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Various Palindromic Number Records | |||

Introduction Various Websources Palindromic Triangles Palindromic Squares |

Palindromic Polygonal Sporadic Number Record Table

Patrick De Geest & R. Xiao's CUDApalin September 18, 2023 | TRIANGLE | 3-gonal | base = 11604355897766717137264209814751 | 32 |
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palindrome | 67330537901016595837936819195144159191863973859561010973503376 | 62 | ||

Patrick De Geest & R. Xiao's CUDApalin July 5, 2024 | SQUARE | 4-gonal | base = 31346828297209660045268842120992233 | 35 |

palindrome | 982623644294744275088611239676071787170676932116880572447492446326289 | 69 | ||

Patrick De Geest & R. Xiao's CUDApalin September 24, 2024 | PENTA | 5-gonal | base = 8380905875275233386185451026 | 28 |

palindrome | 105359374935334388747447001151100744747883433539473953501 | 57 | ||

Patrick De Geest & R. Xiao's CUDApalin September 18, 2023 | HEXA | 6-gonal | base = 5802177948883358568632104907376 | 31 |

palindrome | 67330537901016595837936819195144159191863973859561010973503376 | 62 | ||

Patrick De Geest & R. Xiao's CUDApalin September 25, 2024 | HEPTA | 7-gonal | base = 14158030498610414267313727430 | 29 |

palindrome | 501124568998956639076141901363109141670936659899865421105 | 57 | ||

Patrick De Geest & R. Xiao's CUDApalin September 26, 2024 | OCTA | 8-gonal | base = 7310534542684826648283312143 | 28 |

palindrome | 160331745899364142498750393141393057894241463998547133061 | 57 | ||

Patrick De Geest & R. Xiao's CUDApalin September 27, 2024 | NONA | 9-gonal | base = 5322627062113007506846668678 | 28 |

palindrome | 99156255948182109153103300355300330135190128184955265199 | 56 |

Various Palindromic Sporadic Numbers Record Table

Patrick De Geest & R. Xiao's CUDApalin June 24, 2023 | PRONIC | n(n+1) | base = 542552107088554950844257896843 | 30 |
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palindrome | 294362788906230799339509080886688080905933997032609887263492 | 60 | ||

Patrick De Geest & R. Xiao's CUDApalin October 5, 2024 | Q-PRONIC | n(n+2) | base = 29769956671342479605203208616 | 29 |

palindrome | 886250320213608608256318705636507813652806806312023052688 | 57 | ||

Patrick De Geest & R. Xiao's CUDApalin October 5, 2024 | SQUADD1 | n^2+1 | base = 15642408617501978745151573229 | 29 |

palindrome | 244684947356900165986671614757416176689561009653749486442 | 57 | ||

Patrick De Geest & R. Xiao's CUDApalin October 8, 2024 | SQUDEC2 | n^2–2 | base = 20915126182018287738025776856 | 29 |

palindrome | 437442503209746877820785299464992587028778647902305244734 | 57 | ||

Patrick De Geest & R. Xiao's CUDApalin October 6, 2024 | QO-SQR | n^2+(n+1) | base = 18620621484912128040098254793 | 29 |

palindrome | 346727544484371144216056730919037650612441173484445727643 | 57 | ||

Patrick De Geest & R. Xiao's CUDApalin Jul 26, 2023 | QU-SQR | n+(n+1)^2 | base = 3020454975635151325450374803 | 28 |

palindrome | 9123148259839142587072274593954722707852419389528413219 | 55 |

Various Non Quadratic Palindromic Numbers Record Table

Patrick De Geest -- -- ---- | TETRA | (n)(n+1)(n+2)/6 | base = 336 | 3 |
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palindrome | 6378736 | 7 | ||

Charles Trigg -- -- 1961 | CUBE | n^3 | base = 2201 | 4 |

palindrome | 10662526601 | 11 |

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 case | Change of variables | CUDApalin parameters | Pattern EVEN case | Change of Variables | CUDApalin parameters | |

triangle | n = 2 * m + 1 | `A B C → 2 3 1 ` | triangle | n = 2 * m + 2 | `A B C → 2 5 3 ` | |

penta | n = 2 * m + 1 | `A B C → 6 5 1 ` | penta | n = 2 * m + 2 | `A B C → 6 11 5 ` | |

hepta | n = 2 * m + 1 | `A B C → 10 7 1 ` | hepta | n = 2 * m + 2 | `A B C → 10 17 7 ` | |

nona | n = 2 * m + 1 | `A B C → 14 9 1 ` | nona | n = 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 case | Change of variables | CUDApalin parameters |

square | n = m + 1 | `A B C → 1 2 1 ` |

hexa | n = m + 1 | `A B C → 2 3 1 ` |

octa | n = m + 1 | `A B C → 3 4 1 ` |

deca | n = 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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Patrick De Geest - Belgium - Short Bio - Some Pictures

E-mail address : pdg@worldofnumbers.com