I was going through Knuth's lectures in which he solves 5 problems that have not yet been solved.

First problem:

Monotonic squares: Find all positive integers n such that the decimal digits of n and

n2 are both in nondecreasing order from left to right

While solving this problem, Anil Gangolli pointed out

(6(m)7)2=4(m+1)8(m+1)9

where a(m) is series of a digits (something like Kleene's operator)

eg 3(5) is 33333.

Here is proof for above statement using mathematical induction:

Given: (6(m)7)2=4(m+1)8(m+1)9

TO prove:(6(m+1)7)2=4(m+2)8(m+1)9

Take lhs:(6(m+1)7)2

=(6(m)7+60(m+1))2

so i am now squaring

applying (a+b)2=a2+b2+2ab

= (6(m)7)2 + (60(m+1))2 + 2.6(m)7.60(m+1)

= 4(m+1)8(m)9 + 360(2m+2) +80(m)40(m+1)

Now this can be simplified as

4(m) 4 8(m) 9

8 0(m) 4 0(m) 0

36 0(m) 0 0(m) 0

Adding three equations gives us

44 4(m) 8 8(m) 9

Final equation is

4(m+2) 8(m+1) 9

Hence proved

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