h2g2 The Hitchhiker's Guide to the Galaxy: Earth Edition

Hello, I stumbled upon a paper which might interest you:

http://arxiv.org/abs/0911.3038

Thanks. Unfortunately, until I get software to read the article I won't know what it's about. It's not a high priority, but I'll get around to it eventually.

I can disclose the 'curious property' if you wish. But I wouldn't want to ruin the suspense

Tell me. I might just forget about it, so you needn't worry about there being some suspense you will be eliminating.

It's unforgettable.

3435 = 3^3 + 4^4 + 3^3 + 5^5

Thanks. Now I can forget about it.

Well, I may as well research the problem a little bit.

Ah, nice! The only other 4-digit number of this type with digit values up to 35 is 6534 in base 20.

It took a long time to get the five-digit solutions, and I'm going to lower the size of the maximum digit to 15 to continue, but here are the three- and five-digit solutions: 131 and 313 in base 4, 513 in base 25, 615 in base 91, 126 in base 215, 171 in base 904, 22352 and 23452 in base 6, 13454 in base 7, and 33661 in base 13.

I decided my method has been weak and to start over, but I've gotten 156262 and 1656547 in base 9 as possibly the only six- and seven-digit examples, and 18453278 and 18453487 in base 11 as possibly the only eight-digit ones. I've been approximating the base for a given possible representation with an unsound method, though.

My approximation was fine--same exact results, but my method is still weak. I'll have to go at this again by a different route if I want to have any hope of checking large digits and bases and long lengths.

This looks like well-trod terrain, and I'm leaving it. I pointed it out to someone over at wikipedia, and he linked me to an article which further links to other things. I'm a bit surprised anyone would be writing about it as though it were new. It is not.

Now, I have something interesting to report. If you start with constant 1 and build a polynomial sequence by the rule that the coefficient should be the least positive one that gives a result relstively prime to all of its predecessors, you get no pattern until the 89th degree, and then at least to the 1000th you get all 1s except for 2s at powers congruent to 3 modulo 5.