In continuation of 5 Digit Number, let us define a 5-digit non leading zero base N (N > 3) positive integer x as a split number whenever, 3*x is a perfect square and, when the digits of x are split, the first number is double the second one.

How many split numbers are there whenever 11 ≤ N ≤ 36. What are the respective minimum and maximum values?

(Splitting a base-N 5-digit number into two numbers means 12345 into 1 and 2345 or, 123 and 45.)

The following is true in any base N>3:

x = abcde, a concatenation of digits or
x = 10000a + 1000b + 100c + 10d + e

If 3x is a p.s. then x is a multiple of 3.

The split must be abc and de so we have

100a + 10b + c = 2(10d + e) = 20d + 2e
10000a + 1000b + 100c = 2000d + 200e

x = 2001d + 201e

The divisibility of x by 3 depends on N (as well as d and e)
If N = 0 mod 3 then 2001 = 1 mod 3 and 201 = 1 mod 3
If N = 1 mod 3 then 2001 = 0 mod 3 and 201 = 0 mod 3
If N = 2 mod 3 then 2001 = 2 mod 3 and 201 = 0 mod 3

So:
If N = 0 mod 3, d and e are either both = 0 mod 3 or one of them = 1 and the other = 2 mod 3
If N = 1 mod 3, d and e can be anything
If N = 2 mod 3, d must be a multiple of 3 and e can be any number.

This is probably much easier with a computer but we can at least find how many candidates exist in each base.
I will work on this more if I have time.

Posted by Jer on 2011-01-03 15:13:28