In continuation of
5 Digit Number, let us define a 5digit 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 baseN 5digit number into two numbers means 12345 into 1 and 2345 or, 123 and 45.)
(In reply to
re: A start. by Steve Herman)
Sorry if I wasn't clear. When I wrote:
x = 10000a + 1000b + 100c + 10d + e
My intent was to interpret these numbers as all being in base N:
10 = N
100 = N^2
1000 = N^3
10000 = N^4
It was a bit of a shorthand that served until I started considering divisibility.
You can restate x = 2001d + 201e as
x = (2N^3 + 1)d + (2N^2 + 1)e
and the divisibility rules might be more clear. This seems like a dead end though because it doesn't rule out much.
There are few enough numbers to check anyway that a program like the one Charlie wrote can just churn through all d and e from 0 to N1 and not worry about just checking values that will force x to be divisible by 3.
I will point out that the most restrictions occurred when N was divisible by 3. Most of the bases divisible by 3 have no solutions (all except N=12)

Posted by Jer
on 20110104 11:04:31 