TreueEckhardt2 Posted July 12, 2023 Share Posted July 12, 2023 I have a question about something similar to the twin prime conjecture. If you have two consecutive numbers, is there a limit to how smooth they can be? Here is the basic definition: If the largest prime factor of a number is less than or equal to the nth root of that number, then the number is nth root smooth. I am curious about pairs of consecutive numbers that are both nth root smooth for large values of n. Let’s call them twin nth root smooth numbers. For example, 2400 and 2401 are consecutive numbers. The largest prime factor of 2400 is 5, and 5 is less than the 4^{th} root of 2400, so 2400 is 4^{th} root smooth. The largest prime factor of 2401 is 7, and 7 is equal to the 4^{th} root of 2401, so 2401 is 4^{th} root smooth. Thus, 2400 and 2401 are twin 4^{th} root smooth numbers. I have found an example of twin 10^{th} root smooth numbers. Let A=2^210-1 and B=2^210. A is equal to the following 64-digit number: 1645 504557 321206 042154 969182 557350 504982 735865 633579 863348 609023 The largest prime factor of A is 1,564,921. I found this by using the calculator at the following website: https://www.alpertron.com.ar/ECM.HTM Since the largest prime factor of A is smaller than the 10^{th} root of A, A is 10^{th} root smooth. The largest prime factor of B is 2, and since 2 is smaller than the 10^{th} root of B, B is 10^{th} root smooth. So A and B are twin 10^{th} root smooth numbers. My question is this: Is there a largest value of n for which twin nth root smooth numbers exist? I haven’t been able to find any examples twin nth root smooth numbers for values of n larger than 10, but I don’t suppose that means a lot. Link to comment Share on other sites More sharing options...

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