14.4 Sorted numbers with sorted squares 419
14.4 Sorted numbers with sorted squares ...........
A number issortedif its digits are nondecreasing from left to right. It
is strongly sorted if its square is also sorted. It is known that the only
strongly sorted integers are given in the table below.^1
- 1, 2, 3, 6, 12, 13, 15, 16, 38, 116, 117.
• (^16) n 7.
• (^3) n 4.
• (^3) n 5.
• (^3) m (^6) n 7.
(3n 51 )^2 =(10· (^3) n+5)^2
=100·(3n)^2 + 100·(3n)+25
=1n− 108 n− 19102 +3n 25
=1n− 112 n− 1225
=1n (^2) n+1 5.
Ifx=3m (^6) n 7 , then 3 x=10m− 110 n 1 , and it is easy to find its square.
(3m (^6) n7)^2 =
{
(^1) m (^3) m (^4) n−m+1 (^6) m (^8) n 9 , ifn+1≥m,
(^1) m (^3) n+1 (^5) m−n− 16 n+1 (^8) n 9 , ifn+1<m.
More generally, the product of any two numbers of the form (^3) m (^6) n 7
is sorted.
(^1) Problem 1234,Math. Mag., 59 (1986) 1, solution, 60 (1987)1. See also R. Blecksmith and C. Nicol,
Monotonic numbers,Math. Mag., 66 (1993) 257–262.