With the Palindromic Squares (and Cubes) we have an opposite situation.įinding a next higher number is very easy. Predict a next higher one, whether its basenumber is palindromic or not, Unlike Palindromic Triangulars where it is impossible to What are the conditions for creating a pseudopalindrome with a palindromic square ? 1 can only be followed by an even number : 10, 12, 14, 16 or 18 4 can only be followed by an even number : 40, 42, 44, 46 or 48 5 can only be followed by 2 : 52 6 can only be followed by an odd number : 61, 63, 65, 67 or 69 9 can only be followed by an even number : 90, 92, 94, 96 or 98 The program is very amenable to divide-and-conquer approaches.Īll palindromic square numbers can only start or end with digits 0, 1, 4, 5, 6 or 9.Īlas, my palindromes may not have leading 0's! So the zero option must not be investigated. Together”, as David Griffeath put it, “and get a distributed computation going.” 60 digits is about two days of GPU time, and it’ll go up by a factor of 10 every 4 digits.ĭoable but it’ll be a pretty decent power bill :) “Maybe we could get some palindrome enthusiasts He answered that as for 70 digits the time estimate on that is around ~400 days on one of I asked Robert now that his CUDA is running at warp speed how far it would reach. GPUs though the logic of the code follows the Rust version closely. Recently he generalized the program to handle arbitrary quadratics.ĬUDA is a programming language, or more properly a programming toolkit,įor writing software to run on GPUs rather than CPUs. This world record was achieved using CUDA code written by Robert Xiao and no longer Here is the largest Sporadic Square Palindrome that Patrick De Geest discovered, using CUDA code by Robert Xiao, on. There’s a lot to ponder in this curious prime realm.So far this compilation counts 8726 + 2 square palindromic numbers. Is there a better way than exhaustive search for finding the tallest pyramids with fixed step sizes? Can you prove that fixed step size pyramids are finite? Are there efficient strategies for building taller pyramids? What happens in bases other than 10? Many questions about palidromic prime pyramids remain open. Every other palindrome having an even number of digits is divisible by 11 and so can’t be prime. In the realm of palindromic primes, 11 is the only one with an even number of digits. “For any starting prime we should be able to build as high as we like,” Honaker and Caldwell say, “though the taller the pyramids get the larger our step size must be (on average).” If the step size is allowed to vary from one row to the next, many more possibilities become available. ![]() They even have a formula that looks promising for estimating the average maximum pyramid height for a given step size. īased on further computer investigations, Honaker and Caldwell conjecture that all palindromic prime pyramids with fixed step size are finite. You can find these pyramids and additional information at. There’s one each starting with the primes 5 and 7, both of height 29. Honaker and Caldwell found that there are three pyramids tied for tallest starting with 3, each of height 28. They found the two examples by doing an exhaustive computer search-building every possible pyramid. “In fact, there are two pyramids of this height,” Caldwell and Honaker remark in a Journal of Recreational Mathematics article on the topic. Starting with 2, the tallest pyramid that can be built has 26 levels. You can get a nine-step, truncated pyramid starting with the prime 7159123219517 (found by Felice Russo), but the prospects of doing much better appear limited.Īdding two digits to each side per step, however, is more promising. Possible strategies include starting with larger primes and allowing the addition of more than one digit to each side with each step.Īs it happens, starting with larger primes doesn’t help much. “How can we build them higher?” Honaker and Caldwell ask. None of these pyramids has more than four levels. ![]() It’s fairly easy to come up with single-step palindromic prime pyramids starting with the primes 3, 5, and 7. Caldwell of the University of Tennessee at Martin in a detailed investigation of these curious structures. Honaker later collaborated with mathematician and prime specialist Chris K. The idea of studying palindromic prime pyramids was first proposed by G.L. ![]() These two pyramids are the tallest that can be built beginning with the prime 2 and adding one digit at a time to each side (single step).
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