -- 1961
CUBEn^3base = 22014
palindrome1066252660111


Contributions


These world records were 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.


Patterns arising in the CUDApalin parameters for the odd polygonals

Pattern ODD caseChange of variablesCUDApalin parameters Pattern EVEN caseChange of VariablesCUDApalin parameters
trianglen = 2 * m + 1
A B C   2 3 1 
 trianglen = 2 * m + 2
A B C   2 5 3 
pentan = 2 * m + 1
A B C   6 5 1 
 pentan = 2 * m + 2
A B C   6 11 5 
heptan = 2 * m + 1
A B C   10 7 1 
 heptan = 2 * m + 2
A B C   10 17 7 
nonan = 2 * m + 1
A B C   14 9 1 
 nonan = 2 * m + 2
A B C   14 23 9 

So for the hendeca (etc.) case one has just to extend the previous obvious patterns showing up in the parameters.

The sequence for A_odd is 2, 6, 10, 14, ... [+ 4]
The sequence for B_odd is 3, 5, 7, 9, ... [+ 2]
The sequence for C_odd is 1, 1, 1, 1, ... [+ 0]

The sequence for A_even is 2, 6, 10, 14, ... [+ 4]
The sequence for B_even is 5, 11, 17, 23, ... [+ 6]
The sequence for C_even is 3, 5, 7, 9, ... [+ 2]

Of course there is also a pattern for the even polygonals

Pattern ODD/EVEN caseChange of variablesCUDApalin parameters
squaren = m + 1
A B C   1 2 1 
hexan = m + 1
A B C   2 3 1 
octan = m + 1
A B C   3 4 1 
decan = m + 1
A B C   4 5 1 

So for the dodeca (etc.) case one has just to extend the previous obvious patterns showing up in the parameters.

The sequence for A_odd/even is 1, 2, 3, 4, ... [+ 1]
The sequence for B_odd/even is 2, 3, 4, 5 ... [+ 1]
The sequence for C_odd/even is 1, 1, 1, 1, ... [+ 0]










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( © All rights reserved ) - Last modified : October 18, 2023.
Patrick De Geest - Belgium flag - Short Bio - Some Pictures
E-mail address : pdg@worldofnumbers.com