There exists a number oddity with 3 different 4-digit numbers. One is 9801, where (98 + 01)^2 = 9801. It also works with 3025: (30+25)^2 = 3025.
What is the other number?
What is the smallest 6-digit number that would work?
(in other words, in a 6-digit number abcdef: abcdef=(abc+def)^2)

9801 = 99² = (98 + 01)²
998001 = 999² = (998 +001)²
99980001 = 9999² =(9998 + 0001)²
In general:
[(10^n)-1]² = 10^2n -2(10^n) + 1

9² = 100 - 20 + 1 = 81
99² = 10000 - 200 + 1 = 9801
999² = 1000000 - 2000 +1

And since 10^2n - 2(10^n) = (10^n)[(10^n)-2] and
n-1
[∑9(10^i)] + 8 = (10^n)-2 (easier to show than to prove)
i=1

it holds for all n>1, that [(10^n)-1]² = (10^n)[(10^n)-1]+1

Posted by TomM on 2004-04-03 12:30:46