# The Most Powerful Number

A power-full number is defined as a number with $$n$$ amount of non-zero digits $$D_x$$. The number takes the form $$D_n$$&$$...$$&$$D_2$$&$$D_1$$. The digits in a power-full number satisfies the rule
$$D_{x+1}$$&$$D_x=k_x^{x+1}$$ where $$k_x$$ are positive integers. This rule applies for $$1\le x\lt n$$.

Essentially a power-full number has paired digits that equal some perfect power corresponding to its position in the chain + 1.

Given that the magnitude of a power-full number and $$n$$ determines how powerful it is (the bigger the better), what is the most powerful number?

• Feel free to help with formatting if there is an easier way to convey this puzzle – Adam May 21 at 13:35
• Do you mean kx are positive integers or is there a single k? – Jonathan Allan May 21 at 13:50
• Oops, fixed! @JonathanAllan – Adam May 21 at 14:00
• My edits were to make the spelling consistent; now you have both 'power-full' and 'powerful' in your title and body. Pedants may think the definitions are different ... – Glorfindel May 21 at 14:27
• @Veedrac $1\le x\lt n$ shouldn't include $n$. I thought it would be more confusing to have $1\le x\le n-1$ – Adam May 21 at 15:47

I think the most powerful number is

1649, $$n=4$$ (or, if zeros are allowed: 01649, $$n=5$$)

It's powerful since

49 is a square, 64 a third power, 16 a fourth power (and 01 a fifth power).

In order to have a more powerful number

with $$n=6$$, it needs to start with two digits forming a sixth power, so either 01 or 64. However, there is no two-digit fifth power starting with 1 or 4; we have only 01 and 32.
Another $$n=5$$ powerful number without zeros should start with 32, but there's no two-digit fourth power starting with 2, so that's impossible too.
Another $$n=4$$ powerful number could start with 81, but there are no two-digit third powers starting with one. Similarly, there are no other two-digit third powers starting with 6 than 64, so no improvement can be made there either.

• To complete the demonstration, you should add why you can't have the same n but the leading digit being the number of letters of the 5th word of your answer. (the reason is similar to what you already said) – Evargalo May 21 at 15:09
• @Evargalo you're right, thanks; updated – Glorfindel May 21 at 15:13