Posted: Apr 09, 2010 10:26 am
by Someone
I'm bound to get some sleep soon, so this will be a quick, short post on multi-base palindromes. This subject actually seems to me more ripe with the sense that other investigators were collectively blocked from thinking about it much than with my being pushed toward it, but that's not been thought out.

Example one: 4838419019 [correction edited in late] is the smallest number which is a 5-digit palindrome in four different bases, beginning with base 91. You'll note that it is almost the concatenation of two 5-digit palindromes in base ten, and that base fits as a part of the coincidence. I just recently discovered the factorization 7*691202717 is cute, as you can say that movement of one digit makes concatenation of palindromes in two ways for both this prime and the original number.

Example two: 3360633 is the first number which is a palindrome of length seven in three bases--9, 10, & 11. It is also 8th on the list someone else came up with long ago of palindromes in base 10 which are sums of the composite numbers up to some point, and 6th on the list is 33633. This is just the straightforward mathematics part of the result. The really strange thing was the discovery process and other stuff, starting with the apparent fact that the someone else concerned came up just the smallest bit short of the full discovery. I chose this number as the basis for the name Eegogee at the old forum. If there is any conversation later on, I may expand upon this.

Okay, after I get my post-sleep bearings, I'll talk about some more of this. Those are the big two on that particular subject.

Note: These are peer-reviewed items at OEIS.