Reverse digit-to-digit invariant

Number Theory Level 3

The integer $3435$ is the only so-called perfect digit-to-digit invariant because $3^3+4^4+3^3+5^5 = 3435$ in base $10$ .

Can you find the only 5-digit number $\overline{{a_4}{a_3}{a_2}{a_1}{a_0}}$ such that ${a_4}^{a_0}+{a_3}^{a_1}+{a_2}^{a_2}+{a_1}^{a_3}+{a_0}^{a_4} = \overline{{a_4}{a_3}{a_2}{a_1}{a_0}}?$

Note: You have to use the convention $0^0 = 1$ .

The answer is 48625.

1 solution

Giorgos K.
Mar 27, 2018

Here is a Mathematica code that does not use the digit $0$

Select[Tuples[Range@9,5],Total[#^Reverse@#]==FromDigits@#&]
this returns {{4, 8, 6, 2, 5}}

but if we want to use zero (in Mathematica $0^0$ is Indeterminate expression) and check all 5 digit numbers, we can do it like this:

Select[Range[10000,99999],(t=IntegerDigits@#;Total@Table[If[t[[i]]==t[[s+1-i]]==0,1,t[[i]]^t[[s+1-i]]],{i,s=Length@t}]==#)&]
which returns the same result 48625

It does seem more like a computational problem. I did it in Python.

I wonder if it can be solved with Number Theory arguments and principles?

Steven Perkins - 3 years, 2 months ago

Reverse digit-to-digit invariant

The answer is 48625.

1 solution

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