Absolutely, yeah - I was thinking out loud when I wrote that last post. It's a nice divisibility test, that one, isn't it?Gordon wrote:No, I meant if the digits of x add up to a multiple of 9, then x is a multiple of 9.
I do agree with you, Martijn, that the set of numbers that the constant difference c can be is small. What I still think is an open and interesting question though is whether there's some similarity between the prime sequences that have a 3330 difference. That particular difference of 3330 seems to recur no matter how you vary the number of digits in the prime (though I've not gone beyond length-3 sequences as yet, and I should say that other values of c also recur). So I wonder if the particular kind of permuting being done in those sequences is somehow similar (or perhaps the same). If you could identify 'classes' of prime sequence of this sort - the class with difference 3330, for instanc - that could be maybe interesting. Maybe.
The only progress I've made though is to show that for most permutations of abcd the difference c will be given by something like 90[10(a-c) + (b-c)]. So in effect, for some permutations of a 4-digit prime, you construct the 2-digit number 10(a-c) + (b-c) from its digits and that gives you the multiple of 90 that the difference c will be. And in our cases - for 4817->1847 or 6299->2969 - a-c and b-c are the same, and the number 10(a-c) + (b-c) is 37, and 37.90 is 3330.